I'm trying to understand Stephan Wolphram's thinking. I'd like a comprehensive explanation or introduction to his thought on the 'ruliad 
https://writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad/
The Concept of the Ruliad—Stephen Wolfram Writings
2021-11-10
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<h2>The Entangled Limit of Everything</h2>
<p>I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our <a href="https://www.wolframphysics.org/">Physics Project</a>. And it’s one that I think has extremely deep implications—both in science and beyond.</p>
<p>In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.</p>
<p>We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is.</p>
<p>In the language of our Physics Project, it’s the ultimate limit of all <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/">rulial multiway systems</a>. And as such, it traces out the entangled consequences of progressively applying all possible computational rules. </p>
<p>Here is an example of an <a href="https://www.wolframscience.com/nks/p204--multiway-systems/">ordinary multiway system</a> based on the string replacement rules {A → AB, BB → A} (indicated respectively by blueish and reddish edges):</p>
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<p>At each step, the rules are applied in all possible ways to each state. Often this generates multiple new states, leading to branching in the graph. But, importantly, there can also be merging—from multiple states being transformed to the same state.</p>
<p>The idea of a rulial multiway system is not just to apply particular rules in all possible ways, but to apply all possible rules of a given form. For example, if we consider “<a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/typical-multiway-graph-structures/">1 → 2, 2 → 1 A, B string rules</a>”, the possible rules are</p>
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<p>and the resulting multiway graph is (where now we’re using purple to indicate that there are edges for every possible rule):</p>
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<p>Continuing a little longer, and with a different layout, we get:</p>
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<p>This may already look a little complicated. But the ruliad is something in a sense infinitely more complicated. Its concept is to use not just all rules of a given form, but all possible rules. And to apply these rules to all possible initial conditions. And to run the rules for an infinite number of steps. </p>
<p>The pictures above can be thought of as coarse finite approximations to the ruliad. The full ruliad involves taking the infinite limits of all possible rules, all possible initial conditions and all possible steps. Needless to say, this is a complicated thing to do, and there are many subtleties yet to work out about how to do it.</p>
<p>Perhaps the most obviously difficult issue is how conceivably to enumerate “all possible rules”. But here we can use the <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> to tell us that whatever “basis” we use, what comes out will eventually be effectively equivalent. Above we used <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/">string substitution systems</a>. But here, for example, is a <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/">rulial multiway system made with 2-state 2-color Turing machines</a>:</p>
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<p>And here is a rulial multiway system made from <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-in-our-models/multiway-systems-for-our-models/">hypergraph rewriting of the kind used in our Physics Project</a>, using <a href="https://www.wolframphysics.org/technical-introduction/typical-behaviors/the-number-of-possible-rules/">all rules with signature</a> :</p>
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<p>As another example, consider a <a href="https://www.wolframphysics.org/bulletins/2021/10/multicomputation-with-numbers-the-case-of-simple-multiway-systems">multiway system based on numbers</a>, in which the rules multiply by each possible integer:</p>
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<p>Here’s what happens starting with 1 (and truncating the graph whenever the value exceeds 100): </p>
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<p>Even with this simple setup, the results are surprisingly complicated (though it’s possible to give quite a bit of analysis in this particular case, as described in the <a href="https://writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad/#appendix-the-case-of-the-multiplicad">Appendix</a> at the end of this piece). </p>
<p>The beginning of the multiway graph is nevertheless simple: from 1 we connect to each successive integer. But then things get more complicated. To see what’s going on, let’s look at a fragment of the graph:</p>
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<p>In a sense, everything would be simple if every path in the graph were separate: </p>
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<p>But the basic concept of multiway systems is that equivalent states should be merged—so here the “two ways to get 6” (i.e. 1 × 2 × 3 and 1 × 3 × 2) are combined, and what appears in the multiway graph is:</p>
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<p>For integers, the obvious notion of equivalence is numerical equality. For hypergraphs, it’s isomorphism. But the important point is that equivalence is what makes the multiway graph nontrivial. We can think about what it does as being to entangle paths. Without equivalence, different paths in the multiway system—corresponding to different possible histories—would all be separate. But equivalence entangles them.</p>
<p>The full ruliad is in effect a representation of all possible computations. And what gives it structure is the equivalences that exist between states generated by different computations. In a sense, there are two forces at work: the “forward” effect of the progress of computation, and the “sideways” effect of equivalences that entangle different computations. (Mathematically this can be thought of as being like decomposing the ruliad structure in terms of fibrations and foliations.)</p>
<h2>Experiencing the Ruliad</h2>
<p>In thinking about finding a fundamental theory of physics, one thing always bothered me. Imagine we successfully identify a rule that describes everything about our universe. Then the obvious next question will be: “Why this rule, and not another?” Well, how about if actually the universe in effect <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#why-this-universe-the-relativity-of-rules">just runs every possible rule</a>? What would this mean? It means that in a sense the “full story” of the universe is just the ruliad.</p>
<p>But the ruliad contains everything that is computationally possible. So why then do we have the perception that the universe has specific laws, and that definite things happen in it?</p>
<p>It all has to do with the fact that we are <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">bounded observers</a>, <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/">embedded within the ruliad</a>. We never get to see the full ruliad; we just sample tiny parts of it, parsing them according to our particular <a href="https://www.wolframscience.com/nks/chap-10--processes-of-perception-and-analysis/">methods of perception and analysis</a>. And the crucial point is that for coherent observers like us, there are certain robust features that we will inevitably see in the ruliad. And these features turn out to include <a href="https://www.wolframphysics.org/">fundamental laws of our physics</a>, in particular general relativity and quantum mechanics.</p>
<p>One can imagine an observer very different from us (say some kind of alien intelligence) who would sample different aspects of the ruliad, and deduce different laws. But one of the surprising core discoveries of our <a href="https://www.wolframphysics.org/">Physics Project</a> is that even an observer with quite basic features like us will experience laws of physics that precisely correspond to ones we know.</p>
<p>An analogy (that’s actually ultimately the result of the same underlying phenomenon) may help to illustrate what’s going on. Consider molecules in a gas. The molecules bounce around in a complicated pattern that depends on their detailed properties. But an observer like us doesn’t trace this whole pattern. Instead we only observe certain “coarse-grained” features. And the point is that these features are largely independent of the detailed properties of the molecules—and robustly correspond to our standard laws of physics, like the <a href="https://www.wolframscience.com/nks/p441--irreversibility-and-the-second-law-of-thermodynamics/">Second Law of thermodynamics</a>. But a different kind of observer, sampling and “parsing” the system differently, could in principle identify different features, corresponding to different laws of physics.</p>
<p>One of the conceptual difficulties in thinking about how we perceive the ruliad is that it’s a story of “self-observation”. Essentially by the very definition of the ruliad, we ourselves are part of it. We never get to “see the whole ruliad from the outside”. We only get to “<a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/">experience it from the inside</a>”.</p>
<p>In some ways it’s a bit like our efforts to construct the ruliad. In the end, the ruliad involves infinite rules, infinite initial conditions, and infinite time. But any way of assembling the ruliad from pieces effectively involves making particular choices about how we take those infinite limits. And that’s pretty much like the fact that as entities embedded within the ruliad, we have to make particular choices about how to sample it.</p>
<p>One of the remarkable aspects of the ruliad is that it’s in some sense the <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/#the-laws-of-the-rulial-universe">unique ultimately inevitable and necessary formal object</a>. If one sets up some particular computational system or mathematical theory, there are choices to be made. But in the ruliad there are no choices. Because everything is there. And in a sense every aspect of the structure of the ruliad is just something formally necessary. It requires no outside input; it is just a formal consequence of the meaning of terms, like the abstract fact . </p>
<p>But while the ruliad is unique, the description of it is not. In constructing it, one can imagine using Turing machines or hypergraph rewriting systems or indeed any other kind of computational system. Each will ultimately lead to the same limiting object that is the ruliad, but each of them can be thought of as defining a different coordinate system for describing the ruliad. </p>
<p>The very generality of the ruliad makes it unsurprising that there is vast diversity in how it can be described. And in a sense each possible description is like a possible way of experiencing the ruliad. In analogy to the (deeply related) situation with spacetime in general relativity, we might say that there are many reference frames in which to experience the ruliad—but it’s always the same ruliad underneath.</p>
<p>It’s important to understand that the “ruliad from the outside” could seem very different from any “internal” experience of it by an observer like us. As an example, consider a simple finite approximation to the ruliad, built from string substitution systems. In what we did above, we always started from a specific initial condition. But the full ruliad involves starting from all possible initial conditions. (Of course, one could always just say one starts from a “null” initial condition, then have rules of the form null → everything.) So now let’s consider starting from all possible strings, say of length 4. If we use all possible 2-element-to-2-element rules, the finite approximation to the ruliad that we’ll get will be:</p>
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<p>At some level this is a simple structure, and—as is inevitable for any finite approximation to the ruliad—its <a href="https://reference.wolfram.com/language/ref/TransitiveClosureGraph.html">transitive closure</a> is just the <a href="https://reference.wolfram.com/language/ref/CompleteGraph.html">complete graph</a>:</p>
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<p>So why doesn’t this mean that the ruliad is somehow trivial? A key part of the story is that we never get to “see the ruliad from the outside” like this. We are always part of it, sampling it according to some procedure, or, somewhat equivalently, thinking about constructing it according to some procedure.</p>
<p>As an analogy, consider the real numbers. The whole continuum of all real numbers is “from the outside” in many ways a simple construct. But if we imagine actually trying to construct real numbers, say digit by digit, according to some definite procedure, then we’re dealing precisely with what <a href="https://www.wolframscience.com/nks/notes-12-4--computable-reals/">Turing machines were originally invented to model</a>, and the whole structure of computation is involved. (As we’ll see, our way of thinking about “observers like us” is ultimately quite related to “Turing machines with bounded descriptions”.)</p>
<p>In a sense, at an outside “holistic” level, the ruliad has a certain simple perfection. But as soon as you try to look at “what’s in the ruliad”, you have to parametrize or coordinatize it, and then you’re inevitably exposed to its intricate internal structure.</p>
<h2>Observers Like Us</h2>
<p>One could imagine very different ways in which entities embedded within the ruliad could “experience” it. But what’s most relevant for us is how “<a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">observers like us</a>” do it—and how we manage to synthesize from what’s going on in the ruliad our perception of reality, and our view of how our physical universe works.</p>
<p>Let’s start by talking not about the full ruliad but rather about models in our Physics Project based on specific underlying rules. At the lowest level, we have a “machine-code” description of the universe is which everything just consists of a network of “atoms of space” that is continually being updated—and which we can think of as carrying out a giant, if incoherent, computation, full of <a href="https://www.wolframscience.com/nks/p737--computational-irreducibility/">computational irreducibility</a>. But the remarkable fact is that somehow we, as observers of this, <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/#observers-and-their-physics">manage to pick out of it a certain slice</a> that ends up showing coherent, computationally reducible features—that for example seem to reproduce our known laws of physics.</p>
<p>How does this work? Partly it has to do with features of us as observers, partly with features of how the universe fundamentally works, and partly with an interplay between these. </p>
<p>The first crucial feature of us as observers is that we’re computationally bounded: the way we “parse” the universe involves doing an amount of computation that’s absolutely tiny compared to all the computation going on in the universe. We sample only a tiny part of what’s “really going on underneath”, and we aggregate many details to get the summary that represents our perception of the universe.</p>
<p>But why should that summary have any coherence? Basically it’s because we impose coherence through our definition of how observers like us work. One part of the universe will be affected by others. But to consider part of the universe as an “observer”, there has to be a certain coherence to it. The behavior of the universe somehow has to imprint itself on a “medium” that has a certain coherence and consistency.</p>
<p>Down at the level of atoms of space, everything is always changing. But we can still identify emergent features that have a certain persistence. And it’s out of those features that what we call observers are built. Given only the atoms of space with all their computationally irreducible behavior, it’s not at the outset obvious that any real persistence could exist or be identified. But in our models we expect that there will, for example, be essentially <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#elementary-particles-old-and-new">topological features that correspond to particles</a> that persistently maintain their identity. </p>
<p>And the point is that we can expect to “aggregate up” much further and be able to identify something like a human observer—that we can consider to persistently maintain its identity to the point where phenomena from the universe can be “systematically imprinted” on it. </p>
<p>Down at the level of atoms of space, there’s a whole multiway graph of possible sequences of updates that can occur—with each path in effect corresponding to a different “thread of time” for the universe. But it’s a crucial fact about <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">us as observers of the universe</a> that we don’t perceive all those branching and merging threads of time. Instead, we imagine that we have a single, definite thread of experience—in which everything is sequentialized in time. </p>
<p>I’ve <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">argued elsewhere</a> that this sequentialization in time is a defining characteristic of “human-like consciousness”. And it turns out that one of its consequences is that it implies that the particular perception we will have of the universe must be one in which there are laws of physics that correspond to ones we know.</p>
<p>It’s not obvious, by the way, that if we sequentialize time we can form any consistent view of the universe. But the phenomenon of <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-phenomenon-of-causal-invariance">causal invariance</a>—which seems ultimately to be guaranteed by the fundamental structure of the ruliad—turns out to imply that we can expect a certain generalized relativistic invariance that will inevitably lead to eventual consistency. </p>
<p>The notion of sequentialization in time is closely related to the idea that—even though our individual atoms of space are continually changing—we can view ourselves as having a coherent existence through time. And there’s a similar phenomenon for space. At the outset, it’s not obvious that there can be “pure motion”, in which something can move in space without “fundamentally changing”. But it turns out again to be consistent to view this as how things work for us: that even though we’re “made of different atoms of space” when we’re in different places, we can still imagine that in some sense we maintain the “same identity”.</p>
<p>Down at the level of individual atoms of space, there really isn’t any coherent notion of space. And the fact that we form such a notion seems to be intimately connected to what we might think of as details of us. Most important is that we’re in a sense “intermediate in size” in the universe. We’re large relative to the effective distance between atoms of space (<a href="https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/units-and-scales/">which might be m</a>), yet we’re small compared to the size of the whole universe ( m). And the result is that we tend to aggregate the effects of many atoms of space, but still perceive different features of space (say, different gravitational fields) in different parts of the universe.</p>
<p>The fact that we “naturally form a notion of space” also seems to depend on another issue of scale—that for us the speed of light “seems fast”. It takes our brains perhaps milliseconds to process anything we see. But the point is that this is very long compared to the time it takes light to get to us from objects in our typical local environment. And the result is that we tend to perceive there as being an instantaneous configuration of the world laid out in space, that “separately” changes in time. But if, for example, our brains ran much faster, or we were much bigger than we are, then the speed of light would “seem slower” to us, and we wouldn’t tend to form the notion of an “instantaneous state of space”.</p>
<p>OK, so what about quantum mechanics? The most fundamental feature of quantum mechanics is that it implies that things in the universe follow not just one but many possible paths of history—which we only get to make certain kinds of measurements on. And in our <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#the-inevitability-of-quantum-mechanics">Physics Project this is something natural</a>, and in fact inevitable. Given any particular configuration of the universe, there are many possible updates that can occur. And when we trace out all the possibilities, we get a multiway system, in which different threads of history continually branch and merge.</p>
<p>So how do observers like us fit into this? Being part of the universe, we inevitably branch and merge, just like the rest of the universe. So to understand our experience, what we need to ask is how a <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/#the-quantum-observer">“branching brain” will perceive a “branching universe”</a>. And the story is remarkably similar to what we discussed above for our experience of space and time: it all has to do with imagining ourselves to have a certain definite persistence.</p>
<p>In other words, even if when “viewed from the outside” our brain might be following many different paths of history, “from the inside” we can still potentially assume that everything is conflated into a single thread of history. But will this ultimately be a consistent thing to do? Once again, causal invariance implies that it will. There are specific “quantum effects” where we can tell that there are multiple branches of history being followed, but in the end it’ll be consistent to imagine an “objective reality” about “what happened”.</p>
<p>In our Physics Project we imagine that there are abstract relations between atoms of space, and in the end the pattern of these relations defines the structure of physical space. But what about different branches of history in the multiway graph? Can we think of these as related? The answer is yes. For example, we can say that at a particular time, states on two branches are “adjacent” if they share an immediate ancestor in the multiway graph. And tracing through such connections we can develop a notion of “<a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-concept-of-branchial-graphs/">branchial space</a>”—a kind of space in which states on different branches of history are laid out:</p>
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<p>One can think of branchial space as being defined by the pattern of entanglements between different branches of history. And in our Physics Project it turns out that the fundamental <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#general-relativity-and-quantum-mechanics-are-the-same-idea">laws of quantum mechanics seem to just be a direct translation of the fundamental laws of spacetime</a> into branchial space. And just like the speed of light governs the maximum rate at which effects can propagate in physical space, so similarly in our models there’s a “<a href="https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/correspondence-between-relativity-and-quantum-mechanics/">maximum entanglement speed</a>” at which effects can propagate in branchial space.</p>
<p>So what are we like as observers in branchial space? Just like in physical space, we can presumably be thought of as having a certain size in branchial space. We don’t yet know quite how to measure this size, but it’s surely related to the effective number of quantum degrees of freedom we involve. </p>
<p>In our everyday experience of things like gases, we’re sufficiently large compared to individual molecules that we normally just perceive the gas as some kind of continuum fluid—and in normal circumstances we can’t even tell that it’s made of molecules. Well, it’s presumably the same kind of thing for physical space—where we’re even much larger compared to the atoms of space, and it’s a major challenge to <a href="https://writings.stephenwolfram.com/2021/04/the-wolfram-physics-project-a-one-year-update/#towards-experimental-implications">figure out how to detect their presence</a>. What about for branchial space? As the underlying rules for the system get applied, different branches of history will in effect “move around” in branchial space in complex and computationally irreducible ways. And just like when we observe molecules in a gas, we’ll mostly just observe overall aggregate effects analogous to fluid mechanics—and only in special circumstances will we notice “quantum effects” that reveal the presence of multiple independent threads of history.</p>
<h2>Living in Rulial Space</h2>
<p>We’ve discussed how “observers like us” perceive models of physics of the type that arise in our Physics Project. But how will we perceive the whole ruliad? It begins with a generalization of the story for branchial space. Because now as well as having different branches associated with different updatings according to a particular rule, we have different branches associated with updatings according to different rules.</p>
<p>And just as we can slice an ordinary multiway system at a particular time to get an instantaneous version of branchial space, so now we can slice a rulial multiway system to get an instantaneous version of what we can call rulial space—a space in which different branches can correspond not just to different histories, but to different rules for history.</p>
<p>It’s a fairly complicated setup, with “pure branchial space” inevitably being deeply interwoven with rulial space. But as a first approximation, we can think of rulial space as being somewhat separate, and laid out so that different places in it correspond to the results of applying different rules—with nearby places effectively being associated with “nearby” rules.</p>
<p>And just as we can think of effects propagating in branchial space, so also we can think of them propagating in rulial space. In branchial space we can talk about entanglement cones as the analog of light cones, and a maximum entanglement speed as the analog of the speed of light. In rulial space we can instead talk about “<a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#the-emerging-picture-of-rulial-space">emulation cones</a>”—and a “maximum emulation speed”.</p>
<p>In our rough approximation of rulial space, each point is in effect associated with a particular rule. So how do we “move” from one point to another? Effectively we have to be emulating the behavior of one rule by another. But why should it even be possible to do this? The answer is the <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a>, which states that, in effect, most rules will be equivalent in their computational capabilities—and in particular they will be capable of universal computation, so that any given rule can always “run a program” that will make it emulate any other rule.</p>
<p>One can think of the program as an interpreter or translator that goes from one rule to another. The Principle of Computational Equivalence tells one that such a translator must essentially always exist. But how fast will the translator run? Effectively that’s what distance in rulial space measures. Because to “do a certain translation”, branches in the rulial multiway system have to reach from one rule to another. But they can only do that at the maximum emulation speed. </p>
<p>What does the maximum emulation speed measure? Effectively it corresponds to the <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#the-emerging-picture-of-rulial-space">raw computational processing speed of the universe</a>. We can think of representing computations in some language—say the <a href="https://www.wolfram.com/language/">Wolfram Language</a>. Then the processing speed will be measured in “Wolfram Language tokens processed per second” (“WLT/s”). In some sense, of course, giving a value for this speed is just a way of relating our human units of time (say, seconds) to the “intrinsic unit of time” associated with the computational processing that’s going on in the universe. Or, in other words, it’s a kind of ultimate definition of a second relative to purely formal constructs.</p>
<p>OK, but how does this relate to us as observers embedded within the ruliad? Well, just as we imagine that—along with the rest of the universe—we’re continually branching and merging in branchial space, so also this will be what happens in rulial space. In other words—like the rest of the universe—our brains aren’t following a particular rule; they’re following branching and merging paths that represent all possible rules.</p>
<p>But “from inside” we can still potentially imagine that we have a single thread of experience—effectively conflating what happens on all those different branches. And once again we can ask whether doing this will be consistent. And the answer seems to be that, yes, it can be. And what guarantees this is again a kind of “<a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#why-this-universe-the-relativity-of-rules">rulial relativity</a>” that’s a consequence of causal invariance. There are many details here, which we’ll address to some extent later. But the broad outline is that causal invariance can be thought of as being associated with paths of history that diverge, eventually converging again. But since the ruliad contains paths corresponding to all possible rules, it’s basically inevitable that it will contain what’s needed to “undo” whatever divergence occurs. </p>
<p>So what does this mean? Basically it’s saying that even though the universe is in some sense intrinsically “following all possible rules”—as represented by paths in the ruliad—we as observers of the universe can still “take the point of view” that the universe follows a particular rule. Well, actually, it’s not quite a particular rule. Because just as we’re in some sense “quite big” in physical and presumably branchial space, so also we’re potentially “quite big” in rulial space.</p>
<p>And being extended in rulial space is basically saying that we consider not just one, but a range of possible rules to be what describe the universe. How can it work this way? Well, as observers of the universe, we can try to deduce what the “true rule for the universe” is. But inevitably we have to do this by performing physical experiments, and then using inductive inference to try to figure out what the “rule for the universe is”. But the issue is that as entities embedded within the universe, we can only ever do a finite number of experiments—and with these we’ll never be able to precisely nail down the “true rule”; there’ll always be some uncertainty.</p>
<p>When we think of ourselves as observers of the universe, there’s in a sense lots of “arbitrariness” in the way we’re set up. For example, we exist at a particular location in physical space—in our particular solar system and so on. Presumably we also exist at a particular location in branchial space, though it’s less clear how to “name” that. And in addition we exist at a particular location in rulial space.</p>
<p>What determines that location? Essentially it’s determined by how we operate as observers: the particular sensory system we have, and the particular means of description that we’ve developed in our language and in the history of knowledge in our civilization. In principle we could imagine sensing or describing our universe differently. But the way we do it defines the particular place in rulial space at which we find ourselves.</p>
<p>But what does all this mean in terms of the ruliad? The ruliad is the unique limiting structure formed by following all possible rules in all possible ways. But when we “observe the ruliad” we’re effectively “paying attention to” only particular aspects of it. Some of that “paying attention” we can conveniently describe in terms of our particular “location in the ruliad”. But some is more naturally described by thinking about equivalence classes in the ruliad.</p>
<p>Given two states that exist in the ruliad, we have to ask whether as observers we want to consider them distinct, or whether we want to conflate them, and consider them “the same”. When we discussed the construction of the ruliad, we already had many versions of this issue. Indeed, whenever we said that two paths in the ruliad “merge”, that’s really just saying that we treat the outcomes as equivalent.</p>
<p>“Viewed from the outside”, one could imagine that absolutely nothing is equivalent. Two hypergraphs produced in two different ways (and thus, perhaps, with differently labeled nodes) are “from the outside” in some sense different. But “viewed from the inside”, they pretty much have to be viewed as “the same”, in essence because all their effects will be the same. But at some level, even such conflation of differently labeled hypergraphs can be thought of as an “act of the observer”; something that one can only see works that way if one’s “observing it from inside the system”.</p>
<p>But all the way through our description of the observer, it’s very much the same story: it’s a question of what should be considered equivalent to what. In sequentializing time, we’re effectively saying that “all of space” (or “all of branchial space”, or rulial space) should be considered “equivalent”. There are many subtle issues of equivalence that also arise in the <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#the-formal-structure-of-multicomputation">construction of states in the ruliad from underlying tokens</a>, in defining what rules and initial conditions should be considered the same, and in many other places.</p>
<p>The ruliad is in some sense the most complicated constructible object. But if we as computationally bounded observers are going to perceive things about it, we have to find some way to “cut it down to size”. And we do that by defining equivalence classes, and then paying attention only to those whole classes, not all the details of what’s going on inside them. But a key point is that because we are computationally bounded observers who imagine a certain coherence in their experience, there are strong constraints on what kinds of equivalence classes we can use.</p>
<p>If we return again to the situation of molecules in a gas, we can say that we form equivalence classes in which we look only coarsely at the positions of molecules, in “buckets” defined by simple, bounded computations—and we don’t look at their finer details, with all the computational irreducibility they involve. And it’s because of this way of looking at the system that we conclude that it follows the Second Law of thermodynamics, exhibits fluid behavior, etc.</p>
<p>And it’s very much the same story with the ruliad—and with the laws of physics. If we constrain the kind of way that we observe—or “parse”—the ruliad, then it becomes inevitable that the effective laws we’ll see will have certain features, which turns out apparently to be exactly what’s needed to reproduce known laws of physics. The full ruliad is in a sense very wild; but as observers with certain characteristics, we see a much tamer version of it, and in fact what we see is <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#observers-reference-frames-and-emergent-laws">capable of being described in terms of laws</a> that we can largely write just in terms of existing mathematical constructs.</p>
<p>At the outset, we might have imagined that the ruliad would basically just serve as a kind of dictionary of possible universes—a “universe of all possible universes” in which each possible universe has different laws. But the ruliad is in a sense a much more complicated object. Rather than being a “dictionary” of possible separate universes, it is something that entangles together all possible universes. The Principle of Computational Equivalence implies a certain homogeneity to this entangled structure. But the crucial point is that we don’t “look at this structure from the outside”: we are instead observers embedded within the structure. And what we observe then depends on our characteristics. And it turns out that even very basic features of our consciousness and sensory apparatus in a sense inevitably lead to known laws of physics—and in a sense do so generically, independent of details of just where in rulial space we are, or exactly what slice of the ruliad we take.</p>
<p>So far we’ve primarily talked about the ruliad in terms of physics and the fundamental structure of our physical universe. But the ruliad is actually something still more general than that. Because ultimately it is just created from the abstract concept of following all possible computational rules. And, yes, we can interpret these rules as representing things going on in our universe. But we can also interpret them as representing things going on in some other, less immediately physically realizable system. Or, for that matter, representing something purely formal, and, say, mathematical.</p>
<p>This way of talking about the ruliad might make one think that it should be “considered a possible model” for our universe, or for other things. But the bizarre and surprising point is that it is more than that. It’s not just a possible model that might be one of many. Rather, it is the unique ultimate representation of all possible models, entangled together. As we’ve discussed, there are many subtle choices about how we observe the ruliad. But the ultimate ruliad itself is a unique thing, with no choice about what it is. </p>
<p>As I’ve <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/">discussed at more length elsewhere</a>, the ruliad is in a sense a representation of all possible necessary truths—a formal object whose structure is an inevitable consequence of the very notion of formalization. So how does this relate to the idea that the ruliad also at an ultimate level represents our physical universe? What I’ve <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/">argued elsewhere</a> is that it means that the ultimate structure of our universe is a formal necessity. In other words, it’s a matter of formal necessity that the universe must exist, and have an ultimate ruliad structure. The fact that we perceive the universe to operate in a certain way—with our standard laws of physics, for example—is then a consequence of the particular way observers like us perceive it, which in turn depends on things like where in rulial space we happen to find ourselves.</p>
<p>But beyond physics, what else might the ruliad represent? The ruliad is an ultimate example of <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/">multicomputation, and of what I’ve characterized as the fourth major paradigm for theoretical science</a>. Often in multicomputation, what’s of interest is multiway systems with specific underlying rules. And already at this level, much of the apparatus that we’ve described in connection with the ruliad also applies—and in a sense “trickles down” to give various universal results.</p>
<p>But there are also definitely cases of multicomputation (other than physics) where the full notion of applying all possible rules is relevant. The global structures of <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#metamathematics">metamathematics</a>, <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#economics">economics</a>, <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#linguistics">linguistics</a> and <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#evolutionary-biology">evolutionary biology</a> seem likely to provide examples—and in each case we can expect that at the core is the ruliad, with its unique structure. Of course, this doesn’t mean that what we observe must always be the same, because what we observe depends on our characteristics as an observer—and the characteristics of “being an observer” in metamathematics, for example, are surely different from those for economics or evolutionary biology, or, for that matter, physics. </p>
<p>For sure, the “sensory apparatus” that we effectively use is different in different cases. But there are certain similar human-based features that still seem to apply. Whatever the domain, we always act as computationally bounded observers. And it seems that we also always have a certain coherence, consistently maintaining our “observerhood” through time or across some form of space. And it seems likely that these “human-induced” characteristics alone are sufficient to yield some very global implications for observed behavior.</p>
<h2>The View from Mathematics</h2>
<p>How should we think about the ruliad mathematically? In many ways, the ruliad is more an object of metamathematics than of mathematics itself. For in talking about the effects of all possible rules, it in a sense transcends individual mathematical theories—to describe a kind of metatheory of all possible theories.</p>
<p>Given a particular mathematical axiom system, it’s rather easy to see <a href="https://www.wolframscience.com/nks/p777--implications-for-mathematics-and-its-foundations/">correspondence with a multiway system</a>. There are a variety of ways to set it up, but one approach is to think of states in the multiway system as being expressions in the language used for the axiom system, and then to think of rules in the multiway system as applying transformations on these expressions that implement axioms in the axiom system.</p>
<p>For example, with the (Abelian semigroup) axioms</p>
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<p>here’s a multiway system generated from the expression by applying the (two-way) transformations defined by the axioms in all possible ways to each expression:</p>
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<p>But now from this graph we can read off the “theorem”:</p>
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<p>A proof of this theorem</p>
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<p>is just a path in the multiway graph:</p>
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<p>A significantly less direct but still perfectly valid proof would correspond to the 13-step path:</p>
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<p>It’s a slightly technical point, but perhaps worth mentioning that there are alternative ways to set up the correspondence between axiomatic mathematical systems and multiway systems. One such way is to take the states in the multiway system to be not expressions (like ) but whole propositions (like ). Axioms then show up as states, and the rules for the multiway system are “rules of inference” or “rules of entailment” that define how, say, pairs of propositions “entail” (or “imply”) other ones. (And, yes, this requires a generalized multiway system with not just 1 state → many states, but for example 2 states → 1 state.) Typical automated theorem provers (like <a href="http://reference.wolfram.com/language/ref/FindEquationalProof.html">FindEquationalProof</a>) operate in this kind of setup, attempting to find paths that lead from initial propositions and axioms to some final state that corresponds to an “obviously true” proposition, like .</p>
<p>But whatever the detailed setup, the basic picture is that an axiomatic mathematical system has an associated multiway graph, in which paths correspond to proofs. Given the rules for the multiway system, there is in general no way to guarantee that the path (if it exists) corresponding to the proof of some particular result will be of bounded length, leading to the possibility of undecidability. But even when a path exists, it may require an irreducibly large amount of computation to find it. Still, finding such paths is what automated theorem provers do. For example, we know <a href="https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/">(as I discovered in 2000) that ((<em>b</em> · <em>c</em>) · <em>a</em>) · (<em>b</em> · ((<em>b</em> · <em>a</em>) · <em>b</em>)) = <em>a</em> is the minimal axiom system for Boolean algebra</a>, because <a href="http://reference.wolfram.com/language/ref/FindEquationalProof.html">FindEquationalProof</a> finds a path that proves it.</p>
<p>But this path—and the corresponding proof—is a very “non-human” construct (and, for example, in 21 years essentially no progress has been made in finding a <a href="https://writings.stephenwolfram.com/2018/11/logic-explainability-and-the-future-of-understanding/">“human-understandable narrative” for it</a>). And we can make an analogy here to the situation in physics. The individual rule applications in the multiway graph (or the proof) are like individual updating events applied to the atoms of space—and they show all kinds of complexity and computational irreducibility. But in physics, human observers work at a higher level. And the same, one suspects, is true in mathematics.</p>
<p>Rather than looking at every detail of the multiway graph, human “mathematical observers” (i.e. pure mathematicians) in effect define all sorts of equivalences that conflate together different parts of the graph. If the individual updates in the multiway graph are like molecular dynamics, human pure mathematics seems to operate much more at the “fluid dynamics level”, concentrating on “broad mathematical constructs”, not the “machine code” of specific low-level axiomatic representations. (Of course, there are some situations, for example related to undecidability, where the “molecular dynamics” effectively “breaks through”.)</p>
<p>We’ve outlined above (and <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">discussed at length elsewhere</a>) how physical observers like us “parse” the low-level structure of the physical universe (and the ruliad). How might mathematical observers do it? A large part has to do with the identification of equivalences. And the key idea is that things which are considered equivalent should be assumed to be “the same”, and therefore “conflated for mathematical purposes”.</p>
<p>The most elementary example of something like this is the statement (<a href="https://writings.stephenwolfram.com/2020/09/the-empirical-metamathematics-of-euclid-and-beyond/">already present in Euclid</a>) that if and , then . The extensionality axiom of set theory is a more sophisticated example. And the univalence axiom of homotopy type theory is perhaps the most sophisticated current version. </p>
<p>There’s a very operational version of this that appears in automated theorem proving. Imagine that you’ve proved that and . Then (by the assumed properties of equality) it follows that . One way we could use this result is just to merge the nodes for and . But a “bigger” thing we can do is to add the “completion” as a general rule for generating the multiway system.</p>
<p>Consider, for example, the string substitution multiway system A ↔ AB:</p>
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<p>But notice here that both ABA ↔ ABBA and ABA ↔ ABAB. So now add the “completion” ABBA ↔ ABAB. Here’s the resulting multiway graph: </p>
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<p>Adding the completion has generated lots of new “direct equivalences”. But we can also think of it as having done something else: not only has it defined equivalences between states; it’s also defined equivalences between paths—or in effect between proofs. (Or, put another way, it’s implementing homotopic equivalence between proofs. By the way, it’s an important feature of human mathematics that progress is typically measured in theorems proved; different proofs of the same theorem are normally implicitly considered equivalent in terms of the progress of mathematics.) </p>
<p>In a <a href="https://arxiv.org/pdf/2105.10822.pdf">category theory interpretation</a>, transformations between states in the original multiway graph are like ordinary morphisms (1-morphisms). But when we’re making transformations between “proof paths”, this is like 2-morphisms. And just as we can add transformations between proofs, we can also add transformations between proofs-between-proofs, and so on. The result is that we can build up a whole hierarchy of higher and higher categories, eventually ending with an ∞-category. </p>
<p>But now we can begin to see the connection with the ruliad. The ruliad is in a sense the result of taking the limit of adding more and more possible rules. Above we did this quite explicitly in terms of the original underlying rules, for example by enumerating possible strings or possible integer multipliers. </p>
<p>But we can view successive completions as doing something very similar. Yes, the rules are enumerated in a different order. But in the end there’ll still be an infinite number of distinct rules being used. </p>
<p>Of course there are many mathematical details missing here. But in broad outline, it seems that one can think of the approach to the ruliad as some kind of limit of successively higher categories. But this limit is something that’s been studied (albeit in the upper reaches of pure mathematics), and it’s an object called the ∞-groupoid. (It’s a groupoid because when all the rules are included things inevitably “go both ways”). So, OK, is the ruliad “just” the ∞-groupoid? Not quite. Because there are more rules and more initial conditions in the ruliad, even beyond those added by completions. And in the end the ruliad actually seems to be the ∞-category of ∞-groupoids, or <a href="https://arxiv.org/pdf/2111.03460.pdf">what’s called the (∞,1)-category</a>.</p>
<p>But knowing that the ruliad can be thought of as composed of ∞-groupoids means that we can <a href="https://arxiv.org/pdf/2111.03460.pdf">apply mathematical ideas</a> about the ∞-groupoid to the ruliad. </p>
<p>Probably the most important is Grothendieck’s hypothesis, which asserts that the ∞-groupoid inevitably has a topological and (with a few other conditions) ultimately geometric structure. In other words, even though one might have imagined that one constructed the ∞-groupoid from “pure logic” (or from pure formal axiomatic structures), the assertion is that the limiting object one obtains inevitably exhibits some kind of geometrical or “spatial” structure. </p>
<p>Viewed in terms of the ruliad—and our explicit finite examples of it—this might not seem surprising. And indeed in our Physics Project, the whole concept of the <a href="https://www.wolframphysics.org/technical-introduction/limiting-behavior-and-emergent-geometry/">emergence of space from large-scale hypergraphs</a> is closely related. But here from Grothendieck’s hypothesis we’re basically seeing a general claim that the <a href="https://www.wolframphysics.org/bulletins/2020/08/a-candidate-geometrical-formalism-for-the-foundations-of-mathematics-and-physics/">ruliad must have “inevitable geometry”</a>—and we can then view things like the emergence of space in our Physics Project as a kind of “trickle down” from results about the ruliad. (And in general, a big “application” of geometrical structure is the possibility of “pure motion”.)</p>
<p>What does all this mean about the ruliad and mathematics? In a sense the ruliad represents all possible mathematics—the application of all possible rules, corresponding to all possible axiom systems. And from this “ultimate metamathematics”, <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/#the-relation-to-mathematics">human “mathematical observers”</a> are sampling pieces that correspond to the pure mathematics they consider of interest.</p>
<p>Perhaps these will align with particular axiom systems of the kind automated theorem provers (or proof assistants) use. But things may be “sloppier” than that, with human mathematical observers effectively being extended in rulial space—and capable of making “fluid-dynamics-level” conclusions, even if not “molecular-dynamics-level” ones. </p>
<p>But a key (and in some ways very surprising) point is that the ruliad can be viewed as the basis of both physics and mathematics. In some sense, physics and mathematics are at their core the same thing. They only “appear different” to us because the way we “observe” them is different. </p>
<p>I plan to discuss the implications for mathematics at greater length elsewhere. But suffice it to say here that the existence of a common underlying core—namely the ruliad—for both physics and mathematics immediately allows one to start importing powerful results from physics into mathematics, and vice versa. It also allows one, <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/#the-relation-to-mathematics">as I have done elsewhere</a>, to start comparing the existence of the universe with the (Platonic-style) concept of the fundamental existence of mathematics.</p>
<h2>The View from Computation Theory</h2>
<p>The ruliad can be thought of as an encapsulation of doing all possible computations in all possible ways. What we might think of as a “single computation” might consist of repeatedly applying the rules for a <a href="https://www.wolframscience.com/nks/p78--turing-machines/">Turing machine</a> to “deterministically” generate a sequence of computational steps: </p>
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<p>But one can also consider a <a href="https://www.wolframphysics.org/bulletins/2021/02/multiway-turing-machines/">“multicomputational” system</a>, in which rules can generate multiple states, and the whole evolution of the system can be represented by a multiway graph:</p>
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<p>In traditional computation theory, one talks about rules like these as “nondeterministic”, because they can have multiple outcomes—though one usually imagines that the final answer one wants from a computation can be found as the result of some particular path. (In what I now call the <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/">multicomputational paradigm</a>—that I believe is important for modeling in physics and other places—one instead considers the <a href="https://www.wolframphysics.org/bulletins/2021/02/multiway-turing-machines/">complete multiway graph</a> of entangled possible histories.)</p>
<p>In constructing the ruliad, one is in a sense going to a more extreme version of multicomputation, in which one uses not just a particular rule with multiple outcomes, but all possible rules. In effect, the concept is to use “maximal nondeterminism”, and at each step to independently “pick whatever rule one wants”, tracing out a rulial multiway system that includes all the different possible paths this generates. </p>
<p>For the kind of Turing machines illustrated above, the <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/">rulial multiway graph one gets</a> after one step is: </p>
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<p>After 2 steps the result is:</p>
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<p>The full ruliad is then some kind of infinite limit of this process. But as before, there’s lots of subtlety in how this limit is taken. But we can at least characterize some ways of approaching it using ideas from computational complexity theory. Increasing the number of steps of evolution is like increasing the time complexity one allows. Increasing the “size of states” (e.g. the width of nonzero Turing machine tape) that one includes is like increasing the space complexity one allows. And increasing the complexity of the rule (as measured in the number of bits needed to specify it) is like increasing the algorithmic complexity one allows. The ruliad is what is obtained by taking all these computational resource measures to infinity. </p>
<p>And a critical claim is that regardless of how this is done, the final ruliad construct one gets will always—at least in some sense—be the same. There will be many ways to coordinatize it, or to sample it, but the claim is that it’s always the same object that one’s dealing with. And ultimately the reason for this is the <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a>. Because it implies that whatever “computational parametrization” or “computational description language” one uses for the ruliad, one will almost always get something that can be viewed as “computationally equivalent”. </p>
<p>We’ve talked about building up the ruliad using Turing machines. But what about other models of computation—like cellular automata or register machines or lambda calculus? As soon as there’s computation universality we know that we’ll get results that are at least in principle equivalent, because in a sense there’s only a “finite translation cost” associated with setting up an interpreter from one model of computation to another. Or, put another way, we can always emulate the application of the rule for one system by just a finite number of rule applications for the other system.</p>
<p>But from computation universality alone we have no guarantee that there won’t be “extreme deformations” introduced by this deformation. What the Principle of Computational Equivalence says, however, is that almost always the deformations won’t have to be extreme. And indeed we can expect that particularly when multiple rules are involved, there’ll be rapid convergence almost always to a kind of “uniform equivalence” that ensures that the final structure of the ruliad is always the same.</p>
<p>But the Principle of Computational Equivalence appears to say still more about the ruliad: it says that not only will the ruliad be the same independent of the “computational basis” used to construct it, but also that there’ll be a certain uniformity across the ruliad. Different “regions of the ruliad” might involve different specific rules or different patterns of their application. But the Principle of Computational Equivalence implies that almost always the computations that happen will be equivalent, so that—at least at a certain scale—the structure associated with them will also be equivalent.</p>
<p>Knowing that the ruliad contains so many different computations, one might imagine that it would show no particular uniformity or homogeneity. But the Principle of Computational Equivalence seems to imply that it necessarily does, and moreover that there must be a certain coherence to its structure—that one can interpret (in the style of Grothendieck’s hypothesis) as an inevitable emergent geometry.</p>
<p>An individual computation corresponds to a path in the ruliad, going from its “input state” to its “output state”. In an ordinary deterministic computation, the path is restricted to always use the same rule at each step. In a nondeterministic computation, there can be different rules at different steps. But now we can formulate things like the P vs. NP problem essentially in terms of the geometry of the ruliad.</p>
<p>Here’s a picture of the same finite Turing-machine-based approximation to the ruliad as above—but now with the <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#deterministic-turing-machine-paths-in-rulial-space">paths that correspond to deterministic Turing machine computations</a> marked in red: </p>
<div>
<p>
	</p>
</div>
<p>The P vs. NP problem basically asks roughly whether the deterministic computations (shown here in red) will eventually “fill the ruliad”, or whether the general nondeterministic computations that are part of the ruliad will always “reach further”. Once again, there are many complicated and subtle issues here. But it’s interesting to see how something like the P vs. NP problem might play out in the ruliad. </p>
<p>In physics (and mathematics), we as human observers tend to sample the ruliad in a coarse-grained way, “noticing” only certain aspects of it. So is there an analog of this in computation theory—perhaps associated with certain characteristics of the “computation-theoretic observer”? There’s a potential answer, rather similar to what we’ve already seen in both physics and mathematics. </p>
<p>The basic point is that in computation theory we tend to study classes of computations (say P or NP) rather than individual computations. And in doing this we are in a sense always conflating many different possible inputs and possible outputs—which we assume we do in a computationally bounded way (e.g. through polynomial-time transformations, etc.) </p>
<p>Another thing is that we tend to focus more on the “experience of the end user” than the detailed activities of the “programmer”. In other words, we’re concerned more with what computational results are obtained, with what computational resources, rather than on the details of the program constructed to achieve this. Or, put another way, we tend to think about computation in terms of things like the successive evaluation of functions—and we conflate the different paths by which this is achieved.</p>
<p>Most likely this means that there are “effective laws” that can be derived in this computational view of the ruliad, analogous to laws of physics like general relativity. So what might some other analogies be?</p>
<p>A computation, as we’ve mentioned, corresponds to a path in the ruliad. And whenever there’s a possibility for an infinite path in the ruliad, this is a sign of undecidability: that there may be no finite way to determine whether a computation can reach a particular result. But what about cases when many paths converge to a point at which no further rules apply, or effectively “time stops”? This is the analog of a spacelike singularity—or a black hole—in the ruliad. And in terms of computation theory, it corresponds to something decidable: every computation one does will get to a result in finite time.</p>
<p>One can start asking questions like: <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#the-emerging-picture-of-rulial-space">What is the density of black holes in rulial space</a>? If we construct the ruliad using Turing machines, this is basically analogous to asking “What’s the density of halting Turing machines (+initial conditions) in rulial space?” And this is essentially given by <a href="https://www.wolframscience.com/nks/notes-10-3--algorithmic-information-theory/">Chaitin’s Ω</a>. </p>
<p>But so is there some number Ω that we can just compute for the ruliad? Well, actually, no. Because the undecidability of the halting problem makes Ω noncomputable. One can get approximations to it, but—in the language of the ruliad—those will correspond to using particular samplings or particular reference frames. Or in other words, even the perceived density of “decidability black holes” in the ruliad depends on features of the observer.</p>
<h2>What’s beyond the Ruliad?</h2>
<p>In our Physics Project we usually talk of the universe “evolving through time” (albeit with many entangled threads of history). But if the ruliad and its structure is a matter of formal necessity, doesn’t that mean that the whole ruliad effectively “already exists”—“outside of time”? Well, in some sense it does. But ultimately that would only be relevant to us if we could “look at the ruliad from the outside”.</p>
<p>And as observers like us within the ruliad, we necessarily have a different perception. Because <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">our consciousness</a>—with its computational boundedness—only gets to sample a certain sequence of pieces of the ruliad. If it were not for computational irreducibility, we might get to “jump around” in time. But computational irreducibility, together with our own computational boundedness, implies that our perception must necessarily just experience the passage of time through an irreducible process of computation.</p>
<p>In other words, while in some sense the ruliad may all “already be there” when viewed from the outside, our own perception of it “from the inside” is necessarily a progressive one, that effectively corresponds to the passage of time.</p>
<p>Could we experience the ruliad differently, even while being computationally bounded? If we think of the ruliad as a graph, then our usual “sequence of configurations of space at successive times” way of experiencing the ruliad is like a <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/foliations-and-coordinates-on-causal-graphs/#p-255">breadth-first traversal</a>. But could we for example instead do a depth-first traversal, exploring all time before investigating different parts of space? (And, yes, something like this can happen in general relativity near an event horizon, or in connection with timelike singularities.)</p>
<p>Later, we’ll discuss different ways to perceive the ruliad and the universe. But it seems to be a feature of anything we might call a coherent observer that there needs to be some form of progression in the perception. And so while we might not call it the passage of time, there’ll still be some way in which our exploration of the ruliad has a computationally irreducible process underneath.</p>
<p>A very important <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/#is-this-the-only-universe?">claim about the ruliad is that it’s unique.</a> Yes, it can be coordinatized and sampled in different ways. But ultimately there’s only one ruliad. And we can trace the argument for this to the Principle of Computational Equivalence. In essence there’s only one ruliad because the Principle of Computational Equivalence says that almost all rules lead to computations that are equivalent. In other words, the Principle of Computational Equivalence tells us that there’s only one ultimate equivalence class for computations.</p>
<p>But what if we just imagine a “<a href="https://www.wolframscience.com/nks/p720--the-content-of-the-principle/">hypercomputation</a>” not in that class? For example, imagine a hypercomputation (analogous, for example, to an oracle for a Turing machine) that in a finite number of steps will give us the result from an infinite number of steps of a computationally irreducible process. Such a hypercomputation isn’t part of our usual ruliad. But we could still formally imagine a hyperruliad that includes it—and indeed we could imagine a whole infinite hierarchy of successively larger and more powerful hyperruliads.</p>
<p>But it’s a fundamental claim that we’re making—that can be thought of as a matter of natural science—that in our universe only computation can occur, not hypercomputation. </p>
<p>At a purely formal level, there’s nothing wrong with hyperruliads. They exist as a matter of formal necessity just like the ordinary ruliad does. But the key point is that an observer embedded within the ruliad <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#why-this-universe-the-relativity-of-rules">can never perceive a hyperruliad</a>. As a matter of formal necessity there is, in a sense, a permanent event horizon that prevents anything from any hyperruliad from affecting anything in the ordinary ruliad. </p>
<p>So now we can be a bit more precise about our statement that “hypercomputation doesn’t happen in our universe”. Really we should say that we assert that we as observers operate purely computationally and not hypercomputationally. And this means that we are embedded within the ordinary ruliad, and not the hyperruliad.</p>
<p>Yes, we could imagine some other entity that’s embedded within the hyperruliad, and perceives what it considers to be the universe to operate hypercomputationally. But in a statement that’s in a sense more “about us” than “about the universe”, we assert that that can’t be us, and that we in a sense live purely within the ruliad—which means that for us the <a href="https://www.wolframscience.com/nks/p726--the-validity-of-the-principle/">Principle of Computational Equivalence holds</a>, and we perceive only computation, not hypercomputation.</p>
<h2>Communicating across Rulial Space</h2>
<p>What observers can there be embedded in the ruliad, and how should we characterize them? In physical spacetime we’re used to characterizing observers by their locations in physical space and by things like the spacetime reference frames they construct. And it’s very much the same for observers in the ruliad: we can characterize them by where they are in rulial space, and what rulial reference frames they use.</p>
<p>The Principle of Computational Equivalence tells us that it’s almost always possible to “encode” one “model of how the ruliad works” in any other model—effectively just by setting up a program that emulates the rules for one model using the rules for the other model. But we can think of these different models as being associated with different possible observers in the ruliad. </p>
<p>In other words, we can say that observers “at different places in rulial space” (or “using different rulial reference frames”) are using different description languages for what’s happening in the ruliad. And when an observer “moves” in rulial space, they’re effectively doing a translation from one description language to another. (And, yes, there’s a <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#the-emerging-picture-of-rulial-space">maximum rate of motion ρ</a> in rulial space—which is the rulial analog of the speed of light—and which is effectively determined by the fundamental processing speed of the universe.)</p>
<p>So far this might all seem quite abstract. But there are immediate, everyday examples that effectively correspond to being at different places in rulial space. A simple concrete one is computers with different instruction sets. Another one is different brains with different consciousnesses.</p>
<p>We can think of a single human <a href="https://writings.stephenwolfram.com/2021/03/what-is-consciousness-some-new-perspectives-from-our-physics-project/">consciousness as having a certain thread of experience of the universe</a>. Part of that experience is determined by the physical location of the consciousness and by the sensory apparatus with which it samples the world. But part is determined by the “internal description language” that it uses. And inevitably this internal description language depends both on the detailed physiology of the brain in which it is implemented, and on the past history of experiences that have “defined its way of looking at the world”. In the analogy of artificial neural networks, different networks will tend to have different “internal representations” because this depends not only on the network architecture, but also on the particular training data that the network has “experienced”.</p>
<p>Why can’t one human consciousness “get inside” another? It’s not just a matter of separation in physical space. It’s also that the different consciousnesses—in particular by virtue of their different histories—are inevitably at different locations in rulial space. In principle they could be brought together; but this would require not just motion in physical space, but also motion in rulial space.</p>
<p>But why then do different consciousnesses seem to have compatible views about “what happens in the universe”? Essentially this can be seen as a consequence of rulial relativity—which in turn depends on the inevitable causal invariance of the ruliad, which follows from the Principle of Computational Equivalence. There are certainly many issues to be worked out, but basically what seems to be going on is that because of causal invariance, different rulial reference frames will ultimately yield the same <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#rulial-multiway-causal-graphs">rulial multiway causal graphs</a>, and therefore the same “fundamental description of reality”.</p>
<p>We’ve talked about different consciousnesses. But what about just “different ways of thinking”? Well, it’s definitely more than an analogy to say that different ways of thinking correspond to different positions in rulial space. If there’s lots of common history then there’ll be common ancestry in the rulial multiway graph and one will necessarily end up close in rulial space. But without common history, one can end up with different description languages—or different ways of thinking—that are not nearby in rulial space.</p>
<p>In physical space we expect to effectively use momentum to move our location. And it’s potentially a bizarrely similar story in rulial space. In our models of fundamental physics, <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#what-is-energy-what-is-mass">energy and momentum are essentially related to the density of activity</a> (i.e. elementary updating events) in physical space. And we can similarly define a rulial analog of energy and momentum in terms of activity in rulial space. But it’s exactly this activity that provides connections between different parts of rulial space, or in effect “enables motion” in rulial space. </p>
<p>In other words, if you want to move in rulial space, you can do it by putting in the appropriate computational work to change your conceptual point of view (or, essentially equivalently, your language for describing things). So what about curvature (or the analog of gravity) in rulial space—say generated through an analog of Einstein’s equations from density of activity in rulial space? Presumably this relates to the difficulty—or time it takes—to get from one place in rulial space, and one way of thinking, to another. And conceivably things like “paradigm shifts” between different ways of thinking might be associated with features of rulial space like event horizons.</p>
<p>But let’s say you’re at one place in rulial space, and you want to get to another—or at least “send a signal” there. A typical microscopic change at one point in rulial space will tend to just “spread out in all directions” and “decay quickly”. But if you want to “coherently communicate”, you need some kind of structure that will persist as it propagates through rulial space. And by analogy with the case of physical space, what this presumably means is that you effectively need a “rulial particle”. </p>
<p>In terms of the ruliad, a rulial particle would presumably be some kind of “topological obstruction” or “topologically stable structure” that is at any moment effectively localized in rulial space and maintains its identity as it propagates across rulial space. But what might a rulial particle be in more everyday terms?</p>
<p>Potentially it’s like what we’d normally consider a concept—or something to which we might assign a word in human language. If we have ways of thinking—or consciousnesses—whose details are different, the issue is what will be robust enough to be able to be transported between them. And what everyday experience seems to suggest is that the answer is concepts. Even though one might have a slightly different way of thinking, what one calls “a fish” (or essentially, the concept of a fish) is something that can still robustly be communicated.</p>
<p>It’s interesting to notice that for an observer like us, there seem to be only a finite set of types of “elementary particles” that exist in physical space. And perhaps that’s not unrelated to the fact that observers like us also seem to imagine that there are in some sense only a finite number of “basic concepts” (associated, say, with distinct words in human languages). There’s lots more detail that exists in rulial space—or in the ruliad—but for observers like us, with our type of way of sampling the ruliad, these might be core coherent structures that we perceive. </p>
<h2>So Is There a Fundamental Theory of Physics? </h2>
<p>The concept of the ruliad arose from our efforts to find a fundamental theory of physics. But now that we know about the ruliad, what does it tell us about a fundamental theory?</p>
<p>At the outset, we might have imagined that the end point of our project would be the identification of some particular rule of which we could say “This is the rule for the universe”. But of course then we’d be faced with the question: “Why that rule, and not another?” And perhaps we would imagine just having to say “That’s something that you have to go beyond science to answer”.</p>
<p>But the ruliad implies a quite different—and in my view ultimately much more satisfying—picture. The ruliad itself is a construct of abstract necessity—that in a sense represents the entangled behavior of all possible rules for the universe. But instead of imagining that some particular rule out of all these possibilities is “picked from outside” as “the choice for our universe”, what we suppose is that—as observers embedded within the ruliad—we’re the ones who are implicitly picking the rule by virtue of how we sample and perceive the ruliad.</p>
<p>At first this might seem like it’s a wimp out. We want to know how our universe works. Yet we seem to be saying “we just pick whatever rule we feel like”. But that’s not really the story at all. Because in fact observers that are even vaguely like us are in effect deeply constrained in what rules they can attribute to the universe. There’s still some freedom, but a fundamental result is that for observers like us it seems to be basically inevitable that any rule we can pick will on a large scale reproduce the central known general laws of physics, in particular general relativity and quantum mechanics.</p>
<p>In other words, for observers generally like us it’s a matter of abstract necessity that we must observe general laws of physics that are the ones we know. But what about more specific things, like the particular spectrum of elementary particles, or the particular distribution of matter in the universe? It’s not clear how far “the general” goes—in other words, what is a matter of abstract necessity purely from the structure of the ruliad and general features of observers like us. </p>
<p>But inevitably at some point we will run out of “the general”. And then we’ll be down to specifics. So where do those specifics enter? Ultimately they must be determined by the details of how we sample the ruliad. And a prominent part of that is simply: Where in the ruliad are we? We can ask that about our location in physical space. And we can also ask it about our location in rulial space.</p>
<p>What does all this mean? At some level it’s saying that the way we are as observers is what makes us attribute certain rules to our universe. The ruliad is in a sense the only thing that fundamentally exists—and in fact its existence is a matter of abstract necessity. And our universe as we experience it is some “slice of the ruliad”, with what slice it is being determined by what we’re like as observers.</p>
<p>Let’s look at the logical structure of what we’re saying. First, we’re describing the ruliad, which at the outset doesn’t have anything specifically to do with physics: it’s just a formal construct whose structure is a matter of abstract necessity, and which relates as much to mathematics as it does to physics. But what “puts the physics in” is that we in effect “live in the ruliad”, and our perception of everything is based on “experiencing the ruliad”. But that experience—and the effective laws of physics it entails—inevitably depends on “where we are in the ruliad” and how we’re able to sample it.</p>
<p>And this is where our pieces of “falsifiable natural science” come in. The first “assertion of natural science” that we make is that we are embedded only within the ordinary ruliad, and not a hyperruliad—or in other words that our experience encompasses only computation, and not hypercomputation. </p>
<p>This is closely related to a second assertion, which may in fact be considered to subsume this: that we are computationally bounded observers, or, in other words, that our processes of perception involve bounded computation. Relative to the whole ruliad—and all the computation it entails—we’re asserting that we as observers occupy only a tiny part.</p>
<p>There’s one more assertion as well, again related to computational boundedness: that we as observers have a certain coherence or persistence. In general the ruliad contains all sorts of wild and computationally irreducible behavior. But what we’re asserting is that that part of the ruliad that is associated with us as observers has a certain simplicity or computational reducibility: and that as we evolve through time or move in space, we somehow maintain our identity. </p>
<p>These assertions seem very general, and in some ways almost self-evident—at least as they apply to us. But the important and surprising discovery is that they alone seem to lead us inexorably to crucial features of physics as we know it.</p>
<p>Where does this physics “come from”? It comes partly from the formal structure of the ruliad, and formal features of the multicomputational processes it involves. And it comes partly from the nature of us as observers. So if we ask “Why is the physics of our universe the way it is?”, an important part of the answer is “Because we observe the universe the way we do”. </p>
<p>One might imagine that in some sense physics would give us no choice about how we observe the universe. But that’s not the case. Because in the end our “observation” of the universe is about the “abstract conceptual model” we build up for the universe. And, yes, that’s certainly informed by the particular sensory apparatus we have, and so on. But it’s something we can certainly imagine being different.</p>
<p>We can think of ourselves as using some particular description language for the universe. The structure of that language is constrained by the assertions we gave above. But within such a description language, the laws of physics necessarily work out the way they do. But if we chose a different description language, we’d end up with different laws of physics.</p>
<p>Much of our perception of the universe is based on our raw biological structure—the way our sensory organs (like our eyes) work, as well as the way our brains integrate the inputs we get. But that’s not all there is to it. There’s also a certain base of knowledge in our civilization that informs how we parse our “raw perception”—and in effect what description language we use. Once we have the idea of periodic behavior, say, we can use it to describe things that we’d previously have to talk about in a less “economical” way.</p>
<p>But what if our knowledge changed? Or we had different sensory capabilities? Or we used technology to integrate our sensory input in different ways? Then we’d be able to perceive and describe the universe in different ways.</p>
<p>One’s first impression might be that the ruliad effectively contains many possible “parallel universes”, and that we have selected ourselves into one of these, perhaps as a result of our particular characteristics. But in fact the ruliad isn’t about “parallel universes”, it’s about universes that are entangled at the finest possible level. And an important consequence of this is that it means we’re not “stuck in a particular parallel universe”. Instead, we can expect that by somehow “changing our point of view”, we can effectively find ourselves in a “different universe”.</p>
<p>Put another way, a given description of the universe is roughly represented by being at a certain location in rulial space. But it’s possible to move in rulial space—and end up with a different description, and different effective laws for the universe.</p>
<p>But how difficult is motion in rulial space? It could be that some impressive future technology would allow us to “move far enough” to end up with significantly different laws of physics. But it seems more likely that we’d be able to move only comparatively little—and never be able to “escape the box” of things like computational boundedness, and coherence of the observer.</p>
<p>Of course, even changing a little might lead us to different detailed laws of physics—say attributing a different mass to the electron, or a different value of the electromagnetic coupling constant α. But actually, even in traditional physics, this is <a href="https://www.wolframscience.com/nks/notes-9-14--particle-masses/">already something that happens</a>. When viewed at different energy scales—or in a sense with different technology—these quantities have different effective values (as characterized by the renormalization group). </p>
<p>At first it might seem a little strange to say that as our knowledge or technology change, the laws of physics change. But the whole point is that it’s really our perceived laws of physics. At the level of the raw ruliad there aren’t definite laws of physics. It’s only when we “sample our slice” of the ruliad that we perceive definite laws.</p>
<p>What does all this mean operationally for the search for a fundamental theory of physics? At some level we could just point to the ruliad and declare victory. But this certainly wouldn’t give us specific predictions about the particulars of our perceived universe. To get that we have to go further—and we have to be able to say something about what “slice of the ruliad” we’re dealing with. But the good news is that we don’t seem to have to make many assumptions about ourselves as observers to be able to identify many physical laws that observers like us should perceive. </p>
<p>So can we ever expect to nail down a single, specific rule for the universe, say one a particular observer would attribute to it? Given our characteristics as observers, the answer is undoubtedly no. We’re simply not that small in rulial space. But we’re not that big, either. And, importantly, we’re small enough that we can expect to “do science” and consider the universe to “behave in definite ways”. But just as in physical space we’re vastly larger than the scale associated with the atoms of space, so similarly we’re also undoubtedly vastly larger in rulial space than the individual components of the ruliad—so we can’t expect our experience to all be “concentrated in one thread” of the ruliad, following one particular rule. </p>
<p>As we discussed above, by doing experiments we can use scientific inference to attempt to localize ourselves in rulial space. But we won’t be able to do enough to say “from our point of view, the universe is operating according to this one specific rule, and not another”. Instead, there’ll be a whole collection of rules that are “good enough”, in the sense that they’ll be sufficient to predict the results of experiments we can realistically do.</p>
<p>People have often imagined that, try as we might, we’d never be able to “get to the bottom of physics” and find a specific rule for our universe. And in a sense our inability to localize ourselves in rulial space supports this intuition. But what our Physics Project seems to rather dramatically suggest is that we can “get close enough” in rulial space to have vast predictive power about how our universe must work, or at least how observers like us must perceive it to work. </p>
<h2>Alien Views of the Ruliad</h2>
<p>We’ve discussed how “observers like us” will necessarily “parse the ruliad” in ways that make us perceive the universe to follow the laws of physics as we know them. But how different could things get? We have a definite sense of what constitutes a “reasonable observer” based on our 21st-century human experience—and in particular our biology, our technology and our ways of thinking. </p>
<p>But what other kinds of observers can we imagine? What about, for example, <a href="https://www.wolframscience.com/nks/p827--intelligence-in-the-universe/">animals other than humans</a>—in particular say ones whose sensory experience emphasizes olfaction or echolocation or fluid motion? We can think of such animals as operating in a different rulial reference frame or at a different place in rulial space. But how far away in rulial space will they be? How similar or not will their “world views” (and perceived laws of physics) be to ours? It’s hard to know. Presumably our basic assertions about computational boundedness and coherence still apply. But just how the specifics of something like sequentialization in time play out, say, for an ant colony, seems quite unclear.</p>
<p>Maybe one day we’ll be able to systematically “think like other animals”. But as of now we haven’t been able to “travel that far” in rulial space. We’ve quite thoroughly explored physical space, say on the surface of our planet, but we haven’t explored very far at all in rulial space. We don’t have a way to translate our thinking into some kind of “thinking differently”—and we don’t, for example, <a href="https://writings.stephenwolfram.com/2018/01/showing-off-to-the-universe-beacons-for-the-afterlife-of-our-civilization/">have a common language to get there</a>.</p>
<p>There’s often an assumption (a kind of “human exceptionalism”) that if it wasn’t for details of the human experience—like brains and words—then we’d necessarily be dealing with something fundamentally simpler, that could not, for example, show features that we might identify as intelligence. But the <a href="https://www.wolframscience.com/nks/p822--intelligence-in-the-universe/">Principle of Computational Equivalence tells us this isn’t correct</a>. Because it says that there’s a certain maximal computational sophistication that’s achieved not just by us humans but also by a vast range of other systems. The restrictions of what we’ve chosen to study (in science and elsewhere) have often made us miss this, but in fact computational sophistication—and the direct generalization of our notion of intelligence that’s associated with it—seems quite ubiquitous across many different kinds of systems.</p>
<p>So can those other kinds of systems act as “observers like us”? To do so, they need not just computational sophistication, but also a certain alignment with the features we have that lead to our coherent thread of “conscious experience”. And even given that, to actually “connect with” such systems, we need to be able to reach far enough in rulial space to sufficiently make a translation. </p>
<p>Imagine the <a href="https://www.wolframscience.com/nks/p822--intelligence-in-the-universe/">weather (sometimes said to “have a mind of its own”)</a>. It’s got plenty of computational sophistication. But is there any sense in which it sequentializes time like we do? Or can one only think of all those different parts of our atmosphere “running in their own time”? To know things like this, we effectively have to have a way to “translate” from the operation of the weather to our (current) way of thinking.</p>
<p>And in some sense we can consider the whole enterprise of natural science as being an effort to find a method of translation—or a <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#the-challenge-of-language-design-for-the-universe">common language—between nature and our way of thinking</a>. </p>
<p>We as observers in effect trace out particular trajectories in rulial space; the challenge of natural science is to “reach out” in rulial space and “pull in” more of the ruliad; to be able to define a way to translate more parts of the ruliad to our processes of thinking. Every time we do an experiment, we can think of this as representing a moment of “connection” or “communication” between us and some aspect of nature. The experiment in effect defines a small piece of “common history” between us and nature—which helps “knit together” the parts of rulial space associated with us and with nature. </p>
<p>One of the great mysteries of science has been why—in the vastness of physical space—we’ve never detected something we identify as “alien intelligence”, or an “alien civilization”. We might have thought that it was because we humans have either achieved a unique pinnacle of intelligence or computational ability—or have fundamentally not gotten far enough. But the Principle of Computational Equivalence explodes the idea of this kind of cosmic computational pecking order.</p>
<p>So what could actually be going on? Thinking in terms of the ruliad suggests an answer. Our radio telescopes might be able to detect signals from far away in physical space. But our putative aliens might not only live far away in physical space, but also in rulial space.</p>
<p>Put another way, the “alien civilization” might be sampling aspects of the ruliad—and in effect the universe—that are utterly different from those we’re used to. That different sampling might be happening right down at the level of atoms of space, or it might be that the rulial distance from us to the aliens is small enough that there’s enough “shared description language” that the alien civilization might rise to the level of seeming like some kind of <a href="https://www.wolframscience.com/nks/p836--intelligence-in-the-universe/">“noise” relative to our view of “what’s important in the universe”</a>.</p>
<p>We might wonder how far apart what we could consider “alien civilizations” would be in physical space. But what we now realize is that we also have to consider <a href="https://writings.stephenwolfram.com/2018/01/showing-off-to-the-universe-beacons-for-the-afterlife-of-our-civilization/#the-space-of-all-possible-civilizations">how far apart they might be in rulial space</a>. And just like in exploring physical space we can imagine building better spacecraft or better telescopes, so also we can imagine building better ways to reach across rulial space. </p>
<p>We’re so used to physical space that it seems to us very concrete to reach across it. Of course, in our Physics Project, things like motion in physical space end up—like everything else—being pure computational processes. And from this point of view, reaching across rulial space is ultimately no more abstract—even though today we would describe it in terms of “doing (abstract) computations” rather than “moving in space”.</p>
<p>Relative to our own physical size, the universe already seems like a vast place. But the full ruliad is even incredibly more vast. And we are likely much tinier in rulial space relative to the whole universe than we are in physical space. From the Principle of Computational Equivalence we can expect that there’s ultimately no lack of raw computational sophistication out there—but thinking in terms of the ruliad, the issue is whether what’s going on is close enough to us in rulial space that we can <a href="https://writings.stephenwolfram.com/2018/01/showing-off-to-the-universe-beacons-for-the-afterlife-of-our-civilization/">successfully see it as an “alien civilization”</a>.</p>
<p>One test of rulial distance might be to ask whether our putative aliens perceive the same laws of physics for the universe that we do. We know that at least the general forms of those laws depend only on what seem to us rather loose conditions. But to get good alignment presumably requires at the very least that we and the aliens are somehow “comparable in size” not only in physical space (and branchial space), but also in rulial space. </p>
<p>It’s humbling how difficult it is to imagine the universe from the point of view of an alien at a different place in rulial space. But for example if the alien is big compared to us in rulial space, we can say that they’ll inevitably have a version of science that seems to us much “vaguer” than ours. Because if they maintain a coherent thread of experience, they’ll have to conflate more distant paths in rulial space, on which the universe will do things that are “more different” than what we’re used to. (And, yes, there should be rulial analogs of quantum phenomena, associated for example with conflated paths that diverge far in rulial space.)</p>
<p>What would it mean operationally for there to be an alien civilization perhaps nearby in physical space but at a distance in rulial space? Basically the alien civilization will be “operating” in features of the universe that our parsing of the universe just doesn’t pick up. As a simple analogy, our view of, for example, a box of gas might be that it’s something with a certain temperature and pressure. But a different “parsing” of that system might identify a whole world of detailed motions of molecules that with respect to that parsing can be viewed as a vast “alien civilization”. Of course, the situation is much more extreme when it comes to the whole ruliad, and all the paths of history and configurations of atoms of space that it represents.</p>
<p>Relative to the whole ruliad, our civilization and our experience have carved out an extremely tiny piece. And what we’re thinking of as “alien civilizations” might also have carved out their own tiny pieces. And while we’re all “living in the same ruliad”, we might no more be able to detect each other or communicate (and likely very much less) than we can across vast distances in physical space. </p>
<p>What of the future? The future of our civilization might well be a story of mapping out more of rulial space. If we continue to invent new technology, explore new ideas and generally broaden our ways of thinking and perceiving, we will gradually—albeit in tiny steps—map out more of rulial space. How far can we get? The ultimate limit is determined by the maximum rulial speed. But if we expect to maintain our character as “observers like us”, we’ll no doubt be limited to something much less.</p>
<p>Among other issues, moving in rulial space involves doing computation. (The ultimate scale is set by the “processing power” of the universe—which defines the maximum rulial speed.) But “density of computation” effectively corresponds to a generalized version of mass—and is for example a source of “generalized gravity”. And it could be that to “move any significant distance” in rulial space, we’d have to “experience enough generalized gravity” that we could never maintain things like the kind of coherence we need to be an “observer like us”.</p>
<p>Put another way: yes, it might in principle be possible to “reach out in rulial space” and “contact the rulial aliens”. But it might be that doing so would require us to be so different from the way we currently are that we wouldn’t recognize anything like consciousness or anything that really makes us “identifiably us”. And if this is so, we are in a sense limited to experiencing the ruliad “on our own” from our particular place in rulial space, forever isolated from “alien civilizations” elsewhere in rulial space.</p>
<h2>Conceptual Implications of the Ruliad</h2>
<p>What does the concept of the ruliad mean for the fundamental way we think about things like science? The typical conception of “what science does” is that it’s about us figuring out—as “objectively” as we can—how the world happens to be. But the concept of the ruliad in a sense turns this on its head. </p>
<p>Because it says that at some ultimate level, everything is a matter of abstract necessity. And it’s just our “parsing” of it that defines the subject matter of what we call science. We might have thought that the science of the universe was just something that’s “out there”. But what we’re realizing is that instead in some fundamental sense, it’s all “on us”.</p>
<p>But does that mean that there’s no “objective truth”, and nothing that can robustly be said about the universe without “passing it through us”? Well, no. Because what we’ve discovered through our Physics Project is that actually there are quite global things that can (“objectively”) be said about our universe and the laws it follows, as perceived by observers like us. </p>
<p>We don’t have to know in detail about us humans and the particular ways we perceive things. All we need are some general features—particularly that we are computationally bounded, and that we have a certain persistence and coherence. And this is all it takes to deduce some quite specific statements about how our universe operates, at least as we perceive it.</p>
<p>So in a sense what this means is that there is a large “zone of objectivity”; a large set of choices for how we could be that will still lead us to the same “objective truth” about our universe. But if we go far enough away in our mechanism for “parsing the ruliad”, this will no longer be the case. From our current vantage point, we’d no doubt then be hard-pressed to recognize how we’re “doing the parsing”, but the results we’d get would no longer give us the same laws of physics or general perception of the universe that we’re used to.</p>
<p>This view of things has all sorts of implications for various long-discussed philosophical issues. But it’s also a view that has precise scientific consequences. And these don’t just relate to physics. Because the ruliad is really a general object that represents the entangled behavior of all possible abstract rules. When we think of ourselves as observers embedded within this object, it means that <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/">for us things are actualized</a>, and we have what we call physics. But we can also imagine sampling the ruliad in different ways. </p>
<p>Some of those ways <a href="https://writings.stephenwolfram.com/2021/04/why-does-the-universe-exist-some-perspectives-from-our-physics-project/#the-relation-to-mathematics">correspond to mathematics (or metamathematics)</a>. Some correspond to theoretical computer science. The ruliad is the single object that underlies all of them. And which of them we’re talking about just depends on how we imagine we’re sampling or parsing the ruliad, and how we’re describing what we’re observing.</p>
<p>With this degree of generality and universality, it’s inevitable that the ruliad must be a complicated object; in fact, in a sense it must encapsulate all possible achievable complexity. But what’s important is that we now have a definite concept of the ruliad, as something we can study and analyze.</p>
<p>It’s not simple to do this. The ruliad is at some level an object of great and perhaps supremely elegant abstract regularity. But for us to get any concrete handle on it and its structure, we need to break it down into some kind of “digestible slices” which inevitably lose much of its abstract regularity. </p>
<p>And we’re just at the beginning of seeing how best to “unpack” and “pick through” the ruliad. With explicit computations, we can only chip away at the <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/">very simplest approximations to the ruliad</a>. In a sense it’s a tribute to the naturalness and inevitability of the ruliad that it’s so closely related to some of the most advanced abstract mathematical methods we know so far. But again, even with these methods we’re barely scratching the surface of the ruliad and what it contains.</p>
<p>The theoretical exploration of the ruliad will be a long and difficult journey. But the incredible generality and universality of the ruliad means that every piece of progress is likely to have exceptionally powerful consequences. In some sense the exploration of the ruliad can be seen as the encapsulated expression of everything it means to do theoretical investigation: a kind of ultimately abstract limit of theoretical science and more.</p>
<p>For me, the ruliad in a sense builds on a tower of ideas, that include the computational paradigm in general, the exploration of the computational universe of simple programs, the Principle of Computational Equivalence, our Physics Project and the notion of multicomputation. But even with all of these it’s still a significant further jump in abstraction. And one whose consequences will take considerable time to unfold.</p>
<p>But for now it’s exciting to have at least been able to define this thing I call the ruliad, and to start seeing some of its unprecedentedly broad and deep implications. </p>
<hr />
<h2>Appendix: The Case of the “Multiplicad”</h2>
<p>As a very simple example of something like the ruliad, we can consider what we might call the “multiplicad”: a rulial <a href="https://www.wolframphysics.org/bulletins/2021/10/multicomputation-with-numbers-the-case-of-simple-multiway-systems">multiway system based on integers</a>, in which the rules simply multiply by successive integers: </p>
<div>
<p>
	</p>
</div>
<p>(Note that this kind of pure multiplication is presumably not computation universal, so the limiting object here will not be a coordinatization of the actual full ruliad.)</p>
<p>Just like with the full ruliad, there are many different “directions” in which to build up the multiplicad. We could allow as many multipliers and steps as we want, but limit the total size of numbers generated, here say to 30:</p>
<div>
<p>
	</p>
</div>
<p>As an alternative, we can limit the number of multipliers <em>s</em>, say to . Then the multiplicad would build up like this:</p>
<div>
<p>
	</p>
</div>
<p>In the pictures we’ve drawn so far, we’re effectively always deduplicating different occurrences of the same integer. So, for example, the integer 12 can be generated as 1 × 3 × 4 or 1 × 6 × 2 or 1 × 3 × 2 × 2, etc. And in principle we could show each of these “different 12s” separately. But in our <a href="https://writings.stephenwolfram.com/2021/09/even-beyond-physics-introducing-multicomputation-as-a-fourth-general-paradigm-for-theoretical-science/#the-formal-structure-of-multicomputation">deduplicated graph</a>, only a single 12 appears—with the different possible decompositions of 12 being reflected in the presence of multiple paths that lead to the 12.</p>
<p>Sometimes the structure we get is richer—if much bigger—when we don’t immediately do deduplication. For example, if we allow any number of multipliers (i.e. take ) then after just 1 step we will get all integers—and if we do deduplication, then this will be the end of our graph, because we “already have all the integers”. But if we don’t do deduplication, we’ll get a slightly more complicated picture, that begins like this:</p>
<div>
<p>
	</p>
</div>
<p>The “topological” structure of this graph is now straightforward, but its “labeling” with numbers is less so—and if we ask, for example, where a particular number appears after <em>t</em> steps, this can be more complicated. </p>
<p>Imagine that we are looking only at the subtrees associated with up to <em>s</em> multipliers at the first step—or, equivalently, that we are looking at the rulial multiway system “truncated” with only <em>s</em> rules. Which numbers will appear after steps? The answer is that it will be precisely those numbers that show up in an <em>s</em> × <em>s</em> multiplication table where we start from :</p>
<div>
<p>
	</p>
</div>
<p>Clearly no primes appear here, but some numbers can appear multiple times (e.g. 12 appears 4 times). In general, the number of times that the number will show up is the number of proper divisors it has, or <a href="http://reference.wolfram.com/language/ref/DivisorSigma.html">DivisorSigma</a>[0, <em>n</em>]–2: </p>
<div>
<p>
	</p>
</div>
<div>
<p>
	</p>
</div>
<p>We can continue this, to ask how many times a given number <em>n</em> will occur at a particular step <em>t</em>: </p>
<div>
<p>
	</p>
</div>
<div>
<p>
	</p>
</div>
<div>
<p>
	</p>
</div>
<p>We can think of these results as being determined by the number of times that <em>n</em> appears in an <em>s</em> × <em>s</em> × <em>s</em>… (<em>t</em> times) multiplication array. Alternatively, to know the results for a given number <em>n</em>, we can look at all the ways <em>n</em> can be decomposed into factors. For , for example, we would have:</p>
<div>
<p>
	</p>
</div>
<p>And from this we can deduce that 12 appears once at (i.e. with 1 factor), 4 times at (i.e. with 2 factors) and 3 times at (i.e. with 3 factors). </p>
<p>The full multiplicad is formed by taking the limits and (as well as what is essentially the limit for an infinite set of possible initial conditions). As we can see, our “finite perception” of the multiplicad will be different depending on how we sample it in <em>s</em> and <em>t</em>. </p>
<p>As an example, let’s consider what happens for given <em>s</em> as a function of <em>t</em>. For , we simply have powers of 2:</p>
<div>
<p>
	</p>
</div>
<p>For , where can multiply by both 2 and 3, we get:</p>
<div>
<p>
	</p>
</div>
<p>In studying multiway systems, it is often of interest to ask about the <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/typical-multiway-graph-structures/">growth rates of the number of states</a> reached over the course of <em>t</em> steps (i.e. the growth rates of volumes of geodesic balls). In the case , the number of states reached by step <em>t</em> is just <em>t</em>. For , it’s the triangular numbers <em>t </em>(<em>t –</em> 1)/2:</p>
<div>
<p>
	</p>
</div>
<p>Here are some results for larger <em>s</em>:</p>
<div>
<p>
	</p>
</div>
<p>Each of these sequences is generated by a linear recurrence relation with a kernel given by a sequence of signed binomial coefficients. The values for successive <em>t</em> can be represented by polynomials:</p>
<div>
<p>
	</p>
</div>
<p>The leading term in the growth of number of states is then determined by the orders of these polynomials, which turn out to be just <a href="http://reference.wolfram.com/language/ref/PrimePi.html">PrimePi</a>[<em>s</em>]:</p>
<div>
<p>
	</p>
</div>
<div>
<p>
	</p>
</div>
<p>For the case , we saw above that the multiway graph essentially forms a simple 2D grid. For larger <em>s</em>, the graph becomes more complicated, though it still approximates a grid—but in dimension <a href="http://reference.wolfram.com/language/ref/PrimePi.html">PrimePi</a>[<em>s</em>]. (The reason <a href="http://reference.wolfram.com/language/ref/PrimePi.html">PrimePi</a>[<em>s</em>] appears is that in a sense the combining of primes less than <em>s</em> are the largest “drivers” of structure in the multiway graph.)</p>
<p>In our general analysis of multiway graphs, it is common to consider <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-concept-of-branchial-graphs/">branchial graphs</a>—or for a rulial multiway system what we can call <a href="https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/#rulial-graphs">rulial graphs</a>—obtained by looking at a slice of the multiway graph, effectively for a given <em>t</em>, and asking what states are connected by having a common ancestor. The results for are rather trivial (here shown for , 2, 3):</p>
<div>
<p>
	</p>
</div>
<p>For we get:</p>
<div>
<p>
	</p>
</div>
<p>And for we have:</p>
<div>
<p>
	</p>
</div>
<p>In a sense these pictures show how numbers in the multiplicad can be “laid out in rulial space”. For , the “large-<em>t</em> graph” has a very linear form</p>
<div>
<p>
	</p>
</div>
<p>and the numbers that appear “from left to right” are arranged more or less in numerical order:</p>
<div>
<p>
	</p>
</div>
<p>For , the result is a 2D-like structure:</p>
<div>
<p>
	</p>
</div>
<p>And again the numbers that appear are roughly arranged in a kind of “numerical sequence”:</p>
<div>
<p>
	</p>
</div>
<p>We can then think of this as suggesting that the multiplicad “generates a rulial space” that can be approximately coordinatized purely by the values of the numbers that appear in it. How all this works in the limit is not clear, though somehow the “1D” sequence of numerical values presumably “snakes through” <a href="http://reference.wolfram.com/language/ref/PrimePi.html">PrimePi</a>[<em>s</em>]-dimensional space as some kind of approximation to a space-filling curve. </p>
<p>It should be noted that we’ve only considered one particular way of sampling the rulial multiway graph as a function of <em>t</em>. In general there are many different possible foliations that could be used, all of them giving us in effect a different view of the multiplicad, from a different “reference frame”.</p>
<p>As mentioned at the beginning, the multiplicad is presumably not on its own capable of giving us the full ruliad. But if we change the underlying rules—probably even just <a href="https://www.wolframphysics.org/bulletins/2021/10/multicomputation-with-numbers-the-case-of-simple-multiway-systems/#multiplication-and-addition">inserting addition as well as multiplication</a>—we’ll potentially get a system that is <a href="https://www.wolframscience.com/nks/p673--emulating-cellular-automata-with-other-systems/">capable of universal computation</a>, and which can therefore generate the full ruliad. Needless to say, the particular representation of the ruliad obtained by the kind of “numerical processes” that we’ve used here may be utterly different from any representation that we would recognize from our perception of the physical universe.</p>
<h2>Thanks &amp; Note</h2>
<p>Thanks for discussions of various aspects of the ruliad to Xerxes Arsiwalla, James Boyd, Elise Cawley, Hatem Elshatlawy, Jonathan Gorard and Nik Murzin. Thanks also to Ed Pegg and Joseph Stocke for input about the multiplicad. A <a href="https://arxiv.org/pdf/2111.03460.pdf">new paper by Xerxes Arsiwalla and Jonathan Gorard</a> discusses in a more technical way some ideas and results related to the ruliad.</p>
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 Stephen Wolfram

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 Contents
 * Top
 * The Entangled Limit of Everything
 * Experiencing the Ruliad
 * Observers Like Us
 * Living in Rulial Space
 * The View from Mathematics
 * The View from Computation Theory
 * What's beyond the Ruliad?
 * Communicating across Rulial Space
 * So Is There a Fundamental Theory of Physics?
 * Alien Views of the Ruliad
 * Conceptual Implications of the Ruliad
 * Appendix: The Case of the “Multiplicad”
 * Thanks & Note
 The Concept of the Ruliad

 The Concept of the Ruliad

 November 10, 2021

 The Entangled Limit of Everything

 I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our Physics Project. And it’s one that I think has extremely deep implications—both in science and beyond.

 In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.

 We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is.

 In the language of our Physics Project, it’s the ultimate limit of all rulial multiway systems. And as such, it traces out the entangled consequences of progressively applying all possible computational rules.

 Here is an example of an ordinary multiway system based on the string replacement rules {A → AB, BB → A} (indicated respectively by blueish and reddish edges):

 ✕
 

 At each step, the rules are applied in all possible ways to each state. Often this generates multiple new states, leading to branching in the graph. But, importantly, there can also be merging—from multiple states being transformed to the same state.

 The idea of a rulial multiway system is not just to apply particular rules in all possible ways, but to apply all possible rules of a given form. For example, if we consider “1 → 2, 2 → 1 A, B string rules”, the possible rules are

 ✕
 

 and the resulting multiway graph is (where now we’re using purple to indicate that there are edges for every possible rule):

 ✕
 

 Continuing a little longer, and with a different layout, we get:

 ✕
 

 This may already look a little complicated. But the ruliad is something in a sense infinitely more complicated. Its concept is to use not just all rules of a given form, but all possible rules. And to apply these rules to all possible initial conditions. And to run the rules for an infinite number of steps.

 The pictures above can be thought of as coarse finite approximations to the ruliad. The full ruliad involves taking the infinite limits of all possible rules, all possible initial conditions and all possible steps. Needless to say, this is a complicated thing to do, and there are many subtleties yet to work out about how to do it.

 Perhaps the most obviously difficult issue is how conceivably to enumerate “all possible rules”. But here we can use the Principle of Computational Equivalence to tell us that whatever “basis” we use, what comes out will eventually be effectively equivalent. Above we used string substitution systems. But here, for example, is a rulial multiway system made with 2-state 2-color Turing machines:

 ✕
 

 And here is a rulial multiway system made from hypergraph rewriting of the kind used in our Physics Project, using all rules with signature :

 ✕
 

 As another example, consider a multiway system based on numbers, in which the rules multiply by each possible integer:

 ✕
 

 Here’s what happens starting with 1 (and truncating the graph whenever the value exceeds 100):

 ✕
 

 Even with this simple setup, the results are surprisingly complicated (though it’s possible to give quite a bit of analysis in this particular case, as described in the Appendix at the end of this piece).

 The beginning of the multiway graph is nevertheless simple: from 1 we connect to each successive integer. But then things get more complicated. To see what’s going on, let’s look at a fragment of the graph:

 ✕
 

 In a sense, everything would be simple if every path in the graph were separate:

 ✕
 
 ✕
 

 But the basic concept of multiway systems is that equivalent states should be merged—so here the “two ways to get 6” (i.e. 1 × 2 × 3 and 1 × 3 × 2) are combined, and what appears in the multiway graph is:

 ✕
 

 For integers, the obvious notion of equivalence is numerical equality. For hypergraphs, it’s isomorphism. But the important point is that equivalence is what makes the multiway graph nontrivial. We can think about what it does as being to entangle paths. Without equivalence, different paths in the multiway system—corresponding to different possible histories—would all be separate. But equivalence entangles them.

 The full ruliad is in effect a representation of all possible computations. And what gives it structure is the equivalences that exist between states generated by different computations. In a sense, there are two forces at work: the “forward” effect of the progress of computation, and the “sideways” effect of equivalences that entangle different computations. (Mathematically this can be thought of as being like decomposing the ruliad structure in terms of fibrations and foliations.)

 Experiencing the Ruliad

 In thinking about finding a fundamental theory of physics, one thing always bothered me. Imagine we successfully identify a rule that describes everything about our universe. Then the obvious next question will be: “Why this rule, and not another?” Well, how about if actually the universe in effect just runs every possible rule? What would this mean? It means that in a sense the “full story” of the universe is just the ruliad.

 But the ruliad contains everything that is computationally possible. So why then do we have the perception that the universe has specific laws, and that definite things happen in it?

 It all has to do with the fact that we are bounded observers, embedded within the ruliad. We never get to see the full ruliad; we just sample tiny parts of it, parsing them according to our particular methods of perception and analysis. And the crucial point is that for coherent observers like us, there are certain robust features that we will inevitably see in the ruliad. And these features turn out to include fundamental laws of our physics, in particular general relativity and quantum mechanics.

 One can imagine an observer very different from us (say some kind of alien intelligence) who would sample different aspects of the ruliad, and deduce different laws. But one of the surprising core discoveries of our Physics Project is that even an observer with quite basic features like us will experience laws of physics that precisely correspond to ones we know.

 An analogy (that’s actually ultimately the result of the same underlying phenomenon) may help to illustrate what’s going on. Consider molecules in a gas. The molecules bounce around in a complicated pattern that depends on their detailed properties. But an observer like us doesn’t trace this whole pattern. Instead we only observe certain “coarse-grained” features. And the point is that these features are largely independent of the detailed properties of the molecules—and robustly correspond to our standard laws of physics, like the Second Law of thermodynamics. But a different kind of observer, sampling and “parsing” the system differently, could in principle identify different features, corresponding to different laws of physics.

 One of the conceptual difficulties in thinking about how we perceive the ruliad is that it’s a story of “self-observation”. Essentially by the very definition of the ruliad, we ourselves are part of it. We never get to “see the whole ruliad from the outside”. We only get to “experience it from the inside”.

 In some ways it’s a bit like our efforts to construct the ruliad. In the end, the ruliad involves infinite rules, infinite initial conditions, and infinite time. But any way of assembling the ruliad from pieces effectively involves making particular choices about how we take those infinite limits. And that’s pretty much like the fact that as entities embedded within the ruliad, we have to make particular choices about how to sample it.

 One of the remarkable aspects of the ruliad is that it’s in some sense the unique ultimately inevitable and necessary formal object. If one sets up some particular computational system or mathematical theory, there are choices to be made. But in the ruliad there are no choices. Because everything is there. And in a sense every aspect of the structure of the ruliad is just something formally necessary. It requires no outside input; it is just a formal consequence of the meaning of terms, like the abstract fact .

 But while the ruliad is unique, the description of it is not. In constructing it, one can imagine using Turing machines or hypergraph rewriting systems or indeed any other kind of computational system. Each will ultimately lead to the same limiting object that is the ruliad, but each of them can be thought of as defining a different coordinate system for describing the ruliad.

 The very generality of the ruliad makes it unsurprising that there is vast diversity in how it can be described. And in a sense each possible description is like a possible way of experiencing the ruliad. In analogy to the (deeply related) situation with spacetime in general relativity, we might say that there are many reference frames in which to experience the ruliad—but it’s always the same ruliad underneath.

 It’s important to understand that the “ruliad from the outside” could seem very different from any “internal” experience of it by an observer like us. As an example, consider a simple finite approximation to the ruliad, built from string substitution systems. In what we did above, we always started from a specific initial condition. But the full ruliad involves starting from all possible initial conditions. (Of course, one could always just say one starts from a “null” initial condition, then have rules of the form null → everything.) So now let’s consider starting from all possible strings, say of length 4. If we use all possible 2-element-to-2-element rules, the finite approximation to the ruliad that we’ll get will be:

 ✕
 

 At some level this is a simple structure, and—as is inevitable for any finite approximation to the ruliad—its transitive closure is just the complete graph:

 ✕
 

 So why doesn’t this mean that the ruliad is somehow trivial? A key part of the story is that we never get to “see the ruliad from the outside” like this. We are always part of it, sampling it according to some procedure, or, somewhat equivalently, thinking about constructing it according to some procedure.

 As an analogy, consider the real numbers. The whole continuum of all real numbers is “from the outside” in many ways a simple construct. But if we imagine actually trying to construct real numbers, say digit by digit, according to some definite procedure, then we’re dealing precisely with what Turing machines were originally invented to model, and the whole structure of computation is involved. (As we’ll see, our way of thinking about “observers like us” is ultimately quite related to “Turing machines with bounded descriptions”.)

 In a sense, at an outside “holistic” level, the ruliad has a certain simple perfection. But as soon as you try to look at “what’s in the ruliad”, you have to parametrize or coordinatize it, and then you’re inevitably exposed to its intricate internal structure.

 Observers Like Us

 One could imagine very different ways in which entities embedded within the ruliad could “experience” it. But what’s most relevant for us is how “observers like us” do it—and how we manage to synthesize from what’s going on in the ruliad our perception of reality, and our view of how our physical universe works.

 Let’s start by talking not about the full ruliad but rather about models in our Physics Project based on specific underlying rules. At the lowest level, we have a “machine-code” description of the universe is which everything just consists of a network of “atoms of space” that is continually being updated—and which we can think of as carrying out a giant, if incoherent, computation, full of computational irreducibility. But the remarkable fact is that somehow we, as observers of this, manage to pick out of it a certain slice that ends up showing coherent, computationally reducible features—that for example seem to reproduce our known laws of physics.

 How does this work? Partly it has to do with features of us as observers, partly with features of how the universe fundamentally works, and partly with an interplay between these.

 The first crucial feature of us as observers is that we’re computationally bounded: the way we “parse” the universe involves doing an amount of computation that’s absolutely tiny compared to all the computation going on in the universe. We sample only a tiny part of what’s “really going on underneath”, and we aggregate many details to get the summary that represents our perception of the universe.

 But why should that summary have any coherence? Basically it’s because we impose coherence through our definition of how observers like us work. One part of the universe will be affected by others. But to consider part of the universe as an “observer”, there has to be a certain coherence to it. The behavior of the universe somehow has to imprint itself on a “medium” that has a certain coherence and consistency.

 Down at the level of atoms of space, everything is always changing. But we can still identify emergent features that have a certain persistence. And it’s out of those features that what we call observers are built. Given only the atoms of space with all their computationally irreducible behavior, it’s not at the outset obvious that any real persistence could exist or be identified. But in our models we expect that there will, for example, be essentially topological features that correspond to particles that persistently maintain their identity.

 And the point is that we can expect to “aggregate up” much further and be able to identify something like a human observer—that we can consider to persistently maintain its identity to the point where phenomena from the universe can be “systematically imprinted” on it.

 Down at the level of atoms of space, there’s a whole multiway graph of possible sequences of updates that can occur—with each path in effect corresponding to a different “thread of time” for the universe. But it’s a crucial fact about us as observers of the universe that we don’t perceive all those branching and merging threads of time. Instead, we imagine that we have a single, definite thread of experience—in which everything is sequentialized in time.

 I’ve argued elsewhere that this sequentialization in time is a defining characteristic of “human-like consciousness”. And it turns out that one of its consequences is that it implies that the particular perception we will have of the universe must be one in which there are laws of physics that correspond to ones we know.

 It’s not obvious, by the way, that if we sequentialize time we can form any consistent view of the universe. But the phenomenon of causal invariance—which seems ultimately to be guaranteed by the fundamental structure of the ruliad—turns out to imply that we can expect a certain generalized relativistic invariance that will inevitably lead to eventual consistency.

 The notion of sequentialization in time is closely related to the idea that—even though our individual atoms of space are continually changing—we can view ourselves as having a coherent existence through time. And there’s a similar phenomenon for space. At the outset, it’s not obvious that there can be “pure motion”, in which something can move in space without “fundamentally changing”. But it turns out again to be consistent to view this as how things work for us: that even though we’re “made of different atoms of space” when we’re in different places, we can still imagine that in some sense we maintain the “same identity”.

 Down at the level of individual atoms of space, there really isn’t any coherent notion of space. And the fact that we form such a notion seems to be intimately connected to what we might think of as details of us. Most important is that we’re in a sense “intermediate in size” in the universe. We’re large relative to the effective distance between atoms of space (which might be m), yet we’re small compared to the size of the whole universe ( m). And the result is that we tend to aggregate the effects of many atoms of space, but still perceive different features of space (say, different gravitational fields) in different parts of the universe.

 The fact that we “naturally form a notion of space” also seems to depend on another issue of scale—that for us the speed of light “seems fast”. It takes our brains perhaps milliseconds to process anything we see. But the point is that this is very long compared to the time it takes light to get to us from objects in our typical local environment. And the result is that we tend to perceive there as being an instantaneous configuration of the world laid out in space, that “separately” changes in time. But if, for example, our brains ran much faster, or we were much bigger than we are, then the speed of light would “seem slower” to us, and we wouldn’t tend to form the notion of an “instantaneous state of space”.

 OK, so what about quantum mechanics? The most fundamental feature of quantum mechanics is that it implies that things in the universe follow not just one but many possible paths of history—which we only get to make certain kinds of measurements on. And in our Physics Project this is something natural, and in fact inevitable. Given any particular configuration of the universe, there are many possible updates that can occur. And when we trace out all the possibilities, we get a multiway system, in which different threads of history continually branch and merge.

 So how do observers like us fit into this? Being part of the universe, we inevitably branch and merge, just like the rest of the universe. So to understand our experience, what we need to ask is how a “branching brain” will perceive a “branching universe”. And the story is remarkably similar to what we discussed above for our experience of space and time: it all has to do with imagining ourselves to have a certain definite persistence.

 In other words, even if when “viewed from the outside” our brain might be following many different paths of history, “from the inside” we can still potentially assume that everything is conflated into a single thread of history. But will this ultimately be a consistent thing to do? Once again, causal invariance implies that it will. There are specific “quantum effects” where we can tell that there are multiple branches of history being followed, but in the end it’ll be consistent to imagine an “objective reality” about “what happened”.

 In our Physics Project we imagine that there are abstract relations between atoms of space, and in the end the pattern of these relations defines the structure of physical space. But what about different branches of history in the multiway graph? Can we think of these as related? The answer is yes. For example, we can say that at a particular time, states on two branches are “adjacent” if they share an immediate ancestor in the multiway graph. And tracing through such connections we can develop a notion of “branchial space”—a kind of space in which states on different branches of history are laid out:

 ✕
 

 One can think of branchial space as being defined by the pattern of entanglements between different branches of history. And in our Physics Project it turns out that the fundamental laws of quantum mechanics seem to just be a direct translation of the fundamental laws of spacetime into branchial space. And just like the speed of light governs the maximum rate at which effects can propagate in physical space, so similarly in our models there’s a “maximum entanglement speed” at which effects can propagate in branchial space.

 So what are we like as observers in branchial space? Just like in physical space, we can presumably be thought of as having a certain size in branchial space. We don’t yet know quite how to measure this size, but it’s surely related to the effective number of quantum degrees of freedom we involve.

 In our everyday experience of things like gases, we’re sufficiently large compared to individual molecules that we normally just perceive the gas as some kind of continuum fluid—and in normal circumstances we can’t even tell that it’s made of molecules. Well, it’s presumably the same kind of thing for physical space—where we’re even much larger compared to the atoms of space, and it’s a major challenge to figure out how to detect their presence. What about for branchial space? As the underlying rules for the system get applied, different branches of history will in effect “move around” in branchial space in complex and computationally irreducible ways. And just like when we observe molecules in a gas, we’ll mostly just observe overall aggregate effects analogous to fluid mechanics—and only in special circumstances will we notice “quantum effects” that reveal the presence of multiple independent threads of history.

 Living in Rulial Space

 We’ve discussed how “observers like us” perceive models of physics of the type that arise in our Physics Project. But how will we perceive the whole ruliad? It begins with a generalization of the story for branchial space. Because now as well as having different branches associated with different updatings according to a particular rule, we have different branches associated with updatings according to different rules.

 And just as we can slice an ordinary multiway system at a particular time to get an instantaneous version of branchial space, so now we can slice a rulial multiway system to get an instantaneous version of what we can call rulial space—a space in which different branches can correspond not just to different histories, but to different rules for history.

 It’s a fairly complicated setup, with “pure branchial space” inevitably being deeply interwoven with rulial space. But as a first approximation, we can think of rulial space as being somewhat separate, and laid out so that different places in it correspond to the results of applying different rules—with nearby places effectively being associated with “nearby” rules.

 And just as we can think of effects propagating in branchial space, so also we can think of them propagating in rulial space. In branchial space we can talk about entanglement cones as the analog of light cones, and a maximum entanglement speed as the analog of the speed of light. In rulial space we can instead talk about “emulation cones”—and a “maximum emulation speed”.

 In our rough approximation of rulial space, each point is in effect associated with a particular rule. So how do we “move” from one point to another? Effectively we have to be emulating the behavior of one rule by another. But why should it even be possible to do this? The answer is the Principle of Computational Equivalence, which states that, in effect, most rules will be equivalent in their computational capabilities—and in particular they will be capable of universal computation, so that any given rule can always “run a program” that will make it emulate any other rule.

 One can think of the program as an interpreter or translator that goes from one rule to another. The Principle of Computational Equivalence tells one that such a translator must essentially always exist. But how fast will the translator run? Effectively that’s what distance in rulial space measures. Because to “do a certain translation”, branches in the rulial multiway system have to reach from one rule to another. But they can only do that at the maximum emulation speed.

 What does the maximum emulation speed measure? Effectively it corresponds to the raw computational processing speed of the universe. We can think of representing computations in some language—say the Wolfram Language. Then the processing speed will be measured in “Wolfram Language tokens processed per second” (“WLT/s”). In some sense, of course, giving a value for this speed is just a way of relating our human units of time (say, seconds) to the “intrinsic unit of time” associated with the computational processing that’s going on in the universe. Or, in other words, it’s a kind of ultimate definition of a second relative to purely formal constructs.

 OK, but how does this relate to us as observers embedded within the ruliad? Well, just as we imagine that—along with the rest of the universe—we’re continually branching and merging in branchial space, so also this will be what happens in rulial space. In other words—like the rest of the universe—our brains aren’t following a particular rule; they’re following branching and merging paths that represent all possible rules.

 But “from inside” we can still potentially imagine that we have a single thread of experience—effectively conflating what happens on all those different branches. And once again we can ask whether doing this will be consistent. And the answer seems to be that, yes, it can be. And what guarantees this is again a kind of “rulial relativity” that’s a consequence of causal invariance. There are many details here, which we’ll address to some extent later. But the broad outline is that causal invariance can be thought of as being associated with paths of history that diverge, eventually converging again. But since the ruliad contains paths corresponding to all possible rules, it’s basically inevitable that it will contain what’s needed to “undo” whatever divergence occurs.

 So what does this mean? Basically it’s saying that even though the universe is in some sense intrinsically “following all possible rules”—as represented by paths in the ruliad—we as observers of the universe can still “take the point of view” that the universe follows a particular rule. Well, actually, it’s not quite a particular rule. Because just as we’re in some sense “quite big” in physical and presumably branchial space, so also we’re potentially “quite big” in rulial space.

 And being extended in rulial space is basically saying that we consider not just one, but a range of possible rules to be what describe the universe. How can it work this way? Well, as observers of the universe, we can try to deduce what the “true rule for the universe” is. But inevitably we have to do this by performing physical experiments, and then using inductive inference to try to figure out what the “rule for the universe is”. But the issue is that as entities embedded within the universe, we can only ever do a finite number of experiments—and with these we’ll never be able to precisely nail down the “true rule”; there’ll always be some uncertainty.

 When we think of ourselves as observers of the universe, there’s in a sense lots of “arbitrariness” in the way we’re set up. For example, we exist at a particular location in physical space—in our particular solar system and so on. Presumably we also exist at a particular location in branchial space, though it’s less clear how to “name” that. And in addition we exist at a particular location in rulial space.

 What determines that location? Essentially it’s determined by how we operate as observers: the particular sensory system we have, and the particular means of description that we’ve developed in our language and in the history of knowledge in our civilization. In principle we could imagine sensing or describing our universe differently. But the way we do it defines the particular place in rulial space at which we find ourselves.

 But what does all this mean in terms of the ruliad? The ruliad is the unique limiting structure formed by following all possible rules in all possible ways. But when we “observe the ruliad” we’re effectively “paying attention to” only particular aspects of it. Some of that “paying attention” we can conveniently describe in terms of our particular “location in the ruliad”. But some is more naturally described by thinking about equivalence classes in the ruliad.

 Given two states that exist in the ruliad, we have to ask whether as observers we want to consider them distinct, or whether we want to conflate them, and consider them “the same”. When we discussed the construction of the ruliad, we already had many versions of this issue. Indeed, whenever we said that two paths in the ruliad “merge”, that’s really just saying that we treat the outcomes as equivalent.

 “Viewed from the outside”, one could imagine that absolutely nothing is equivalent. Two hypergraphs produced in two different ways (and thus, perhaps, with differently labeled nodes) are “from the outside” in some sense different. But “viewed from the inside”, they pretty much have to be viewed as “the same”, in essence because all their effects will be the same. But at some level, even such conflation of differently labeled hypergraphs can be thought of as an “act of the observer”; something that one can only see works that way if one’s “observing it from inside the system”.

 But all the way through our description of the observer, it’s very much the same story: it’s a question of what should be considered equivalent to what. In sequentializing time, we’re effectively saying that “all of space” (or “all of branchial space”, or rulial space) should be considered “equivalent”. There are many subtle issues of equivalence that also arise in the construction of states in the ruliad from underlying tokens, in defining what rules and initial conditions should be considered the same, and in many other places.

 The ruliad is in some sense the most complicated constructible object. But if we as computationally bounded observers are going to perceive things about it, we have to find some way to “cut it down to size”. And we do that by defining equivalence classes, and then paying attention only to those whole classes, not all the details of what’s going on inside them. But a key point is that because we are computationally bounded observers who imagine a certain coherence in their experience, there are strong constraints on what kinds of equivalence classes we can use.

 If we return again to the situation of molecules in a gas, we can say that we form equivalence classes in which we look only coarsely at the positions of molecules, in “buckets” defined by simple, bounded computations—and we don’t look at their finer details, with all the computational irreducibility they involve. And it’s because of this way of looking at the system that we conclude that it follows the Second Law of thermodynamics, exhibits fluid behavior, etc.

 And it’s very much the same story with the ruliad—and with the laws of physics. If we constrain the kind of way that we observe—or “parse”—the ruliad, then it becomes inevitable that the effective laws we’ll see will have certain features, which turns out apparently to be exactly what’s needed to reproduce known laws of physics. The full ruliad is in a sense very wild; but as observers with certain characteristics, we see a much tamer version of it, and in fact what we see is capable of being described in terms of laws that we can largely write just in terms of existing mathematical constructs.

 At the outset, we might have imagined that the ruliad would basically just serve as a kind of dictionary of possible universes—a “universe of all possible universes” in which each possible universe has different laws. But the ruliad is in a sense a much more complicated object. Rather than being a “dictionary” of possible separate universes, it is something that entangles together all possible universes. The Principle of Computational Equivalence implies a certain homogeneity to this entangled structure. But the crucial point is that we don’t “look at this structure from the outside”: we are instead observers embedded within the structure. And what we observe then depends on our characteristics. And it turns out that even very basic features of our consciousness and sensory apparatus in a sense inevitably lead to known laws of physics—and in a sense do so generically, independent of details of just where in rulial space we are, or exactly what slice of the ruliad we take.

 So far we’ve primarily talked about the ruliad in terms of physics and the fundamental structure of our physical universe. But the ruliad is actually something still more general than that. Because ultimately it is just created from the abstract concept of following all possible computational rules. And, yes, we can interpret these rules as representing things going on in our universe. But we can also interpret them as representing things going on in some other, less immediately physically realizable system. Or, for that matter, representing something purely formal, and, say, mathematical.

 This way of talking about the ruliad might make one think that it should be “considered a possible model” for our universe, or for other things. But the bizarre and surprising point is that it is more than that. It’s not just a possible model that might be one of many. Rather, it is the unique ultimate representation of all possible models, entangled together. As we’ve discussed, there are many subtle choices about how we observe the ruliad. But the ultimate ruliad itself is a unique thing, with no choice about what it is.

 As I’ve discussed at more length elsewhere, the ruliad is in a sense a representation of all possible necessary truths—a formal object whose structure is an inevitable consequence of the very notion of formalization. So how does this relate to the idea that the ruliad also at an ultimate level represents our physical universe? What I’ve argued elsewhere is that it means that the ultimate structure of our universe is a formal necessity. In other words, it’s a matter of formal necessity that the universe must exist, and have an ultimate ruliad structure. The fact that we perceive the universe to operate in a certain way—with our standard laws of physics, for example—is then a consequence of the particular way observers like us perceive it, which in turn depends on things like where in rulial space we happen to find ourselves.

 But beyond physics, what else might the ruliad represent? The ruliad is an ultimate example of multicomputation, and of what I’ve characterized as the fourth major paradigm for theoretical science. Often in multicomputation, what’s of interest is multiway systems with specific underlying rules. And already at this level, much of the apparatus that we’ve described in connection with the ruliad also applies—and in a sense “trickles down” to give various universal results.

 But there are also definitely cases of multicomputation (other than physics) where the full notion of applying all possible rules is relevant. The global structures of metamathematics, economics, linguistics and evolutionary biology seem likely to provide examples—and in each case we can expect that at the core is the ruliad, with its unique structure. Of course, this doesn’t mean that what we observe must always be the same, because what we observe depends on our characteristics as an observer—and the characteristics of “being an observer” in metamathematics, for example, are surely different from those for economics or evolutionary biology, or, for that matter, physics.

 For sure, the “sensory apparatus” that we effectively use is different in different cases. But there are certain similar human-based features that still seem to apply. Whatever the domain, we always act as computationally bounded observers. And it seems that we also always have a certain coherence, consistently maintaining our “observerhood” through time or across some form of space. And it seems likely that these “human-induced” characteristics alone are sufficient to yield some very global implications for observed behavior.

 The View from Mathematics

 How should we think about the ruliad mathematically? In many ways, the ruliad is more an object of metamathematics than of mathematics itself. For in talking about the effects of all possible rules, it in a sense transcends individual mathematical theories—to describe a kind of metatheory of all possible theories.

 Given a particular mathematical axiom system, it’s rather easy to see correspondence with a multiway system. There are a variety of ways to set it up, but one approach is to think of states in the multiway system as being expressions in the language used for the axiom system, and then to think of rules in the multiway system as applying transformations on these expressions that implement axioms in the axiom system.

 For example, with the (Abelian semigroup) axioms

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 here’s a multiway system generated from the expression by applying the (two-way) transformations defined by the axioms in all possible ways to each expression:

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 But now from this graph we can read off the “theorem”:

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 A proof of this theorem

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 is just a path in the multiway graph:

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 A significantly less direct but still perfectly valid proof would correspond to the 13-step path:

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 It’s a slightly technical point, but perhaps worth mentioning that there are alternative ways to set up the correspondence between axiomatic mathematical systems and multiway systems. One such way is to take the states in the multiway system to be not expressions (like ) but whole propositions (like ). Axioms then show up as states, and the rules for the multiway system are “rules of inference” or “rules of entailment” that define how, say, pairs of propositions “entail” (or “imply”) other ones. (And, yes, this requires a generalized multiway system with not just 1 state → many states, but for example 2 states → 1 state.) Typical automated theorem provers (like FindEquationalProof) operate in this kind of setup, attempting to find paths that lead from initial propositions and axioms to some final state that corresponds to an “obviously true” proposition, like .

 But whatever the detailed setup, the basic picture is that an axiomatic mathematical system has an associated multiway graph, in which paths correspond to proofs. Given the rules for the multiway system, there is in general no way to guarantee that the path (if it exists) corresponding to the proof of some particular result will be of bounded length, leading to the possibility of undecidability. But even when a path exists, it may require an irreducibly large amount of computation to find it. Still, finding such paths is what automated theorem provers do. For example, we know (as I discovered in 2000) that ((b · c) · a) · (b · ((b · a) · b)) = a is the minimal axiom system for Boolean algebra, because FindEquationalProof finds a path that proves it.

 But this path—and the corresponding proof—is a very “non-human” construct (and, for example, in 21 years essentially no progress has been made in finding a “human-understandable narrative” for it). And we can make an analogy here to the situation in physics. The individual rule applications in the multiway graph (or the proof) are like individual updating events applied to the atoms of space—and they show all kinds of complexity and computational irreducibility. But in physics, human observers work at a higher level. And the same, one suspects, is true in mathematics.

 Rather than looking at every detail of the multiway graph, human “mathematical observers” (i.e. pure mathematicians) in effect define all sorts of equivalences that conflate together different parts of the graph. If the individual updates in the multiway graph are like molecular dynamics, human pure mathematics seems to operate much more at the “fluid dynamics level”, concentrating on “broad mathematical constructs”, not the “machine code” of specific low-level axiomatic representations. (Of course, there are some situations, for example related to undecidability, where the “molecular dynamics” effectively “breaks through”.)

 We’ve outlined above (and discussed at length elsewhere) how physical observers like us “parse” the low-level structure of the physical universe (and the ruliad). How might mathematical observers do it? A large part has to do with the identification of equivalences. And the key idea is that things which are considered equivalent should be assumed to be “the same”, and therefore “conflated for mathematical purposes”.

 The most elementary example of something like this is the statement (already present in Euclid) that if and , then . The extensionality axiom of set theory is a more sophisticated example. And the univalence axiom of homotopy type theory is perhaps the most sophisticated current version.

 There’s a very operational version of this that appears in automated theorem proving. Imagine that you’ve proved that and . Then (by the assumed properties of equality) it follows that . One way we could use this result is just to merge the nodes for and . But a “bigger” thing we can do is to add the “completion” as a general rule for generating the multiway system.

 Consider, for example, the string substitution multiway system A ↔ AB:

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 But notice here that both ABA ↔ ABBA and ABA ↔ ABAB. So now add the “completion” ABBA ↔ ABAB. Here’s the resulting multiway graph:

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 Adding the completion has generated lots of new “direct equivalences”. But we can also think of it as having done something else: not only has it defined equivalences between states; it’s also defined equivalences between paths—or in effect between proofs. (Or, put another way, it’s implementing homotopic equivalence between proofs. By the way, it’s an important feature of human mathematics that progress is typically measured in theorems proved; different proofs of the same theorem are normally implicitly considered equivalent in terms of the progress of mathematics.)

 In a category theory interpretation, transformations between states in the original multiway graph are like ordinary morphisms (1-morphisms). But when we’re making transformations between “proof paths”, this is like 2-morphisms. And just as we can add transformations between proofs, we can also add transformations between proofs-between-proofs, and so on. The result is that we can build up a whole hierarchy of higher and higher categories, eventually ending with an ∞-category.

 But now we can begin to see the connection with the ruliad. The ruliad is in a sense the result of taking the limit of adding more and more possible rules. Above we did this quite explicitly in terms of the original underlying rules, for example by enumerating possible strings or possible integer multipliers.

 But we can view successive completions as doing something very similar. Yes, the rules are enumerated in a different order. But in the end there’ll still be an infinite number of distinct rules being used.

 Of course there are many mathematical details missing here. But in broad outline, it seems that one can think of the approach to the ruliad as some kind of limit of successively higher categories. But this limit is something that’s been studied (albeit in the upper reaches of pure mathematics), and it’s an object called the ∞-groupoid. (It’s a groupoid because when all the rules are included things inevitably “go both ways”). So, OK, is the ruliad “just” the ∞-groupoid? Not quite. Because there are more rules and more initial conditions in the ruliad, even beyond those added by completions. And in the end the ruliad actually seems to be the ∞-category of ∞-groupoids, or what’s called the (∞,1)-category.

 But knowing that the ruliad can be thought of as composed of ∞-groupoids means that we can apply mathematical ideas about the ∞-groupoid to the ruliad.

 Probably the most important is Grothendieck’s hypothesis, which asserts that the ∞-groupoid inevitably has a topological and (with a few other conditions) ultimately geometric structure. In other words, even though one might have imagined that one constructed the ∞-groupoid from “pure logic” (or from pure formal axiomatic structures), the assertion is that the limiting object one obtains inevitably exhibits some kind of geometrical or “spatial” structure.

 Viewed in terms of the ruliad—and our explicit finite examples of it—this might not seem surprising. And indeed in our Physics Project, the whole concept of the emergence of space from large-scale hypergraphs is closely related. But here from Grothendieck’s hypothesis we’re basically seeing a general claim that the ruliad must have “inevitable geometry”—and we can then view things like the emergence of space in our Physics Project as a kind of “trickle down” from results about the ruliad. (And in general, a big “application” of geometrical structure is the possibility of “pure motion”.)

 What does all this mean about the ruliad and mathematics? In a sense the ruliad represents all possible mathematics—the application of all possible rules, corresponding to all possible axiom systems. And from this “ultimate metamathematics”, human “mathematical observers” are sampling pieces that correspond to the pure mathematics they consider of interest.

 Perhaps these will align with particular axiom systems of the kind automated theorem provers (or proof assistants) use. But things may be “sloppier” than that, with human mathematical observers effectively being extended in rulial space—and capable of making “fluid-dynamics-level” conclusions, even if not “molecular-dynamics-level” ones.

 But a key (and in some ways very surprising) point is that the ruliad can be viewed as the basis of both physics and mathematics. In some sense, physics and mathematics are at their core the same thing. They only “appear different” to us because the way we “observe” them is different.

 I plan to discuss the implications for mathematics at greater length elsewhere. But suffice it to say here that the existence of a common underlying core—namely the ruliad—for both physics and mathematics immediately allows one to start importing powerful results from physics into mathematics, and vice versa. It also allows one, as I have done elsewhere, to start comparing the existence of the universe with the (Platonic-style) concept of the fundamental existence of mathematics.

 The View from Computation Theory

 The ruliad can be thought of as an encapsulation of doing all possible computations in all possible ways. What we might think of as a “single computation” might consist of repeatedly applying the rules for a Turing machine to “deterministically” generate a sequence of computational steps:

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 But one can also consider a “multicomputational” system, in which rules can generate multiple states, and the whole evolution of the system can be represented by a multiway graph:

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 In traditional computation theory, one talks about rules like these as “nondeterministic”, because they can have multiple outcomes—though one usually imagines that the final answer one wants from a computation can be found as the result of some particular path. (In what I now call the multicomputational paradigm—that I believe is important for modeling in physics and other places—one instead considers the complete multiway graph of entangled possible histories.)

 In constructing the ruliad, one is in a sense going to a more extreme version of multicomputation, in which one uses not just a particular rule with multiple outcomes, but all possible rules. In effect, the concept is to use “maximal nondeterminism”, and at each step to independently “pick whatever rule one wants”, tracing out a rulial multiway system that includes all the different possible paths this generates.

 For the kind of Turing machines illustrated above, the rulial multiway graph one gets after one step is:

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 After 2 steps the result is:

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 The full ruliad is then some kind of infinite limit of this process. But as before, there’s lots of subtlety in how this limit is taken. But we can at least characterize some ways of approaching it using ideas from computational complexity theory. Increasing the number of steps of evolution is like increasing the time complexity one allows. Increasing the “size of states” (e.g. the width of nonzero Turing machine tape) that one includes is like increasing the space complexity one allows. And increasing the complexity of the rule (as measured in the number of bits needed to specify it) is like increasing the algorithmic complexity one allows. The ruliad is what is obtained by taking all these computational resource measures to infinity.

 And a critical claim is that regardless of how this is done, the final ruliad construct one gets will always—at least in some sense—be the same. There will be many ways to coordinatize it, or to sample it, but the claim is that it’s always the same object that one’s dealing with. And ultimately the reason for this is the Principle of Computational Equivalence. Because it implies that whatever “computational parametrization” or “computational description language” one uses for the ruliad, one will almost always get something that can be viewed as “computationally equivalent”.

 We’ve talked about building up the ruliad using Turing machines. But what about other models of computation—like cellular automata or register machines or lambda calculus? As soon as there’s computation universality we know that we’ll get results that are at least in principle equivalent, because in a sense there’s only a “finite translation cost” associated with setting up an interpreter from one model of computation to another. Or, put another way, we can always emulate the application of the rule for one system by just a finite number of rule applications for the other system.

 But from computation universality alone we have no guarantee that there won’t be “extreme deformations” introduced by this deformation. What the Principle of Computational Equivalence says, however, is that almost always the deformations won’t have to be extreme. And indeed we can expect that particularly when multiple rules are involved, there’ll be rapid convergence almost always to a kind of “uniform equivalence” that ensures that the final structure of the ruliad is always the same.

 But the Principle of Computational Equivalence appears to say still more about the ruliad: it says that not only will the ruliad be the same independent of the “computational basis” used to construct it, but also that there’ll be a certain uniformity across the ruliad. Different “regions of the ruliad” might involve different specific rules or different patterns of their application. But the Principle of Computational Equivalence implies that almost always the computations that happen will be equivalent, so that—at least at a certain scale—the structure associated with them will also be equivalent.

 Knowing that the ruliad contains so many different computations, one might imagine that it would show no particular uniformity or homogeneity. But the Principle of Computational Equivalence seems to imply that it necessarily does, and moreover that there must be a certain coherence to its structure—that one can interpret (in the style of Grothendieck’s hypothesis) as an inevitable emergent geometry.

 An individual computation corresponds to a path in the ruliad, going from its “input state” to its “output state”. In an ordinary deterministic computation, the path is restricted to always use the same rule at each step. In a nondeterministic computation, there can be different rules at different steps. But now we can formulate things like the P vs. NP problem essentially in terms of the geometry of the ruliad.

 Here’s a picture of the same finite Turing-machine-based approximation to the ruliad as above—but now with the paths that correspond to deterministic Turing machine computations marked in red:

  ✕
 

 The P vs. NP problem basically asks roughly whether the deterministic computations (shown here in red) will eventually “fill the ruliad”, or whether the general nondeterministic computations that are part of the ruliad will always “reach further”. Once again, there are many complicated and subtle issues here. But it’s interesting to see how something like the P vs. NP problem might play out in the ruliad.

 In physics (and mathematics), we as human observers tend to sample the ruliad in a coarse-grained way, “noticing” only certain aspects of it. So is there an analog of this in computation theory—perhaps associated with certain characteristics of the “computation-theoretic observer”? There’s a potential answer, rather similar to what we’ve already seen in both physics and mathematics.

 The basic point is that in computation theory we tend to study classes of computations (say P or NP) rather than individual computations. And in doing this we are in a sense always conflating many different possible inputs and possible outputs—which we assume we do in a computationally bounded way (e.g. through polynomial-time transformations, etc.)

 Another thing is that we tend to focus more on the “experience of the end user” than the detailed activities of the “programmer”. In other words, we’re concerned more with what computational results are obtained, with what computational resources, rather than on the details of the program constructed to achieve this. Or, put another way, we tend to think about computation in terms of things like the successive evaluation of functions—and we conflate the different paths by which this is achieved.

 Most likely this means that there are “effective laws” that can be derived in this computational view of the ruliad, analogous to laws of physics like general relativity. So what might some other analogies be?

 A computation, as we’ve mentioned, corresponds to a path in the ruliad. And whenever there’s a possibility for an infinite path in the ruliad, this is a sign of undecidability: that there may be no finite way to determine whether a computation can reach a particular result. But what about cases when many paths converge to a point at which no further rules apply, or effectively “time stops”? This is the analog of a spacelike singularity—or a black hole—in the ruliad. And in terms of computation theory, it corresponds to something decidable: every computation one does will get to a result in finite time.

 One can start asking questions like: What is the density of black holes in rulial space? If we construct the ruliad using Turing machines, this is basically analogous to asking “What’s the density of halting Turing machines (+initial conditions) in rulial space?” And this is essentially given by Chaitin’s Ω.

 But so is there some number Ω that we can just compute for the ruliad? Well, actually, no. Because the undecidability of the halting problem makes Ω noncomputable. One can get approximations to it, but—in the language of the ruliad—those will correspond to using particular samplings or particular reference frames. Or in other words, even the perceived density of “decidability black holes” in the ruliad depends on features of the observer.

 What’s beyond the Ruliad?

 In our Physics Project we usually talk of the universe “evolving through time” (albeit with many entangled threads of history). But if the ruliad and its structure is a matter of formal necessity, doesn’t that mean that the whole ruliad effectively “already exists”—“outside of time”? Well, in some sense it does. But ultimately that would only be relevant to us if we could “look at the ruliad from the outside”.

 And as observers like us within the ruliad, we necessarily have a different perception. Because our consciousness—with its computational boundedness—only gets to sample a certain sequence of pieces of the ruliad. If it were not for computational irreducibility, we might get to “jump around” in time. But computational irreducibility, together with our own computational boundedness, implies that our perception must necessarily just experience the passage of time through an irreducible process of computation.

 In other words, while in some sense the ruliad may all “already be there” when viewed from the outside, our own perception of it “from the inside” is necessarily a progressive one, that effectively corresponds to the passage of time.

 Could we experience the ruliad differently, even while being computationally bounded? If we think of the ruliad as a graph, then our usual “sequence of configurations of space at successive times” way of experiencing the ruliad is like a breadth-first traversal. But could we for example instead do a depth-first traversal, exploring all time before investigating different parts of space? (And, yes, something like this can happen in general relativity near an event horizon, or in connection with timelike singularities.)

 Later, we’ll discuss different ways to perceive the ruliad and the universe. But it seems to be a feature of anything we might call a coherent observer that there needs to be some form of progression in the perception. And so while we might not call it the passage of time, there’ll still be some way in which our exploration of the ruliad has a computationally irreducible process underneath.

 A very important claim about the ruliad is that it’s unique. Yes, it can be coordinatized and sampled in different ways. But ultimately there’s only one ruliad. And we can trace the argument for this to the Principle of Computational Equivalence. In essence there’s only one ruliad because the Principle of Computational Equivalence says that almost all rules lead to computations that are equivalent. In other words, the Principle of Computational Equivalence tells us that there’s only one ultimate equivalence class for computations.

 But what if we just imagine a “hypercomputation” not in that class? For example, imagine a hypercomputation (analogous, for example, to an oracle for a Turing machine) that in a finite number of steps will give us the result from an infinite number of steps of a computationally irreducible process. Such a hypercomputation isn’t part of our usual ruliad. But we could still formally imagine a hyperruliad that includes it—and indeed we could imagine a whole infinite hierarchy of successively larger and more powerful hyperruliads.

 But it’s a fundamental claim that we’re making—that can be thought of as a matter of natural science—that in our universe only computation can occur, not hypercomputation.

 At a purely formal level, there’s nothing wrong with hyperruliads. They exist as a matter of formal necessity just like the ordinary ruliad does. But the key point is that an observer embedded within the ruliad can never perceive a hyperruliad. As a matter of formal necessity there is, in a sense, a permanent event horizon that prevents anything from any hyperruliad from affecting anything in the ordinary ruliad.

 So now we can be a bit more precise about our statement that “hypercomputation doesn’t happen in our universe”. Really we should say that we assert that we as observers operate purely computationally and not hypercomputationally. And this means that we are embedded within the ordinary ruliad, and not the hyperruliad.

 Yes, we could imagine some other entity that’s embedded within the hyperruliad, and perceives what it considers to be the universe to operate hypercomputationally. But in a statement that’s in a sense more “about us” than “about the universe”, we assert that that can’t be us, and that we in a sense live purely within the ruliad—which means that for us the Principle of Computational Equivalence holds, and we perceive only computation, not hypercomputation.

 Communicating across Rulial Space

 What observers can there be embedded in the ruliad, and how should we characterize them? In physical spacetime we’re used to characterizing observers by their locations in physical space and by things like the spacetime reference frames they construct. And it’s very much the same for observers in the ruliad: we can characterize them by where they are in rulial space, and what rulial reference frames they use.

 The Principle of Computational Equivalence tells us that it’s almost always possible to “encode” one “model of how the ruliad works” in any other model—effectively just by setting up a program that emulates the rules for one model using the rules for the other model. But we can think of these different models as being associated with different possible observers in the ruliad.

 In other words, we can say that observers “at different places in rulial space” (or “using different rulial reference frames”) are using different description languages for what’s happening in the ruliad. And when an observer “moves” in rulial space, they’re effectively doing a translation from one description language to another. (And, yes, there’s a maximum rate of motion ρ in rulial space—which is the rulial analog of the speed of light—and which is effectively determined by the fundamental processing speed of the universe.)

 So far this might all seem quite abstract. But there are immediate, everyday examples that effectively correspond to being at different places in rulial space. A simple concrete one is computers with different instruction sets. Another one is different brains with different consciousnesses.

 We can think of a single human consciousness as having a certain thread of experience of the universe. Part of that experience is determined by the physical location of the consciousness and by the sensory apparatus with which it samples the world. But part is determined by the “internal description language” that it uses. And inevitably this internal description language depends both on the detailed physiology of the brain in which it is implemented, and on the past history of experiences that have “defined its way of looking at the world”. In the analogy of artificial neural networks, different networks will tend to have different “internal representations” because this depends not only on the network architecture, but also on the particular training data that the network has “experienced”.

 Why can’t one human consciousness “get inside” another? It’s not just a matter of separation in physical space. It’s also that the different consciousnesses—in particular by virtue of their different histories—are inevitably at different locations in rulial space. In principle they could be brought together; but this would require not just motion in physical space, but also motion in rulial space.

 But why then do different consciousnesses seem to have compatible views about “what happens in the universe”? Essentially this can be seen as a consequence of rulial relativity—which in turn depends on the inevitable causal invariance of the ruliad, which follows from the Principle of Computational Equivalence. There are certainly many issues to be worked out, but basically what seems to be going on is that because of causal invariance, different rulial reference frames will ultimately yield the same rulial multiway causal graphs, and therefore the same “fundamental description of reality”.

 We’ve talked about different consciousnesses. But what about just “different ways of thinking”? Well, it’s definitely more than an analogy to say that different ways of thinking correspond to different positions in rulial space. If there’s lots of common history then there’ll be common ancestry in the rulial multiway graph and one will necessarily end up close in rulial space. But without common history, one can end up with different description languages—or different ways of thinking—that are not nearby in rulial space.

 In physical space we expect to effectively use momentum to move our location. And it’s potentially a bizarrely similar story in rulial space. In our models of fundamental physics, energy and momentum are essentially related to the density of activity (i.e. elementary updating events) in physical space. And we can similarly define a rulial analog of energy and momentum in terms of activity in rulial space. But it’s exactly this activity that provides connections between different parts of rulial space, or in effect “enables motion” in rulial space.

 In other words, if you want to move in rulial space, you can do it by putting in the appropriate computational work to change your conceptual point of view (or, essentially equivalently, your language for describing things). So what about curvature (or the analog of gravity) in rulial space—say generated through an analog of Einstein’s equations from density of activity in rulial space? Presumably this relates to the difficulty—or time it takes—to get from one place in rulial space, and one way of thinking, to another. And conceivably things like “paradigm shifts” between different ways of thinking might be associated with features of rulial space like event horizons.

 But let’s say you’re at one place in rulial space, and you want to get to another—or at least “send a signal” there. A typical microscopic change at one point in rulial space will tend to just “spread out in all directions” and “decay quickly”. But if you want to “coherently communicate”, you need some kind of structure that will persist as it propagates through rulial space. And by analogy with the case of physical space, what this presumably means is that you effectively need a “rulial particle”.

 In terms of the ruliad, a rulial particle would presumably be some kind of “topological obstruction” or “topologically stable structure” that is at any moment effectively localized in rulial space and maintains its identity as it propagates across rulial space. But what might a rulial particle be in more everyday terms?

 Potentially it’s like what we’d normally consider a concept—or something to which we might assign a word in human language. If we have ways of thinking—or consciousnesses—whose details are different, the issue is what will be robust enough to be able to be transported between them. And what everyday experience seems to suggest is that the answer is concepts. Even though one might have a slightly different way of thinking, what one calls “a fish” (or essentially, the concept of a fish) is something that can still robustly be communicated.

 It’s interesting to notice that for an observer like us, there seem to be only a finite set of types of “elementary particles” that exist in physical space. And perhaps that’s not unrelated to the fact that observers like us also seem to imagine that there are in some sense only a finite number of “basic concepts” (associated, say, with distinct words in human languages). There’s lots more detail that exists in rulial space—or in the ruliad—but for observers like us, with our type of way of sampling the ruliad, these might be core coherent structures that we perceive.

 So Is There a Fundamental Theory of Physics?

 The concept of the ruliad arose from our efforts to find a fundamental theory of physics. But now that we know about the ruliad, what does it tell us about a fundamental theory?

 At the outset, we might have imagined that the end point of our project would be the identification of some particular rule of which we could say “This is the rule for the universe”. But of course then we’d be faced with the question: “Why that rule, and not another?” And perhaps we would imagine just having to say “That’s something that you have to go beyond science to answer”.

 But the ruliad implies a quite different—and in my view ultimately much more satisfying—picture. The ruliad itself is a construct of abstract necessity—that in a sense represents the entangled behavior of all possible rules for the universe. But instead of imagining that some particular rule out of all these possibilities is “picked from outside” as “the choice for our universe”, what we suppose is that—as observers embedded within the ruliad—we’re the ones who are implicitly picking the rule by virtue of how we sample and perceive the ruliad.

 At first this might seem like it’s a wimp out. We want to know how our universe works. Yet we seem to be saying “we just pick whatever rule we feel like”. But that’s not really the story at all. Because in fact observers that are even vaguely like us are in effect deeply constrained in what rules they can attribute to the universe. There’s still some freedom, but a fundamental result is that for observers like us it seems to be basically inevitable that any rule we can pick will on a large scale reproduce the central known general laws of physics, in particular general relativity and quantum mechanics.

 In other words, for observers generally like us it’s a matter of abstract necessity that we must observe general laws of physics that are the ones we know. But what about more specific things, like the particular spectrum of elementary particles, or the particular distribution of matter in the universe? It’s not clear how far “the general” goes—in other words, what is a matter of abstract necessity purely from the structure of the ruliad and general features of observers like us.

 But inevitably at some point we will run out of “the general”. And then we’ll be down to specifics. So where do those specifics enter? Ultimately they must be determined by the details of how we sample the ruliad. And a prominent part of that is simply: Where in the ruliad are we? We can ask that about our location in physical space. And we can also ask it about our location in rulial space.

 What does all this mean? At some level it’s saying that the way we are as observers is what makes us attribute certain rules to our universe. The ruliad is in a sense the only thing that fundamentally exists—and in fact its existence is a matter of abstract necessity. And our universe as we experience it is some “slice of the ruliad”, with what slice it is being determined by what we’re like as observers.

 Let’s look at the logical structure of what we’re saying. First, we’re describing the ruliad, which at the outset doesn’t have anything specifically to do with physics: it’s just a formal construct whose structure is a matter of abstract necessity, and which relates as much to mathematics as it does to physics. But what “puts the physics in” is that we in effect “live in the ruliad”, and our perception of everything is based on “experiencing the ruliad”. But that experience—and the effective laws of physics it entails—inevitably depends on “where we are in the ruliad” and how we’re able to sample it.

 And this is where our pieces of “falsifiable natural science” come in. The first “assertion of natural science” that we make is that we are embedded only within the ordinary ruliad, and not a hyperruliad—or in other words that our experience encompasses only computation, and not hypercomputation.

 This is closely related to a second assertion, which may in fact be considered to subsume this: that we are computationally bounded observers, or, in other words, that our processes of perception involve bounded computation. Relative to the whole ruliad—and all the computation it entails—we’re asserting that we as observers occupy only a tiny part.

 There’s one more assertion as well, again related to computational boundedness: that we as observers have a certain coherence or persistence. In general the ruliad contains all sorts of wild and computationally irreducible behavior. But what we’re asserting is that that part of the ruliad that is associated with us as observers has a certain simplicity or computational reducibility: and that as we evolve through time or move in space, we somehow maintain our identity.

 These assertions seem very general, and in some ways almost self-evident—at least as they apply to us. But the important and surprising discovery is that they alone seem to lead us inexorably to crucial features of physics as we know it.

 Where does this physics “come from”? It comes partly from the formal structure of the ruliad, and formal features of the multicomputational processes it involves. And it comes partly from the nature of us as observers. So if we ask “Why is the physics of our universe the way it is?”, an important part of the answer is “Because we observe the universe the way we do”.

 One might imagine that in some sense physics would give us no choice about how we observe the universe. But that’s not the case. Because in the end our “observation” of the universe is about the “abstract conceptual model” we build up for the universe. And, yes, that’s certainly informed by the particular sensory apparatus we have, and so on. But it’s something we can certainly imagine being different.

 We can think of ourselves as using some particular description language for the universe. The structure of that language is constrained by the assertions we gave above. But within such a description language, the laws of physics necessarily work out the way they do. But if we chose a different description language, we’d end up with different laws of physics.

 Much of our perception of the universe is based on our raw biological structure—the way our sensory organs (like our eyes) work, as well as the way our brains integrate the inputs we get. But that’s not all there is to it. There’s also a certain base of knowledge in our civilization that informs how we parse our “raw perception”—and in effect what description language we use. Once we have the idea of periodic behavior, say, we can use it to describe things that we’d previously have to talk about in a less “economical” way.

 But what if our knowledge changed? Or we had different sensory capabilities? Or we used technology to integrate our sensory input in different ways? Then we’d be able to perceive and describe the universe in different ways.

 One’s first impression might be that the ruliad effectively contains many possible “parallel universes”, and that we have selected ourselves into one of these, perhaps as a result of our particular characteristics. But in fact the ruliad isn’t about “parallel universes”, it’s about universes that are entangled at the finest possible level. And an important consequence of this is that it means we’re not “stuck in a particular parallel universe”. Instead, we can expect that by somehow “changing our point of view”, we can effectively find ourselves in a “different universe”.

 Put another way, a given description of the universe is roughly represented by being at a certain location in rulial space. But it’s possible to move in rulial space—and end up with a different description, and different effective laws for the universe.

 But how difficult is motion in rulial space? It could be that some impressive future technology would allow us to “move far enough” to end up with significantly different laws of physics. But it seems more likely that we’d be able to move only comparatively little—and never be able to “escape the box” of things like computational boundedness, and coherence of the observer.

 Of course, even changing a little might lead us to different detailed laws of physics—say attributing a different mass to the electron, or a different value of the electromagnetic coupling constant α. But actually, even in traditional physics, this is already something that happens. When viewed at different energy scales—or in a sense with different technology—these quantities have different effective values (as characterized by the renormalization group).

 At first it might seem a little strange to say that as our knowledge or technology change, the laws of physics change. But the whole point is that it’s really our perceived laws of physics. At the level of the raw ruliad there aren’t definite laws of physics. It’s only when we “sample our slice” of the ruliad that we perceive definite laws.

 What does all this mean operationally for the search for a fundamental theory of physics? At some level we could just point to the ruliad and declare victory. But this certainly wouldn’t give us specific predictions about the particulars of our perceived universe. To get that we have to go further—and we have to be able to say something about what “slice of the ruliad” we’re dealing with. But the good news is that we don’t seem to have to make many assumptions about ourselves as observers to be able to identify many physical laws that observers like us should perceive.

 So can we ever expect to nail down a single, specific rule for the universe, say one a particular observer would attribute to it? Given our characteristics as observers, the answer is undoubtedly no. We’re simply not that small in rulial space. But we’re not that big, either. And, importantly, we’re small enough that we can expect to “do science” and consider the universe to “behave in definite ways”. But just as in physical space we’re vastly larger than the scale associated with the atoms of space, so similarly we’re also undoubtedly vastly larger in rulial space than the individual components of the ruliad—so we can’t expect our experience to all be “concentrated in one thread” of the ruliad, following one particular rule.

 As we discussed above, by doing experiments we can use scientific inference to attempt to localize ourselves in rulial space. But we won’t be able to do enough to say “from our point of view, the universe is operating according to this one specific rule, and not another”. Instead, there’ll be a whole collection of rules that are “good enough”, in the sense that they’ll be sufficient to predict the results of experiments we can realistically do.

 People have often imagined that, try as we might, we’d never be able to “get to the bottom of physics” and find a specific rule for our universe. And in a sense our inability to localize ourselves in rulial space supports this intuition. But what our Physics Project seems to rather dramatically suggest is that we can “get close enough” in rulial space to have vast predictive power about how our universe must work, or at least how observers like us must perceive it to work.

 Alien Views of the Ruliad

 We’ve discussed how “observers like us” will necessarily “parse the ruliad” in ways that make us perceive the universe to follow the laws of physics as we know them. But how different could things get? We have a definite sense of what constitutes a “reasonable observer” based on our 21st-century human experience—and in particular our biology, our technology and our ways of thinking.

 But what other kinds of observers can we imagine? What about, for example, animals other than humans—in particular say ones whose sensory experience emphasizes olfaction or echolocation or fluid motion? We can think of such animals as operating in a different rulial reference frame or at a different place in rulial space. But how far away in rulial space will they be? How similar or not will their “world views” (and perceived laws of physics) be to ours? It’s hard to know. Presumably our basic assertions about computational boundedness and coherence still apply. But just how the specifics of something like sequentialization in time play out, say, for an ant colony, seems quite unclear.

 Maybe one day we’ll be able to systematically “think like other animals”. But as of now we haven’t been able to “travel that far” in rulial space. We’ve quite thoroughly explored physical space, say on the surface of our planet, but we haven’t explored very far at all in rulial space. We don’t have a way to translate our thinking into some kind of “thinking differently”—and we don’t, for example, have a common language to get there.

 There’s often an assumption (a kind of “human exceptionalism”) that if it wasn’t for details of the human experience—like brains and words—then we’d necessarily be dealing with something fundamentally simpler, that could not, for example, show features that we might identify as intelligence. But the Principle of Computational Equivalence tells us this isn’t correct. Because it says that there’s a certain maximal computational sophistication that’s achieved not just by us humans but also by a vast range of other systems. The restrictions of what we’ve chosen to study (in science and elsewhere) have often made us miss this, but in fact computational sophistication—and the direct generalization of our notion of intelligence that’s associated with it—seems quite ubiquitous across many different kinds of systems.

 So can those other kinds of systems act as “observers like us”? To do so, they need not just computational sophistication, but also a certain alignment with the features we have that lead to our coherent thread of “conscious experience”. And even given that, to actually “connect with” such systems, we need to be able to reach far enough in rulial space to sufficiently make a translation.

 Imagine the weather (sometimes said to “have a mind of its own”). It’s got plenty of computational sophistication. But is there any sense in which it sequentializes time like we do? Or can one only think of all those different parts of our atmosphere “running in their own time”? To know things like this, we effectively have to have a way to “translate” from the operation of the weather to our (current) way of thinking.

 And in some sense we can consider the whole enterprise of natural science as being an effort to find a method of translation—or a common language—between nature and our way of thinking.

 We as observers in effect trace out particular trajectories in rulial space; the challenge of natural science is to “reach out” in rulial space and “pull in” more of the ruliad; to be able to define a way to translate more parts of the ruliad to our processes of thinking. Every time we do an experiment, we can think of this as representing a moment of “connection” or “communication” between us and some aspect of nature. The experiment in effect defines a small piece of “common history” between us and nature—which helps “knit together” the parts of rulial space associated with us and with nature.

 One of the great mysteries of science has been why—in the vastness of physical space—we’ve never detected something we identify as “alien intelligence”, or an “alien civilization”. We might have thought that it was because we humans have either achieved a unique pinnacle of intelligence or computational ability—or have fundamentally not gotten far enough. But the Principle of Computational Equivalence explodes the idea of this kind of cosmic computational pecking order.

 So what could actually be going on? Thinking in terms of the ruliad suggests an answer. Our radio telescopes might be able to detect signals from far away in physical space. But our putative aliens might not only live far away in physical space, but also in rulial space.

 Put another way, the “alien civilization” might be sampling aspects of the ruliad—and in effect the universe—that are utterly different from those we’re used to. That different sampling might be happening right down at the level of atoms of space, or it might be that the rulial distance from us to the aliens is small enough that there’s enough “shared description language” that the alien civilization might rise to the level of seeming like some kind of “noise” relative to our view of “what’s important in the universe”.

 We might wonder how far apart what we could consider “alien civilizations” would be in physical space. But what we now realize is that we also have to consider how far apart they might be in rulial space. And just like in exploring physical space we can imagine building better spacecraft or better telescopes, so also we can imagine building better ways to reach across rulial space.

 We’re so used to physical space that it seems to us very concrete to reach across it. Of course, in our Physics Project, things like motion in physical space end up—like everything else—being pure computational processes. And from this point of view, reaching across rulial space is ultimately no more abstract—even though today we would describe it in terms of “doing (abstract) computations” rather than “moving in space”.

 Relative to our own physical size, the universe already seems like a vast place. But the full ruliad is even incredibly more vast. And we are likely much tinier in rulial space relative to the whole universe than we are in physical space. From the Principle of Computational Equivalence we can expect that there’s ultimately no lack of raw computational sophistication out there—but thinking in terms of the ruliad, the issue is whether what’s going on is close enough to us in rulial space that we can successfully see it as an “alien civilization”.

 One test of rulial distance might be to ask whether our putative aliens perceive the same laws of physics for the universe that we do. We know that at least the general forms of those laws depend only on what seem to us rather loose conditions. But to get good alignment presumably requires at the very least that we and the aliens are somehow “comparable in size” not only in physical space (and branchial space), but also in rulial space.

 It’s humbling how difficult it is to imagine the universe from the point of view of an alien at a different place in rulial space. But for example if the alien is big compared to us in rulial space, we can say that they’ll inevitably have a version of science that seems to us much “vaguer” than ours. Because if they maintain a coherent thread of experience, they’ll have to conflate more distant paths in rulial space, on which the universe will do things that are “more different” than what we’re used to. (And, yes, there should be rulial analogs of quantum phenomena, associated for example with conflated paths that diverge far in rulial space.)

 What would it mean operationally for there to be an alien civilization perhaps nearby in physical space but at a distance in rulial space? Basically the alien civilization will be “operating” in features of the universe that our parsing of the universe just doesn’t pick up. As a simple analogy, our view of, for example, a box of gas might be that it’s something with a certain temperature and pressure. But a different “parsing” of that system might identify a whole world of detailed motions of molecules that with respect to that parsing can be viewed as a vast “alien civilization”. Of course, the situation is much more extreme when it comes to the whole ruliad, and all the paths of history and configurations of atoms of space that it represents.

 Relative to the whole ruliad, our civilization and our experience have carved out an extremely tiny piece. And what we’re thinking of as “alien civilizations” might also have carved out their own tiny pieces. And while we’re all “living in the same ruliad”, we might no more be able to detect each other or communicate (and likely very much less) than we can across vast distances in physical space.

 What of the future? The future of our civilization might well be a story of mapping out more of rulial space. If we continue to invent new technology, explore new ideas and generally broaden our ways of thinking and perceiving, we will gradually—albeit in tiny steps—map out more of rulial space. How far can we get? The ultimate limit is determined by the maximum rulial speed. But if we expect to maintain our character as “observers like us”, we’ll no doubt be limited to something much less.

 Among other issues, moving in rulial space involves doing computation. (The ultimate scale is set by the “processing power” of the universe—which defines the maximum rulial speed.) But “density of computation” effectively corresponds to a generalized version of mass—and is for example a source of “generalized gravity”. And it could be that to “move any significant distance” in rulial space, we’d have to “experience enough generalized gravity” that we could never maintain things like the kind of coherence we need to be an “observer like us”.

 Put another way: yes, it might in principle be possible to “reach out in rulial space” and “contact the rulial aliens”. But it might be that doing so would require us to be so different from the way we currently are that we wouldn’t recognize anything like consciousness or anything that really makes us “identifiably us”. And if this is so, we are in a sense limited to experiencing the ruliad “on our own” from our particular place in rulial space, forever isolated from “alien civilizations” elsewhere in rulial space.

 Conceptual Implications of the Ruliad

 What does the concept of the ruliad mean for the fundamental way we think about things like science? The typical conception of “what science does” is that it’s about us figuring out—as “objectively” as we can—how the world happens to be. But the concept of the ruliad in a sense turns this on its head.

 Because it says that at some ultimate level, everything is a matter of abstract necessity. And it’s just our “parsing” of it that defines the subject matter of what we call science. We might have thought that the science of the universe was just something that’s “out there”. But what we’re realizing is that instead in some fundamental sense, it’s all “on us”.

 But does that mean that there’s no “objective truth”, and nothing that can robustly be said about the universe without “passing it through us”? Well, no. Because what we’ve discovered through our Physics Project is that actually there are quite global things that can (“objectively”) be said about our universe and the laws it follows, as perceived by observers like us.

 We don’t have to know in detail about us humans and the particular ways we perceive things. All we need are some general features—particularly that we are computationally bounded, and that we have a certain persistence and coherence. And this is all it takes to deduce some quite specific statements about how our universe operates, at least as we perceive it.

 So in a sense what this means is that there is a large “zone of objectivity”; a large set of choices for how we could be that will still lead us to the same “objective truth” about our universe. But if we go far enough away in our mechanism for “parsing the ruliad”, this will no longer be the case. From our current vantage point, we’d no doubt then be hard-pressed to recognize how we’re “doing the parsing”, but the results we’d get would no longer give us the same laws of physics or general perception of the universe that we’re used to.

 This view of things has all sorts of implications for various long-discussed philosophical issues. But it’s also a view that has precise scientific consequences. And these don’t just relate to physics. Because the ruliad is really a general object that represents the entangled behavior of all possible abstract rules. When we think of ourselves as observers embedded within this object, it means that for us things are actualized, and we have what we call physics. But we can also imagine sampling the ruliad in different ways.

 Some of those ways correspond to mathematics (or metamathematics). Some correspond to theoretical computer science. The ruliad is the single object that underlies all of them. And which of them we’re talking about just depends on how we imagine we’re sampling or parsing the ruliad, and how we’re describing what we’re observing.

 With this degree of generality and universality, it’s inevitable that the ruliad must be a complicated object; in fact, in a sense it must encapsulate all possible achievable complexity. But what’s important is that we now have a definite concept of the ruliad, as something we can study and analyze.

 It’s not simple to do this. The ruliad is at some level an object of great and perhaps supremely elegant abstract regularity. But for us to get any concrete handle on it and its structure, we need to break it down into some kind of “digestible slices” which inevitably lose much of its abstract regularity.

 And we’re just at the beginning of seeing how best to “unpack” and “pick through” the ruliad. With explicit computations, we can only chip away at the very simplest approximations to the ruliad. In a sense it’s a tribute to the naturalness and inevitability of the ruliad that it’s so closely related to some of the most advanced abstract mathematical methods we know so far. But again, even with these methods we’re barely scratching the surface of the ruliad and what it contains.

 The theoretical exploration of the ruliad will be a long and difficult journey. But the incredible generality and universality of the ruliad means that every piece of progress is likely to have exceptionally powerful consequences. In some sense the exploration of the ruliad can be seen as the encapsulated expression of everything it means to do theoretical investigation: a kind of ultimately abstract limit of theoretical science and more.

 For me, the ruliad in a sense builds on a tower of ideas, that include the computational paradigm in general, the exploration of the computational universe of simple programs, the Principle of Computational Equivalence, our Physics Project and the notion of multicomputation. But even with all of these it’s still a significant further jump in abstraction. And one whose consequences will take considerable time to unfold.

 But for now it’s exciting to have at least been able to define this thing I call the ruliad, and to start seeing some of its unprecedentedly broad and deep implications.

 Appendix: The Case of the “Multiplicad”

 As a very simple example of something like the ruliad, we can consider what we might call the “multiplicad”: a rulial multiway system based on integers, in which the rules simply multiply by successive integers:

  ✕
  

  (Note that this kind of pure multiplication is presumably not computation universal, so the limiting object here will not be a coordinatization of the actual full ruliad.)

  Just like with the full ruliad, there are many different “directions” in which to build up the multiplicad. We could allow as many multipliers and steps as we want, but limit the total size of numbers generated, here say to 30:

  ✕
  

  As an alternative, we can limit the number of multipliers s, say to . Then the multiplicad would build up like this:

  ✕
  

  In the pictures we’ve drawn so far, we’re effectively always deduplicating different occurrences of the same integer. So, for example, the integer 12 can be generated as 1 × 3 × 4 or 1 × 6 × 2 or 1 × 3 × 2 × 2, etc. And in principle we could show each of these “different 12s” separately. But in our deduplicated graph, only a single 12 appears—with the different possible decompositions of 12 being reflected in the presence of multiple paths that lead to the 12.

  Sometimes the structure we get is richer—if much bigger—when we don’t immediately do deduplication. For example, if we allow any number of multipliers (i.e. take ) then after just 1 step we will get all integers—and if we do deduplication, then this will be the end of our graph, because we “already have all the integers”. But if we don’t do deduplication, we’ll get a slightly more complicated picture, that begins like this:

  ✕
  

  The “topological” structure of this graph is now straightforward, but its “labeling” with numbers is less so—and if we ask, for example, where a particular number appears after t steps, this can be more complicated.

  Imagine that we are looking only at the subtrees associated with up to s multipliers at the first step—or, equivalently, that we are looking at the rulial multiway system “truncated” with only s rules. Which numbers will appear after steps? The answer is that it will be precisely those numbers that show up in an s × s multiplication table where we start from :

  ✕
  

  Clearly no primes appear here, but some numbers can appear multiple times (e.g. 12 appears 4 times). In general, the number of times that the number will show up is the number of proper divisors it has, or DivisorSigma[0, n]–2:

  ✕
  
  ✕
  

  We can continue this, to ask how many times a given number n will occur at a particular step t:

  ✕
  
  ✕
  
  ✕
  

  We can think of these results as being determined by the number of times that n appears in an s × s × s… (t times) multiplication array. Alternatively, to know the results for a given number n, we can look at all the ways n can be decomposed into factors. For , for example, we would have:

  ✕
  

  And from this we can deduce that 12 appears once at (i.e. with 1 factor), 4 times at (i.e. with 2 factors) and 3 times at (i.e. with 3 factors).

  The full multiplicad is formed by taking the limits and (as well as what is essentially the limit for an infinite set of possible initial conditions). As we can see, our “finite perception” of the multiplicad will be different depending on how we sample it in s and t.

  As an example, let’s consider what happens for given s as a function of t. For , we simply have powers of 2:

  ✕
  

  For , where can multiply by both 2 and 3, we get:

  ✕
  

  In studying multiway systems, it is often of interest to ask about the growth rates of the number of states reached over the course of t steps (i.e. the growth rates of volumes of geodesic balls). In the case , the number of states reached by step t is just t. For , it’s the triangular numbers t (t – 1)/2:

  ✕
  

  Here are some results for larger s:

  ✕
  

  Each of these sequences is generated by a linear recurrence relation with a kernel given by a sequence of signed binomial coefficients. The values for successive t can be represented by polynomials:

  ✕
  

  The leading term in the growth of number of states is then determined by the orders of these polynomials, which turn out to be just PrimePi[s]:

  ✕
  
  ✕
  

  For the case , we saw above that the multiway graph essentially forms a simple 2D grid. For larger s, the graph becomes more complicated, though it still approximates a grid—but in dimension PrimePi[s]. (The reason PrimePi[s] appears is that in a sense the combining of primes less than s are the largest “drivers” of structure in the multiway graph.)

  In our general analysis of multiway graphs, it is common to consider branchial graphs—or for a rulial multiway system what we can call rulial graphs—obtained by looking at a slice of the multiway graph, effectively for a given t, and asking what states are connected by having a common ancestor. The results for are rather trivial (here shown for , 2, 3):

  ✕
  

  For we get:

  ✕
  

  And for we have:

  ✕
  

  In a sense these pictures show how numbers in the multiplicad can be “laid out in rulial space”. For , the “large-t graph” has a very linear form

  ✕
  

  and the numbers that appear “from left to right” are arranged more or less in numerical order:

  ✕
  

  For , the result is a 2D-like structure:

  ✕
  

  And again the numbers that appear are roughly arranged in a kind of “numerical sequence”:

  ✕
  

  We can then think of this as suggesting that the multiplicad “generates a rulial space” that can be approximately coordinatized purely by the values of the numbers that appear in it. How all this works in the limit is not clear, though somehow the “1D” sequence of numerical values presumably “snakes through” PrimePi[s]-dimensional space as some kind of approximation to a space-filling curve.

  It should be noted that we’ve only considered one particular way of sampling the rulial multiway graph as a function of t. In general there are many different possible foliations that could be used, all of them giving us in effect a different view of the multiplicad, from a different “reference frame”.

  As mentioned at the beginning, the multiplicad is presumably not on its own capable of giving us the full ruliad. But if we change the underlying rules—probably even just inserting addition as well as multiplication—we’ll potentially get a system that is capable of universal computation, and which can therefore generate the full ruliad. Needless to say, the particular representation of the ruliad obtained by the kind of “numerical processes” that we’ve used here may be utterly different from any representation that we would recognize from our perception of the physical universe.

  Thanks & Note

  Thanks for discussions of various aspects of the ruliad to Xerxes Arsiwalla, James Boyd, Elise Cawley, Hatem Elshatlawy, Jonathan Gorard and Nik Murzin. Thanks also to Ed Pegg and Joseph Stocke for input about the multiplicad. A new paper by Xerxes Arsiwalla and Jonathan Gorard discusses in a more technical way some ideas and results related to the ruliad.

  Cite this as
  Stephen Wolfram (2021), "The Concept of the Ruliad," Stephen Wolfram Writings. writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad.
  Text
  Stephen Wolfram (2021), "The Concept of the Ruliad," Stephen Wolfram Writings. writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad.
  CMS
  Wolfram, Stephen. "The Concept of the Ruliad." Stephen Wolfram Writings. November 10, 2021. writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad.
  APA
  Wolfram, S. (2021, November 10). The concept of the Ruliad. Stephen Wolfram Writings. writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad.

  Posted in: Computational Science, Language and Communication, Mathematics, New Kind of Science, Philosophy, Physics

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  10 comments

  1. > In thinking about finding a fundamental theory of physics … the obvious next question will be: “Why this rule, and not another?” Well, how about if actually the universe in effect just runs every possible rule?

  This is where you lose me. It’s an argument from beauty. The (unstated) claim is that running all rules is a better theory than running one seemingly arbitrary rule. But implicit in this proposal is an underlying mechanism that is magic and inexplicable.

  A better approach is to try to discover why the universe is running on one particular rule–one mechanism–rather than any other. Here’s one valid point, I think: multiway systems are excluded as candidates for the most fundamental mechanism because they require a further underlying mechanism that identifies identical states and coalesces them into a single state.

  Earlier physicists made the mistake of thinking that the universe was fundamentally mathematical–that equations were the ultimate answer. Unfortunately now you’re making the mistake of thinking that the universe is fundamentally computational, and that “running every possible rule” is a good and sufficient answer. But the truth is that the universe is a single, concrete mechanism of some type. Computational rules are just a model for this mechanism, and we need to find the computational system that best models the mechanism.

  Martin
  November 12, 2021 at 12:44 pm

  2. Borges Library in Mathematical Terms?

  Johann Ernst Oberklammer
  November 13, 2021 at 1:37 pm

  3. I am fascinated.
  But I would like to tell you that the use of the term “consciousness”, at such a fundamental stage of your reasoning, is a potential source of multiple misunderstandings.
  From an evolutionary point of view, consciousness is primarily a unifying mechanism that imposes itself on sufficiently complex psyches to allow for subject uniqueness, ego-history and long-term planning. It is neither the seat of the will, nor the upper stage of intelligence. You probably know this very well. The sequencing of experience is not the work of consciousness either. One could develop multiple arguments of this kind.
  For all these reasons, I feel that you should look for another term, while there is still time, because I am afraid that with this simple word, “consciousness”, you are sabotaging the transmission of your thought.
  This is said with great admiration…

  Marc PELTIER
  December 9, 2021 at 10:30 am

  4. @martin I think you are just misunderstanding how it works. You state the following:

  “A better approach is to try to discover why the universe is running on one particular rule–one mechanism”

  The thing is that, even if the universe is running one rule, that rule must be turing complete tand it thus means it has to be running all possible rules…Basically the theory is lead inevitably to a concept of the Ruliad and I’m sure Wolfram realized this and it had to be the case.

  You can imagine it like this. Lets just say the universe is running Rule 30. Rule 30 is Turing Complete meaning it can formalize all computational languages…meaning that Rule 30, can be programmed to run rule 1,2,3…256…meaning that this universe that is running rule 30, is also running all possible programs anyway

  So the universe probably doesn’t have a single rule that it starts from…if it’s a Turing machine with 0’s and 1’s…it can and will run every possible rule, with respect to how observers choose to parse that Turing machine for their own purposes. Wolfram’s theory and theory of the Ruliad literally FALLS out of thinking about Turing complete computation in the most absolute basic sense.

  Lastly as a nail in the coffin, you can imagine a universe that is running say, Rule 1 which isn’t Turing Complete…then by definition we wouldn’t have a Turing Complete Universe (we would have a universe that is just rule 1 and we would see no other rules or behaviors)… So as a formal proof the rule that the universe would be running has to be Turing complete and if that is the case, then it is running all possible rules.

  justice
  January 22, 2022 at 10:13 am

  5. This is it. This is what all religions, all mystics, all spiritual paths, etc. have been trying to demonstrate: That the multi-verse is just one thing that is ‘attempting’ to ‘get to know itself’ by creating illusions of separation, with each of these illusory separate entities interacting with each other in order to systematically attempt every possible configuration of itself. The patterns that appear over and over again in our universe are just a manifestation of the path that our universe ‘decided’ to take in this journey. The patterns in other universes might look slightly different and therefore result in slightly different laws of physics/mathematics/chemistry/biology/etc., but at the end of the day it’s all the same process, just with differing levels of complexity and randomness depending upon the dimensional level the field is trying to conceptualize from, and related to that, the number of individuals interacting at the dimensional level.

  I would recommend reading “The Ra Contact.” From a limited perspective, it seems like absolute nonesense. But with the understanding of the Law of One, which Wolfram and his team are so beautifully demonstrating using this simple computational systems, you can see how the repeating patterns can be extrapolated to arrive at what the authors claim the so-called Ra entity told them about the nature of reality and the future of the individual, the nation, the race, the planet, the solar system, and so on.

  Mostly Multitudes
  February 4, 2022 at 3:12 pm

  6. Great work that I understand the following way. The space is a container of ‘stuff’ and the time is the one for ‘events’. And if all time exist in one piece for outside observer than all possible events do exist for him in the very moment of observation.

  The ‘stuff’ can’t be in two different places at the same time due to the law of mass and energy conservation. On micro and macro level events happen sequentially. The faster those elements move the more events happens to them on the same period of time. On quantum level all events happen at the same time. Quantum elements have such speed of interaction that their time converges with space. Time has interesting topology.

  Vladimir
  February 7, 2022 at 3:03 pm

  7. I have long believed that the difference between the universe we inhabit and nothingness is the abstract concept of chance. Thus, given the infinite parameters that are nothingness, what are the odds that a positive “something” never comes into being? I have merely taken it on faith that the odds of that are zero; something must beat nothing, even if the odds are long, and it only takes one “something” to get it all going. Pure abstraction, I realize.

  I had a problem with the first go-round on Wolfram’s theories, both in NKS and the new physics project, precisely because the selection of one rule seemed both arbitrary but also dependent on some antecedent force to choose the rule. It’s fun to imagine an overseer or creator, but it just begs the question of origin all over. So I like the refinement which is that every conceivable rule is always being applied, and that forward progress is therefore inevitable.

  As I grapple with this elegant theory, I choose several analogies or frameworks to do so, but for now, the one helping me the most is this: the Ruliad represents the extraordinary unlikelihood of nothingness, but it does so in a formal way by positing abstract concepts as the underlying reality from which all things spring. The upshot is that everything is abstract in the sense that it arises from the Ruliad by a long chain of causation, but conscious beings capable of abstract thought then, by that very capability of thought, necessarily (if sort of wrongly) conceive of themselves as “real,” tangible, solid, whatever you want to call our physical universe. If I touch my arm, that is real to my conscious self, but of course my arm ultimately boils down to ephemeral structures in a tiny realm far removed from the reality I perceive. It is no more real than the underlying abstraction that begets the steps that set in motion the tiniest possible persistent structures.

  I believe Wolfram is right to conclude that the Ruliad has consequences for other fields, particularly philosophy and math. The Ruliad provides a positivist framework for exploration.

  What strikes me forcefully — what I find so insightful — is that the mind-boggling difference in scale between whatever constitutes a step in the Ruliad-based computation, on the one hand, and the large-scale structures in our universe, on the other, is really nothing more than a relative difference in scale and is by no means infinite. The unsatisfactory conclusion of some physicists that our universe springs from “nothing” is, under Wolfram’s theory, wrong: there is a kind of granularity underlying everything, and its essence is the smallest steps of the computation occurring in the abstract space of the Ruliad. By that process, we are not in an infinite null; nor is our observed universe infinite precisely because it is is merely the instantiation, moment by moment, of ongoing operations in the Ruliad. (It is interesting to observe that these operations are not occurring in “time” but are, instead, the begetters of larger structures which exist in what we perceive as time; I wonder if events in the Ruliad would allow someone in one slice to dip into different slices and different “time” zones? hmmmmmm.)

  My worry, however, is that conceiving of the Ruliad as “true” may be trivial: if every rule is always running, yet inaccessible to us by our captivity in our slice, then we are irrevocably trapped in our universe. I imagine it being like a computer running a program in which computer-generated beings are sentient: they may know themselves, but they will never know the microchip instantiating their being. The gulf between begetter and begotten is insuperable. Perhaps, sadly, we are doomed never to peer into realms which beget us. I hope not!

  patrick sutton
  February 20, 2022 at 9:26 pm

  8. Earlier physicists made the mistake of thinking that the universe was fundamentally mathematical–that equations were the ultimate answer. Unfortunately now you’re making the mistake of thinking that the universe is fundamentally computational, and that “running every possible rule” is a good and sufficient answer. But the truth is that the universe is a single, concrete mechanism of some type. Computational rules are just a model for this mechanism, and we need to find the computational system that best models the mechanism.

  konya reklam ajansı
  March 22, 2022 at 11:13 am

  9. If Wolfram’s ideas are correct, then they preclude the current fad for branching universes existing in parallel. Instead, they allow the more startling possibility, which Wolfram discusses, that many slices are always already present as output from the underlying computational engine. We happen to have a set of senses which perceive certain kinds of structures in the hypergraph, and those are the only ones we need.

  That gravity (dense causal activity) necessarily exists in every slice is intriguing since one candidate I can think of that our senses cannot perceive is dark matter. It may represent a slice with more complex features than just what physicists posit as a single, undifferentiated type of particle (and thus a single persistent hypergraph feature). Dark matter may arise from a few persistent structure of a more fundamental type which itself generates a range of other persistent, larger structures.

  Thus, our slice of the universe may arise from, say, one kind of persistent underlying “particle” in the hypergraph, one much finer than anything our relatively coarse apparatuses could ever detect; another slice may arise from a different such particle resulting in what we indirectly observe as mass affected by the same very busy hypergraph.

  All speculation, of course, but no one else has offered a better candidate for dark matter. I personally find compelling and beautiful the idea of multiple slices existing together, undetected by one another except, at best, obliquely through the lens of gravitational effects. It would even seem that their “clocks” relative to ours could be very different since they arise from different structures within the hypergraph — structures that change at differing rates analogous to two computers running similar programs at different speeds.

  patrick sutton
  May 23, 2022 at 4:43 pm

  10. This describes how reality feels to me when immersed in a mystical experience. Pretty wild to see it formalized.

  Dave
  January 1, 2023 at 7:21 am

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The Entangled Limit of Everything
I call it the ruliad. Think of it as the entangled limit of everything that is computationally possible: the result of following all possible computational rules in all possible ways. It’s yet another surprising construct that’s arisen from our Physics Project. And it’s one that I think has extremely deep implications—both in science and beyond.
In many ways, the ruliad is a strange and profoundly abstract thing. But it’s something very universal—a kind of ultimate limit of all abstraction and generalization. And it encapsulates not only all formal possibilities but also everything about our physical universe—and everything we experience can be thought of as sampling that part of the ruliad that corresponds to our particular way of perceiving and interpreting the universe.
We’re going to be able to say many things about the ruliad without engaging in all its technical details. (And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) But to ground things here, let’s start with a slightly technical discussion of what the ruliad is.
In the language of our Physics Project, it’s the ultimate limit of all rulial multiway systems. And as such, it traces out the entangled consequences of progressively applying all possible computational rules. 
Here is an example of an ordinary multiway system based on the string replacement rules {A → AB, BB → A} (indicated respectively by blueish and reddish edges):
	
At each step, the rules are applied in all possible ways to each state. Often this generates multiple new states, leading to branching in the graph. But, importantly, there can also be merging—from multiple states being transformed to the same state.
The idea of a rulial multiway system is not just to apply particular rules in all possible ways, but to apply all possible rules of a given form. For example, if we consider “1 → 2, 2 → 1 A, B string rules”, the possible rules are
	
and the resulting multiway graph is (where now we’re using purple to indicate that there are edges for every possible rule):
	
Continuing a little longer, and with a different layout, we get:
	
This may already look a little complicated. But the ruliad is something in a sense infinitely more complicated. Its concept is to use not just all rules of a given form, but all possible rules. And to apply these rules to all possible initial conditions. And to run the rules for an infinite number of steps. 
The pictures above can be thought of as coarse finite approximations to the ruliad. The full ruliad involves taking the infinite limits of all possible rules, all possible initial conditions and all possible steps. Needless to say, this is a complicated thing to do, and there are many subtleties yet to work out about how to do it.
Perhaps the most obviously difficult issue is how conceivably to enumerate “all possible rules”. But here we can use the Principle of Computational Equivalence to tell us that whatever “basis” we use, what comes out will eventually be effectively equivalent. Above we used string substitution systems. But here, for example, is a rulial multiway system made with 2-state 2-color Turing machines:
	
And here is a rulial multiway system made from hypergraph rewriting of the kind used in our Physics Project, using all rules with signature :
	
As another example, consider a multiway system based on numbers, in which the rules multiply by each possible integer:
	
Here’s what happens starting with 1 (and truncating the graph whenever the value exceeds 100): 
	
Even with this simple setup, the results are surprisingly complicated (though it’s possible to give quite a bit of analysis in this particular case, as described in the Appendix at the end of this piece). 
The beginning of the multiway graph is nevertheless simple: from 1 we connect to each successive integer. But then things get more complicated. To see what’s going on, let’s look at a fragment of the graph:
	
In a sense, everything would be simple if every path in the graph were separate: 
	
	
But the basic concept of multiway systems is that equivalent states should be merged—so here the “two ways to get 6” (i.e. 1 × 2 × 3 and 1 × 3 × 2) are combined, and what appears in the multiway graph is:
	
For integers, the obvious notion of equivalence is numerical equality. For hypergraphs, it’s isomorphism. But the important point is that equivalence is what makes the multiway graph nontrivial. We can think about what it does as being to entangle paths. Without equivalence, different paths in the multiway system—corresponding to different possible histories—would all be separate. But equivalence entangles them.
The full ruliad is in effect a representation of all possible computations. And what gives it structure is the equivalences that exist between states generated by different computations. In a sense, there are two forces at work: the “forward” effect of the progress of computation, and the “sideways” effect of equivalences that entangle different computations. (Mathematically this can be thought of as being like decomposing the ruliad structure in terms of fibrations and foliations.)
Experiencing the Ruliad
In thinking about finding a fundamental theory of physics, one thing always bothered me. Imagine we successfully identify a rule that describes everything about our universe. Then the obvious next question will be: “Why this rule, and not another?” Well, how about if actually the universe in effect just runs every possible rule? What would this mean? It means that in a sense the “full story” of the universe is just the ruliad.
But the ruliad contains everything that is computationally possible. So why then do we have the perception that the universe has specific laws, and that definite things happen in it?
It all has to do with the fact that we are bounded observers, embedded within the ruliad. We never get to see the full ruliad; we just sample tiny parts of it, parsing them according to our particular methods of perception and analysis. And the crucial point is that for coherent observers like us, there are certain robust features that we will inevitably see in the ruliad. And these features turn out to include fundamental laws of our physics, in particular general relativity and quantum mechanics.
One can imagine an observer very different from us (say some kind of alien intelligence) who would sample different aspects of the ruliad, and deduce different laws. But one of the surprising core discoveries of our Physics Project is that even an observer with quite basic features like us will experience laws of physics that precisely correspond to ones we know.
An analogy (that’s actually ultimately the result of the same underlying phenomenon) may help to illustrate what’s going on. Consider molecules in a gas. The molecules bounce around in a complicated pattern that depends on their detailed properties. But an observer like us doesn’t trace this whole pattern. Instead we only observe certain “coarse-grained” features. And the point is that these features are largely independent of the detailed properties of the molecules—and robustly correspond to our standard laws of physics, like the Second Law of thermodynamics. But a different kind of observer, sampling and “parsing” the system differently, could in principle identify different features, corresponding to different laws of physics.
One of the conceptual difficulties in thinking about how we perceive the ruliad is that it’s a story of “self-observation”. Essentially by the very definition of the ruliad, we ourselves are part of it. We never get to “see the whole ruliad from the outside”. We only get to “experience it from the inside”.
In some ways it’s a bit like our efforts to construct the ruliad. In the end, the ruliad involves infinite rules, infinite initial conditions, and infinite time. But any way of assembling the ruliad from pieces effectively involves making particular choices about how we take those infinite limits. And that’s pretty much like the fact that as entities embedded within the ruliad, we have to make particular choices about how to sample it.
One of the remarkable aspects of the ruliad is that it’s in some sense the unique ultimately inevitable and necessary formal object. If one sets up some particular computational system or mathematical theory, there are choices to be made. But in the ruliad there are no choices. Because everything is there. And in a sense every aspect of the structure of the ruliad is just something formally necessary. It requires no outside input; it is just a formal consequence of the meaning of terms, like the abstract fact . 
But while the ruliad is unique, the description of it is not. In constructing it, one can imagine using Turing machines or hypergraph rewriting systems or indeed any other kind of computational system. Each will ultimately lead to the same limiting object that is the ruliad, but each of them can be thought of as defining a different coordinate system for describing the ruliad. 
The very generality of the ruliad makes it unsurprising that there is vast diversity in how it can be described. And in a sense each possible description is like a possible way of experiencing the ruliad. In analogy to the (deeply related) situation with spacetime in general relativity, we might say that there are many reference frames in which to experience the ruliad—but it’s always the same ruliad underneath.
It’s important to understand that the “ruliad from the outside” could seem very different from any “internal” experience of it by an observer like us. As an example, consider a simple finite approximation to the ruliad, built from string substitution systems. In what we did above, we always started from a specific initial condition. But the full ruliad involves starting from all possible initial conditions. (Of course, one could always just say one starts from a “null” initial condition, then have rules of the form null → everything.) So now let’s consider starting from all possible strings, say of length 4. If we use all possible 2-element-to-2-element rules, the finite approximation to the ruliad that we’ll get will be:
	
At some level this is a simple structure, and—as is inevitable for any finite approximation to the ruliad—its transitive closure is just the complete graph:
	
So why doesn’t this mean that the ruliad is somehow trivial? A key part of the story is that we never get to “see the ruliad from the outside” like this. We are always part of it, sampling it according to some procedure, or, somewhat equivalently, thinking about constructing it according to some procedure.
As an analogy, consider the real numbers. The whole continuum of all real numbers is “from the outside” in many ways a simple construct. But if we imagine actually trying to construct real numbers, say digit by digit, according to some definite procedure, then we’re dealing precisely with what Turing machines were originally invented to model, and the whole structure of computation is involved. (As we’ll see, our way of thinking about “observers like us” is ultimately quite related to “Turing machines with bounded descriptions”.)
In a sense, at an outside “holistic” level, the ruliad has a certain simple perfection. But as soon as you try to look at “what’s in the ruliad”, you have to parametrize or coordinatize it, and then you’re inevitably exposed to its intricate internal structure.
Observers Like Us
One could imagine very different ways in which entities embedded within the ruliad could “experience” it. But what’s most relevant for us is how “observers like us” do it—and how we manage to synthesize from what’s going on in the ruliad our perception of reality, and our view of how our physical universe works.
Let’s start by talking not about the full ruliad but rather about models in our Physics Project based on specific underlying rules. At the lowest level, we have a “machine-code” description of the universe is which everything just consists of a network of “atoms of space” that is continually being updated—and which we can think of as carrying out a giant, if incoherent, computation, full of computational irreducibility. But the remarkable fact is that somehow we, as observers of this, manage to pick out of it a certain slice that ends up showing coherent, computationally reducible features—that for example seem to reproduce our known laws of physics.
How does this work? Partly it has to do with features of us as observers, partly with features of how the universe fundamentally works, and partly with an interplay between these. 
The first crucial feature of us as observers is that we’re computationally bounded: the way we “parse” the universe involves doing an amount of computation that’s absolutely tiny compared to all the computation going on in the universe. We sample only a tiny part of what’s “really going on underneath”, and we aggregate many details to get the summary that represents our perception of the universe.
But why should that summary have any coherence? Basically it’s because we impose coherence through our definition of how observers like us work. One part of the universe will be affected by others. But to consider part of the universe as an “observer”, there has to be a certain coherence to it. The behavior of the universe somehow has to imprint itself on a “medium” that has a certain coherence and consistency.
Down at the level of atoms of space, everything is always changing. But we can still identify emergent features that have a certain persistence. And it’s out of those features that what we call observers are built. Given only the atoms of space with all their computationally irreducible behavior, it’s not at the outset obvious that any real persistence could exist or be identified. But in our models we expect that there will, for example, be essentially topological features that correspond to particles that persistently maintain their identity. 
And the point is that we can expect to “aggregate up” much further and be able to identify something like a human observer—that we can consider to persistently maintain its identity to the point where phenomena from the universe can be “systematically imprinted” on it. 
Down at the level of atoms of space, there’s a whole multiway graph of possible sequences of updates that can occur—with each path in effect corresponding to a different “thread of time” for the universe. But it’s a crucial fact about us as observers of the universe that we don’t perceive all those branching and merging threads of time. Instead, we imagine that we have a single, definite thread of experience—in which everything is sequentialized in time. 
I’ve argued elsewhere that this sequentialization in time is a defining characteristic of “human-like consciousness”. And it turns out that one of its consequences is that it implies that the particular perception we will have of the universe must be one in which there are laws of physics that correspond to ones we know.
It’s not obvious, by the way, that if we sequentialize time we can form any consistent view of the universe. But the phenomenon of causal invariance—which seems ultimately to be guaranteed by the fundamental structure of the ruliad—turns out to imply that we can expect a certain generalized relativistic invariance that will inevitably lead to eventual consistency. 
The notion of sequentialization in time is closely related to the idea that—even though our individual atoms of space are continually changing—we can view ourselves as having a coherent existence through time. And there’s a similar phenomenon for space. At the outset, it’s not obvious that there can be “pure motion”, in which something can move in space without “fundamentally changing”. But it turns out again to be consistent to view this as how things work for us: that even though we’re “made of different atoms of space” when we’re in different places, we can still imagine that in some sense we maintain the “same identity”.
Down at the level of individual atoms of space, there really isn’t any coherent notion of space. And the fact that we form such a notion seems to be intimately connected to what we might think of as details of us. Most important is that we’re in a sense “intermediate in size” in the universe. We’re large relative to the effective distance between atoms of space (which might be m), yet we’re small compared to the size of the whole universe ( m). And the result is that we tend to aggregate the effects of many atoms of space, but still perceive different features of space (say, different gravitational fields) in different parts of the universe.
The fact that we “naturally form a notion of space” also seems to depend on another issue of scale—that for us the speed of light “seems fast”. It takes our brains perhaps milliseconds to process anything we see. But the point is that this is very long compared to the time it takes light to get to us from objects in our typical local environment. And the result is that we tend to perceive there as being an instantaneous configuration of the world laid out in space, that “separately” changes in time. But if, for example, our brains ran much faster, or we were much bigger than we are, then the speed of light would “seem slower” to us, and we wouldn’t tend to form the notion of an “instantaneous state of space”.
OK, so what about quantum mechanics? The most fundamental feature of quantum mechanics is that it implies that things in the universe follow not just one but many possible paths of history—which we only get to make certain kinds of measurements on. And in our Physics Project this is something natural, and in fact inevitable. Given any particular configuration of the universe, there are many possible updates that can occur. And when we trace out all the possibilities, we get a multiway system, in which different threads of history continually branch and merge.
So how do observers like us fit into this? Being part of the universe, we inevitably branch and merge, just like the rest of the universe. So to understand our experience, what we need to ask is how a “branching brain” will perceive a “branching universe”. And the story is remarkably similar to what we discussed above for our experience of space and time: it all has to do with imagining ourselves to have a certain definite persistence.
In other words, even if when “viewed from the outside” our brain might be following many different paths of history, “from the inside” we can still potentially assume that everything is conflated into a single thread of history. But will this ultimately be a consistent thing to do? Once again, causal invariance implies that it will. There are specific “quantum effects” where we can tell that there are multiple branches of history being followed, but in the end it’ll be consistent to imagine an “objective reality” about “what happened”.
In our Physics Project we imagine that there are abstract relations between atoms of space, and in the end the pattern of these relations defines the structure of physical space. But what about different branches of history in the multiway graph? Can we think of these as related? The answer is yes. For example, we can say that at a particular time, states on two branches are “adjacent” if they share an immediate ancestor in the multiway graph. And tracing through such connections we can develop a notion of “branchial space”—a kind of space in which states on different branches of history are laid out:
	
One can think of branchial space as being defined by the pattern of entanglements between different branches of history. And in our Physics Project it turns out that the fundamental laws of quantum mechanics seem to just be a direct translation of the fundamental laws of spacetime into branchial space. And just like the speed of light governs the maximum rate at which effects can propagate in physical space, so similarly in our models there’s a “maximum entanglement speed” at which effects can propagate in branchial space.
So what are we like as observers in branchial space? Just like in physical space, we can presumably be thought of as having a certain size in branchial space. We don’t yet know quite how to measure this size, but it’s surely related to the effective number of quantum degrees of freedom we involve. 
In our everyday experience of things like gases, we’re sufficiently large compared to individual molecules that we normally just perceive the gas as some kind of continuum fluid—and in normal circumstances we can’t even tell that it’s made of molecules. Well, it’s presumably the same kind of thing for physical space—where we’re even much larger compared to the atoms of space, and it’s a major challenge to figure out how to detect their presence. What about for branchial space? As the underlying rules for the system get applied, different branches of history will in effect “move around” in branchial space in complex and computationally irreducible ways. And just like when we observe molecules in a gas, we’ll mostly just observe overall aggregate effects analogous to fluid mechanics—and only in special circumstances will we notice “quantum effects” that reveal the presence of multiple independent threads of history.
Living in Rulial Space
We’ve discussed how “observers like us” perceive models of physics of the type that arise in our Physics Project. But how will we perceive the whole ruliad? It begins with a generalization of the story for branchial space. Because now as well as having different branches associated with different updatings according to a particular rule, we have different branches associated with updatings according to different rules.
And just as we can slice an ordinary multiway system at a particular time to get an instantaneous version of branchial space, so now we can slice a rulial multiway system to get an instantaneous version of what we can call rulial space—a space in which different branches can correspond not just to different histories, but to different rules for history.
It’s a fairly complicated setup, with “pure branchial space” inevitably being deeply interwoven with rulial space. But as a first approximation, we can think of rulial space as being somewhat separate, and laid out so that different places in it correspond to the results of applying different rules—with nearby places effectively being associated with “nearby” rules.
And just as we can think of effects propagating in branchial space, so also we can think of them propagating in rulial space. In branchial space we can talk about entanglement cones as the analog of light cones, and a maximum entanglement speed as the analog of the speed of light. In rulial space we can instead talk about “emulation cones”—and a “maximum emulation speed”.
In our rough approximation of rulial space, each point is in effect associated with a particular rule. So how do we “move” from one point to another? Effectively we have to be emulating the behavior of one rule by another. But why should it even be possible to do this? The answer is the Principle of Computational Equivalence, which states that, in effect, most rules will be equivalent in their computational capabilities—and in particular they will be capable of universal computation, so that any given rule can always “run a program” that will make it emulate any other rule.
One can think of the program as an interpreter or translator that goes from one rule to another. The Principle of Computational Equivalence tells one that such a translator must essentially always exist. But how fast will the translator run? Effectively that’s what distance in rulial space measures. Because to “do a certain translation”, branches in the rulial multiway system have to reach from one rule to another. But they can only do that at the maximum emulation speed. 
What does the maximum emulation speed measure? Effectively it corresponds to the raw computational processing speed of the universe. We can think of representing computations in some language—say the Wolfram Language. Then the processing speed will be measured in “Wolfram Language tokens processed per second” (“WLT/s”). In some sense, of course, giving a value for this speed is just a way of relating our human units of time (say, seconds) to the “intrinsic unit of time” associated with the computational processing that’s going on in the universe. Or, in other words, it’s a kind of ultimate definition of a second relative to purely formal constructs.
OK, but how does this relate to us as observers embedded within the ruliad? Well, just as we imagine that—along with the rest of the universe—we’re continually branching and merging in branchial space, so also this will be what happens in rulial space. In other words—like the rest of the universe—our brains aren’t following a particular rule; they’re following branching and merging paths that represent all possible rules.
But “from inside” we can still potentially imagine that we have a single thread of experience—effectively conflating what happens on all those different branches. And once again we can ask whether doing this will be consistent. And the answer seems to be that, yes, it can be. And what guarantees this is again a kind of “rulial relativity” that’s a consequence of causal invariance. There are many details here, which we’ll address to some extent later. But the broad outline is that causal invariance can be thought of as being associated with paths of history that diverge, eventually converging again. But since the ruliad contains paths corresponding to all possible rules, it’s basically inevitable that it will contain what’s needed to “undo” whatever divergence occurs. 
So what does this mean? Basically it’s saying that even though the universe is in some sense intrinsically “following all possible rules”—as represented by paths in the ruliad—we as observers of the universe can still “take the point of view” that the universe follows a particular rule. Well, actually, it’s not quite a particular rule. Because just as we’re in some sense “quite big” in physical and presumably branchial space, so also we’re potentially “quite big” in rulial space.
And being extended in rulial space is basically saying that we consider not just one, but a range of possible rules to be what describe the universe. How can it work this way? Well, as observers of the universe, we can try to deduce what the “true rule for the universe” is. But inevitably we have to do this by performing physical experiments, and then using inductive inference to try to figure out what the “rule for the universe is”. But the issue is that as entities embedded within the universe, we can only ever do a finite number of experiments—and with these we’ll never be able to precisely nail down the “true rule”; there’ll always be some uncertainty.
When we think of ourselves as observers of the universe, there’s in a sense lots of “arbitrariness” in the way we’re set up. For example, we exist at a particular location in physical space—in our particular solar system and so on. Presumably we also exist at a particular location in branchial space, though it’s less clear how to “name” that. And in addition we exist at a particular location in rulial space.
What determines that location? Essentially it’s determined by how we operate as observers: the particular sensory system we have, and the particular means of description that we’ve developed in our language and in the history of knowledge in our civilization. In principle we could imagine sensing or describing our universe differently. But the way we do it defines the particular place in rulial space at which we find ourselves.
But what does all this mean in terms of the ruliad? The ruliad is the unique limiting structure formed by following all possible rules in all possible ways. But when we “observe the ruliad” we’re effectively “paying attention to” only particular aspects of it. Some of that “paying attention” we can conveniently describe in terms of our particular “location in the ruliad”. But some is more naturally described by thinking about equivalence classes in the ruliad.
Given two states that exist in the ruliad, we have to ask whether as observers we want to consider them distinct, or whether we want to conflate them, and consider them “the same”. When we discussed the construction of the ruliad, we already had many versions of this issue. Indeed, whenever we said that two paths in the ruliad “merge”, that’s really just saying that we treat the outcomes as equivalent.
“Viewed from the outside”, one could imagine that absolutely nothing is equivalent. Two hypergraphs produced in two different ways (and thus, perhaps, with differently labeled nodes) are “from the outside” in some sense different. But “viewed from the inside”, they pretty much have to be viewed as “the same”, in essence because all their effects will be the same. But at some level, even such conflation of differently labeled hypergraphs can be thought of as an “act of the observer”; something that one can only see works that way if one’s “observing it from inside the system”.
But all the way through our description of the observer, it’s very much the same story: it’s a question of what should be considered equivalent to what. In sequentializing time, we’re effectively saying that “all of space” (or “all of branchial space”, or rulial space) should be considered “equivalent”. There are many subtle issues of equivalence that also arise in the construction of states in the ruliad from underlying tokens, in defining what rules and initial conditions should be considered the same, and in many other places.
The ruliad is in some sense the most complicated constructible object. But if we as computationally bounded observers are going to perceive things about it, we have to find some way to “cut it down to size”. And we do that by defining equivalence classes, and then paying attention only to those whole classes, not all the details of what’s going on inside them. But a key point is that because we are computationally bounded observers who imagine a certain coherence in their experience, there are strong constraints on what kinds of equivalence classes we can use.
If we return again to the situation of molecules in a gas, we can say that we form equivalence classes in which we look only coarsely at the positions of molecules, in “buckets” defined by simple, bounded computations—and we don’t look at their finer details, with all the computational irreducibility they involve. And it’s because of this way of looking at the system that we conclude that it follows the Second Law of thermodynamics, exhibits fluid behavior, etc.
And it’s very much the same story with the ruliad—and with the laws of physics. If we constrain the kind of way that we observe—or “parse”—the ruliad, then it becomes inevitable that the effective laws we’ll see will have certain features, which turns out apparently to be exactly what’s needed to reproduce known laws of physics. The full ruliad is in a sense very wild; but as observers with certain characteristics, we see a much tamer version of it, and in fact what we see is capable of being described in terms of laws that we can largely write just in terms of existing mathematical constructs.
At the outset, we might have imagined that the ruliad would basically just serve as a kind of dictionary of possible universes—a “universe of all possible universes” in which each possible universe has different laws. But the ruliad is in a sense a much more complicated object. Rather than being a “dictionary” of possible separate universes, it is something that entangles together all possible universes. The Principle of Computational Equivalence implies a certain homogeneity to this entangled structure. But the crucial point is that we don’t “look at this structure from the outside”: we are instead observers embedded within the structure. And what we observe then depends on our characteristics. And it turns out that even very basic features of our consciousness and sensory apparatus in a sense inevitably lead to known laws of physics—and in a sense do so generically, independent of details of just where in rulial space we are, or exactly what slice of the ruliad we take.
So far we’ve primarily talked about the ruliad in terms of physics and the fundamental structure of our physical universe. But the ruliad is actually something still more general than that. Because ultimately it is just created from the abstract concept of following all possible computational rules. And, yes, we can interpret these rules as representing things going on in our universe. But we can also interpret them as representing things going on in some other, less immediately physically realizable system. Or, for that matter, representing something purely formal, and, say, mathematical.
This way of talking about the ruliad might make one think that it should be “considered a possible model” for our universe, or for other things. But the bizarre and surprising point is that it is more than that. It’s not just a possible model that might be one of many. Rather, it is the unique ultimate representation of all possible models, entangled together. As we’ve discussed, there are many subtle choices about how we observe the ruliad. But the ultimate ruliad itself is a unique thing, with no choice about what it is. 
As I’ve discussed at more length elsewhere, the ruliad is in a sense a representation of all possible necessary truths—a formal object whose structure is an inevitable consequence of the very notion of formalization. So how does this relate to the idea that the ruliad also at an ultimate level represents our physical universe? What I’ve argued elsewhere is that it means that the ultimate structure of our universe is a formal necessity. In other words, it’s a matter of formal necessity that the universe must exist, and have an ultimate ruliad structure. The fact that we perceive the universe to operate in a certain way—with our standard laws of physics, for example—is then a consequence of the particular way observers like us perceive it, which in turn depends on things like where in rulial space we happen to find ourselves.
But beyond physics, what else might the ruliad represent? The ruliad is an ultimate example of multicomputation, and of what I’ve characterized as the fourth major paradigm for theoretical science. Often in multicomputation, what’s of interest is multiway systems with specific underlying rules. And already at this level, much of the apparatus that we’ve described in connection with the ruliad also applies—and in a sense “trickles down” to give various universal results.
But there are also definitely cases of multicomputation (other than physics) where the full notion of applying all possible rules is relevant. The global structures of metamathematics, economics, linguistics and evolutionary biology seem likely to provide examples—and in each case we can expect that at the core is the ruliad, with its unique structure. Of course, this doesn’t mean that what we observe must always be the same, because what we observe depends on our characteristics as an observer—and the characteristics of “being an observer” in metamathematics, for example, are surely different from those for economics or evolutionary biology, or, for that matter, physics. 
For sure, the “sensory apparatus” that we effectively use is different in different cases. But there are certain similar human-based features that still seem to apply. Whatever the domain, we always act as computationally bounded observers. And it seems that we also always have a certain coherence, consistently maintaining our “observerhood” through time or across some form of space. And it seems likely that these “human-induced” characteristics alone are sufficient to yield some very global implications for observed behavior.
The View from Mathematics
How should we think about the ruliad mathematically? In many ways, the ruliad is more an object of metamathematics than of mathematics itself. For in talking about the effects of all possible rules, it in a sense transcends individual mathematical theories—to describe a kind of metatheory of all possible theories.
Given a particular mathematical axiom system, it’s rather easy to see correspondence with a multiway system. There are a variety of ways to set it up, but one approach is to think of states in the multiway system as being expressions in the language used for the axiom system, and then to think of rules in the multiway system as applying transformations on these expressions that implement axioms in the axiom system.
For example, with the (Abelian semigroup) axioms
	
here’s a multiway system generated from the expression by applying the (two-way) transformations defined by the axioms in all possible ways to each expression:
	
But now from this graph we can read off the “theorem”:
	
A proof of this theorem
	
is just a path in the multiway graph:
	
A significantly less direct but still perfectly valid proof would correspond to the 13-step path:
	
It’s a slightly technical point, but perhaps worth mentioning that there are alternative ways to set up the correspondence between axiomatic mathematical systems and multiway systems. One such way is to take the states in the multiway system to be not expressions (like ) but whole propositions (like ). Axioms then show up as states, and the rules for the multiway system are “rules of inference” or “rules of entailment” that define how, say, pairs of propositions “entail” (or “imply”) other ones. (And, yes, this requires a generalized multiway system with not just 1 state → many states, but for example 2 states → 1 state.) Typical automated theorem provers (like FindEquationalProof) operate in this kind of setup, attempting to find paths that lead from initial propositions and axioms to some final state that corresponds to an “obviously true” proposition, like .
But whatever the detailed setup, the basic picture is that an axiomatic mathematical system has an associated multiway graph, in which paths correspond to proofs. Given the rules for the multiway system, there is in general no way to guarantee that the path (if it exists) corresponding to the proof of some particular result will be of bounded length, leading to the possibility of undecidability. But even when a path exists, it may require an irreducibly large amount of computation to find it. Still, finding such paths is what automated theorem provers do. For example, we know (as I discovered in 2000) that ((b · c) · a) · (b · ((b · a) · b)) = a is the minimal axiom system for Boolean algebra, because FindEquationalProof finds a path that proves it.
But this path—and the corresponding proof—is a very “non-human” construct (and, for example, in 21 years essentially no progress has been made in finding a “human-understandable narrative” for it). And we can make an analogy here to the situation in physics. The individual rule applications in the multiway graph (or the proof) are like individual updating events applied to the atoms of space—and they show all kinds of complexity and computational irreducibility. But in physics, human observers work at a higher level. And the same, one suspects, is true in mathematics.
Rather than looking at every detail of the multiway graph, human “mathematical observers” (i.e. pure mathematicians) in effect define all sorts of equivalences that conflate together different parts of the graph. If the individual updates in the multiway graph are like molecular dynamics, human pure mathematics seems to operate much more at the “fluid dynamics level”, concentrating on “broad mathematical constructs”, not the “machine code” of specific low-level axiomatic representations. (Of course, there are some situations, for example related to undecidability, where the “molecular dynamics” effectively “breaks through”.)
We’ve outlined above (and discussed at length elsewhere) how physical observers like us “parse” the low-level structure of the physical universe (and the ruliad). How might mathematical observers do it? A large part has to do with the identification of equivalences. And the key idea is that things which are considered equivalent should be assumed to be “the same”, and therefore “conflated for mathematical purposes”.
The most elementary example of something like this is the statement (already present in Euclid) that if and , then . The extensionality axiom of set theory is a more sophisticated example. And the univalence axiom of homotopy type theory is perhaps the most sophisticated current version. 
There’s a very operational version of this that appears in automated theorem proving. Imagine that you’ve proved that and . Then (by the assumed properties of equality) it follows that . One way we could use this result is just to merge the nodes for and . But a “bigger” thing we can do is to add the “completion” as a general rule for generating the multiway system.
Consider, for example, the string substitution multiway system A ↔ AB:
	
But notice here that both ABA ↔ ABBA and ABA ↔ ABAB. So now add the “completion” ABBA ↔ ABAB. Here’s the resulting multiway graph: 
	
Adding the completion has generated lots of new “direct equivalences”. But we can also think of it as having done something else: not only has it defined equivalences between states; it’s also defined equivalences between paths—or in effect between proofs. (Or, put another way, it’s implementing homotopic equivalence between proofs. By the way, it’s an important feature of human mathematics that progress is typically measured in theorems proved; different proofs of the same theorem are normally implicitly considered equivalent in terms of the progress of mathematics.) 
In a category theory interpretation, transformations between states in the original multiway graph are like ordinary morphisms (1-morphisms). But when we’re making transformations between “proof paths”, this is like 2-morphisms. And just as we can add transformations between proofs, we can also add transformations between proofs-between-proofs, and so on. The result is that we can build up a whole hierarchy of higher and higher categories, eventually ending with an ∞-category. 
But now we can begin to see the connection with the ruliad. The ruliad is in a sense the result of taking the limit of adding more and more possible rules. Above we did this quite explicitly in terms of the original underlying rules, for example by enumerating possible strings or possible integer multipliers. 
But we can view successive completions as doing something very similar. Yes, the rules are enumerated in a different order. But in the end there’ll still be an infinite number of distinct rules being used. 
Of course there are many mathematical details missing here. But in broad outline, it seems that one can think of the approach to the ruliad as some kind of limit of successively higher categories. But this limit is something that’s been studied (albeit in the upper reaches of pure mathematics), and it’s an object called the ∞-groupoid. (It’s a groupoid because when all the rules are included things inevitably “go both ways”). So, OK, is the ruliad “just” the ∞-groupoid? Not quite. Because there are more rules and more initial conditions in the ruliad, even beyond those added by completions. And in the end the ruliad actually seems to be the ∞-category of ∞-groupoids, or what’s called the (∞,1)-category.
But knowing that the ruliad can be thought of as composed of ∞-groupoids means that we can apply mathematical ideas about the ∞-groupoid to the ruliad. 
Probably the most important is Grothendieck’s hypothesis, which asserts that the ∞-groupoid inevitably has a topological and (with a few other conditions) ultimately geometric structure. In other words, even though one might have imagined that one constructed the ∞-groupoid from “pure logic” (or from pure formal axiomatic structures), the assertion is that the limiting object one obtains inevitably exhibits some kind of geometrical or “spatial” structure. 
Viewed in terms of the ruliad—and our explicit finite examples of it—this might not seem surprising. And indeed in our Physics Project, the whole concept of the emergence of space from large-scale hypergraphs is closely related. But here from Grothendieck’s hypothesis we’re basically seeing a general claim that the ruliad must have “inevitable geometry”—and we can then view things like the emergence of space in our Physics Project as a kind of “trickle down” from results about the ruliad. (And in general, a big “application” of geometrical structure is the possibility of “pure motion”.)
What does all this mean about the ruliad and mathematics? In a sense the ruliad represents all possible mathematics—the application of all possible rules, corresponding to all possible axiom systems. And from this “ultimate metamathematics”, human “mathematical observers” are sampling pieces that correspond to the pure mathematics they consider of interest.
Perhaps these will align with particular axiom systems of the kind automated theorem provers (or proof assistants) use. But things may be “sloppier” than that, with human mathematical observers effectively being extended in rulial space—and capable of making “fluid-dynamics-level” conclusions, even if not “molecular-dynamics-level” ones. 
But a key (and in some ways very surprising) point is that the ruliad can be viewed as the basis of both physics and mathematics. In some sense, physics and mathematics are at their core the same thing. They only “appear different” to us because the way we “observe” them is different. 
I plan to discuss the implications for mathematics at greater length elsewhere. But suffice it to say here that the existence of a common underlying core—namely the ruliad—for both physics and mathematics immediately allows one to start importing powerful results from physics into mathematics, and vice versa. It also allows one, as I have done elsewhere, to start comparing the existence of the universe with the (Platonic-style) concept of the fundamental existence of mathematics.
The View from Computation Theory
The ruliad can be thought of as an encapsulation of doing all possible computations in all possible ways. What we might think of as a “single computation” might consist of repeatedly applying the rules for a Turing machine to “deterministically” generate a sequence of computational steps: 
	
But one can also consider a “multicomputational” system, in which rules can generate multiple states, and the whole evolution of the system can be represented by a multiway graph:
	
In traditional computation theory, one talks about rules like these as “nondeterministic”, because they can have multiple outcomes—though one usually imagines that the final answer one wants from a computation can be found as the result of some particular path. (In what I now call the multicomputational paradigm—that I believe is important for modeling in physics and other places—one instead considers the complete multiway graph of entangled possible histories.)
In constructing the ruliad, one is in a sense going to a more extreme version of multicomputation, in which one uses not just a particular rule with multiple outcomes, but all possible rules. In effect, the concept is to use “maximal nondeterminism”, and at each step to independently “pick whatever rule one wants”, tracing out a rulial multiway system that includes all the different possible paths this generates. 
For the kind of Turing machines illustrated above, the rulial multiway graph one gets after one step is: 
	
After 2 steps the result is:
	
The full ruliad is then some kind of infinite limit of this process. But as before, there’s lots of subtlety in how this limit is taken. But we can at least characterize some ways of approaching it using ideas from computational complexity theory. Increasing the number of steps of evolution is like increasing the time complexity one allows. Increasing the “size of states” (e.g. the width of nonzero Turing machine tape) that one includes is like increasing the space complexity one allows. And increasing the complexity of the rule (as measured in the number of bits needed to specify it) is like increasing the algorithmic complexity one allows. The ruliad is what is obtained by taking all these computational resource measures to infinity. 
And a critical claim is that regardless of how this is done, the final ruliad construct one gets will always—at least in some sense—be the same. There will be many ways to coordinatize it, or to sample it, but the claim is that it’s always the same object that one’s dealing with. And ultimately the reason for this is the Principle of Computational Equivalence. Because it implies that whatever “computational parametrization” or “computational description language” one uses for the ruliad, one will almost always get something that can be viewed as “computationally equivalent”. 
We’ve talked about building up the ruliad using Turing machines. But what about other models of computation—like cellular automata or register machines or lambda calculus? As soon as there’s computation universality we know that we’ll get results that are at least in principle equivalent, because in a sense there’s only a “finite translation cost” associated with setting up an interpreter from one model of computation to another. Or, put another way, we can always emulate the application of the rule for one system by just a finite number of rule applications for the other system.
But from computation universality alone we have no guarantee that there won’t be “extreme deformations” introduced by this deformation. What the Principle of Computational Equivalence says, however, is that almost always the deformations won’t have to be extreme. And indeed we can expect that particularly when multiple rules are involved, there’ll be rapid convergence almost always to a kind of “uniform equivalence” that ensures that the final structure of the ruliad is always the same.
But the Principle of Computational Equivalence appears to say still more about the ruliad: it says that not only will the ruliad be the same independent of the “computational basis” used to construct it, but also that there’ll be a certain uniformity across the ruliad. Different “regions of the ruliad” might involve different specific rules or different patterns of their application. But the Principle of Computational Equivalence implies that almost always the computations that happen will be equivalent, so that—at least at a certain scale—the structure associated with them will also be equivalent.
Knowing that the ruliad contains so many different computations, one might imagine that it would show no particular uniformity or homogeneity. But the Principle of Computational Equivalence seems to imply that it necessarily does, and moreover that there must be a certain coherence to its structure—that one can interpret (in the style of Grothendieck’s hypothesis) as an inevitable emergent geometry.
An individual computation corresponds to a path in the ruliad, going from its “input state” to its “output state”. In an ordinary deterministic computation, the path is restricted to always use the same rule at each step. In a nondeterministic computation, there can be different rules at different steps. But now we can formulate things like the P vs. NP problem essentially in terms of the geometry of the ruliad.
Here’s a picture of the same finite Turing-machine-based approximation to the ruliad as above—but now with the paths that correspond to deterministic Turing machine computations marked in red: 
	
The P vs. NP problem basically asks roughly whether the deterministic computations (shown here in red) will eventually “fill the ruliad”, or whether the general nondeterministic computations that are part of the ruliad will always “reach further”. Once again, there are many complicated and subtle issues here. But it’s interesting to see how something like the P vs. NP problem might play out in the ruliad. 
In physics (and mathematics), we as human observers tend to sample the ruliad in a coarse-grained way, “noticing” only certain aspects of it. So is there an analog of this in computation theory—perhaps associated with certain characteristics of the “computation-theoretic observer”? There’s a potential answer, rather similar to what we’ve already seen in both physics and mathematics. 
The basic point is that in computation theory we tend to study classes of computations (say P or NP) rather than individual computations. And in doing this we are in a sense always conflating many different possible inputs and possible outputs—which we assume we do in a computationally bounded way (e.g. through polynomial-time transformations, etc.) 
Another thing is that we tend to focus more on the “experience of the end user” than the detailed activities of the “programmer”. In other words, we’re concerned more with what computational results are obtained, with what computational resources, rather than on the details of the program constructed to achieve this. Or, put another way, we tend to think about computation in terms of things like the successive evaluation of functions—and we conflate the different paths by which this is achieved.
Most likely this means that there are “effective laws” that can be derived in this computational view of the ruliad, analogous to laws of physics like general relativity. So what might some other analogies be?
A computation, as we’ve mentioned, corresponds to a path in the ruliad. And whenever there’s a possibility for an infinite path in the ruliad, this is a sign of undecidability: that there may be no finite way to determine whether a computation can reach a particular result. But what about cases when many paths converge to a point at which no further rules apply, or effectively “time stops”? This is the analog of a spacelike singularity—or a black hole—in the ruliad. And in terms of computation theory, it corresponds to something decidable: every computation one does will get to a result in finite time.
One can start asking questions like: What is the density of black holes in rulial space? If we construct the ruliad using Turing machines, this is basically analogous to asking “What’s the density of halting Turing machines (+initial conditions) in rulial space?” And this is essentially given by Chaitin’s Ω. 
But so is there some number Ω that we can just compute for the ruliad? Well, actually, no. Because the undecidability of the halting problem makes Ω noncomputable. One can get approximations to it, but—in the language of the ruliad—those will correspond to using particular samplings or particular reference frames. Or in other words, even the perceived density of “decidability black holes” in the ruliad depends on features of the observer.
What’s beyond the Ruliad?
In our Physics Project we usually talk of the universe “evolving through time” (albeit with many entangled threads of history). But if the ruliad and its structure is a matter of formal necessity, doesn’t that mean that the whole ruliad effectively “already exists”—“outside of time”? Well, in some sense it does. But ultimately that would only be relevant to us if we could “look at the ruliad from the outside”.
And as observers like us within the ruliad, we necessarily have a different perception. Because our consciousness—with its computational boundedness—only gets to sample a certain sequence of pieces of the ruliad. If it were not for computational irreducibility, we might get to “jump around” in time. But computational irreducibility, together with our own computational boundedness, implies that our perception must necessarily just experience the passage of time through an irreducible process of computation.
In other words, while in some sense the ruliad may all “already be there” when viewed from the outside, our own perception of it “from the inside” is necessarily a progressive one, that effectively corresponds to the passage of time.
Could we experience the ruliad differently, even while being computationally bounded? If we think of the ruliad as a graph, then our usual “sequence of configurations of space at successive times” way of experiencing the ruliad is like a breadth-first traversal. But could we for example instead do a depth-first traversal, exploring all time before investigating different parts of space? (And, yes, something like this can happen in general relativity near an event horizon, or in connection with timelike singularities.)
Later, we’ll discuss different ways to perceive the ruliad and the universe. But it seems to be a feature of anything we might call a coherent observer that there needs to be some form of progression in the perception. And so while we might not call it the passage of time, there’ll still be some way in which our exploration of the ruliad has a computationally irreducible process underneath.
A very important claim about the ruliad is that it’s unique. Yes, it can be coordinatized and sampled in different ways. But ultimately there’s only one ruliad. And we can trace the argument for this to the Principle of Computational Equivalence. In essence there’s only one ruliad because the Principle of Computational Equivalence says that almost all rules lead to computations that are equivalent. In other words, the Principle of Computational Equivalence tells us that there’s only one ultimate equivalence class for computations.
But what if we just imagine a “hypercomputation” not in that class? For example, imagine a hypercomputation (analogous, for example, to an oracle for a Turing machine) that in a finite number of steps will give us the result from an infinite number of steps of a computationally irreducible process. Such a hypercomputation isn’t part of our usual ruliad. But we could still formally imagine a hyperruliad that includes it—and indeed we could imagine a whole infinite hierarchy of successively larger and more powerful hyperruliads.
But it’s a fundamental claim that we’re making—that can be thought of as a matter of natural science—that in our universe only computation can occur, not hypercomputation. 
At a purely formal level, there’s nothing wrong with hyperruliads. They exist as a matter of formal necessity just like the ordinary ruliad does. But the key point is that an observer embedded within the ruliad can never perceive a hyperruliad. As a matter of formal necessity there is, in a sense, a permanent event horizon that prevents anything from any hyperruliad from affecting anything in the ordinary ruliad. 
So now we can be a bit more precise about our statement that “hypercomputation doesn’t happen in our universe”. Really we should say that we assert that we as observers operate purely computationally and not hypercomputationally. And this means that we are embedded within the ordinary ruliad, and not the hyperruliad.
Yes, we could imagine some other entity that’s embedded within the hyperruliad, and perceives what it considers to be the universe to operate hypercomputationally. But in a statement that’s in a sense more “about us” than “about the universe”, we assert that that can’t be us, and that we in a sense live purely within the ruliad—which means that for us the Principle of Computational Equivalence holds, and we perceive only computation, not hypercomputation.
Communicating across Rulial Space
What observers can there be embedded in the ruliad, and how should we characterize them? In physical spacetime we’re used to characterizing observers by their locations in physical space and by things like the spacetime reference frames they construct. And it’s very much the same for observers in the ruliad: we can characterize them by where they are in rulial space, and what rulial reference frames they use.
The Principle of Computational Equivalence tells us that it’s almost always possible to “encode” one “model of how the ruliad works” in any other model—effectively just by setting up a program that emulates the rules for one model using the rules for the other model. But we can think of these different models as being associated with different possible observers in the ruliad. 
In other words, we can say that observers “at different places in rulial space” (or “using different rulial reference frames”) are using different description languages for what’s happening in the ruliad. And when an observer “moves” in rulial space, they’re effectively doing a translation from one description language to another. (And, yes, there’s a maximum rate of motion ρ in rulial space—which is the rulial analog of the speed of light—and which is effectively determined by the fundamental processing speed of the universe.)
So far this might all seem quite abstract. But there are immediate, everyday examples that effectively correspond to being at different places in rulial space. A simple concrete one is computers with different instruction sets. Another one is different brains with different consciousnesses.
We can think of a single human consciousness as having a certain thread of experience of the universe. Part of that experience is determined by the physical location of the consciousness and by the sensory apparatus with which it samples the world. But part is determined by the “internal description language” that it uses. And inevitably this internal description language depends both on the detailed physiology of the brain in which it is implemented, and on the past history of experiences that have “defined its way of looking at the world”. In the analogy of artificial neural networks, different networks will tend to have different “internal representations” because this depends not only on the network architecture, but also on the particular training data that the network has “experienced”.
Why can’t one human consciousness “get inside” another? It’s not just a matter of separation in physical space. It’s also that the different consciousnesses—in particular by virtue of their different histories—are inevitably at different locations in rulial space. In principle they could be brought together; but this would require not just motion in physical space, but also motion in rulial space.
But why then do different consciousnesses seem to have compatible views about “what happens in the universe”? Essentially this can be seen as a consequence of rulial relativity—which in turn depends on the inevitable causal invariance of the ruliad, which follows from the Principle of Computational Equivalence. There are certainly many issues to be worked out, but basically what seems to be going on is that because of causal invariance, different rulial reference frames will ultimately yield the same rulial multiway causal graphs, and therefore the same “fundamental description of reality”.
We’ve talked about different consciousnesses. But what about just “different ways of thinking”? Well, it’s definitely more than an analogy to say that different ways of thinking correspond to different positions in rulial space. If there’s lots of common history then there’ll be common ancestry in the rulial multiway graph and one will necessarily end up close in rulial space. But without common history, one can end up with different description languages—or different ways of thinking—that are not nearby in rulial space.
In physical space we expect to effectively use momentum to move our location. And it’s potentially a bizarrely similar story in rulial space. In our models of fundamental physics, energy and momentum are essentially related to the density of activity (i.e. elementary updating events) in physical space. And we can similarly define a rulial analog of energy and momentum in terms of activity in rulial space. But it’s exactly this activity that provides connections between different parts of rulial space, or in effect “enables motion” in rulial space. 
In other words, if you want to move in rulial space, you can do it by putting in the appropriate computational work to change your conceptual point of view (or, essentially equivalently, your language for describing things). So what about curvature (or the analog of gravity) in rulial space—say generated through an analog of Einstein’s equations from density of activity in rulial space? Presumably this relates to the difficulty—or time it takes—to get from one place in rulial space, and one way of thinking, to another. And conceivably things like “paradigm shifts” between different ways of thinking might be associated with features of rulial space like event horizons.
But let’s say you’re at one place in rulial space, and you want to get to another—or at least “send a signal” there. A typical microscopic change at one point in rulial space will tend to just “spread out in all directions” and “decay quickly”. But if you want to “coherently communicate”, you need some kind of structure that will persist as it propagates through rulial space. And by analogy with the case of physical space, what this presumably means is that you effectively need a “rulial particle”. 
In terms of the ruliad, a rulial particle would presumably be some kind of “topological obstruction” or “topologically stable structure” that is at any moment effectively localized in rulial space and maintains its identity as it propagates across rulial space. But what might a rulial particle be in more everyday terms?
Potentially it’s like what we’d normally consider a concept—or something to which we might assign a word in human language. If we have ways of thinking—or consciousnesses—whose details are different, the issue is what will be robust enough to be able to be transported between them. And what everyday experience seems to suggest is that the answer is concepts. Even though one might have a slightly different way of thinking, what one calls “a fish” (or essentially, the concept of a fish) is something that can still robustly be communicated.
It’s interesting to notice that for an observer like us, there seem to be only a finite set of types of “elementary particles” that exist in physical space. And perhaps that’s not unrelated to the fact that observers like us also seem to imagine that there are in some sense only a finite number of “basic concepts” (associated, say, with distinct words in human languages). There’s lots more detail that exists in rulial space—or in the ruliad—but for observers like us, with our type of way of sampling the ruliad, these might be core coherent structures that we perceive. 
So Is There a Fundamental Theory of Physics? 
The concept of the ruliad arose from our efforts to find a fundamental theory of physics. But now that we know about the ruliad, what does it tell us about a fundamental theory?
At the outset, we might have imagined that the end point of our project would be the identification of some particular rule of which we could say “This is the rule for the universe”. But of course then we’d be faced with the question: “Why that rule, and not another?” And perhaps we would imagine just having to say “That’s something that you have to go beyond science to answer”.
But the ruliad implies a quite different—and in my view ultimately much more satisfying—picture. The ruliad itself is a construct of abstract necessity—that in a sense represents the entangled behavior of all possible rules for the universe. But instead of imagining that some particular rule out of all these possibilities is “picked from outside” as “the choice for our universe”, what we suppose is that—as observers embedded within the ruliad—we’re the ones who are implicitly picking the rule by virtue of how we sample and perceive the ruliad.
At first this might seem like it’s a wimp out. We want to know how our universe works. Yet we seem to be saying “we just pick whatever rule we feel like”. But that’s not really the story at all. Because in fact observers that are even vaguely like us are in effect deeply constrained in what rules they can attribute to the universe. There’s still some freedom, but a fundamental result is that for observers like us it seems to be basically inevitable that any rule we can pick will on a large scale reproduce the central known general laws of physics, in particular general relativity and quantum mechanics.
In other words, for observers generally like us it’s a matter of abstract necessity that we must observe general laws of physics that are the ones we know. But what about more specific things, like the particular spectrum of elementary particles, or the particular distribution of matter in the universe? It’s not clear how far “the general” goes—in other words, what is a matter of abstract necessity purely from the structure of the ruliad and general features of observers like us. 
But inevitably at some point we will run out of “the general”. And then we’ll be down to specifics. So where do those specifics enter? Ultimately they must be determined by the details of how we sample the ruliad. And a prominent part of that is simply: Where in the ruliad are we? We can ask that about our location in physical space. And we can also ask it about our location in rulial space.
What does all this mean? At some level it’s saying that the way we are as observers is what makes us attribute certain rules to our universe. The ruliad is in a sense the only thing that fundamentally exists—and in fact its existence is a matter of abstract necessity. And our universe as we experience it is some “slice of the ruliad”, with what slice it is being determined by what we’re like as observers.
Let’s look at the logical structure of what we’re saying. First, we’re describing the ruliad, which at the outset doesn’t have anything specifically to do with physics: it’s just a formal construct whose structure is a matter of abstract necessity, and which relates as much to mathematics as it does to physics. But what “puts the physics in” is that we in effect “live in the ruliad”, and our perception of everything is based on “experiencing the ruliad”. But that experience—and the effective laws of physics it entails—inevitably depends on “where we are in the ruliad” and how we’re able to sample it.
And this is where our pieces of “falsifiable natural science” come in. The first “assertion of natural science” that we make is that we are embedded only within the ordinary ruliad, and not a hyperruliad—or in other words that our experience encompasses only computation, and not hypercomputation. 
This is closely related to a second assertion, which may in fact be considered to subsume this: that we are computationally bounded observers, or, in other words, that our processes of perception involve bounded computation. Relative to the whole ruliad—and all the computation it entails—we’re asserting that we as observers occupy only a tiny part.
There’s one more assertion as well, again related to computational boundedness: that we as observers have a certain coherence or persistence. In general the ruliad contains all sorts of wild and computationally irreducible behavior. But what we’re asserting is that that part of the ruliad that is associated with us as observers has a certain simplicity or computational reducibility: and that as we evolve through time or move in space, we somehow maintain our identity. 
These assertions seem very general, and in some ways almost self-evident—at least as they apply to us. But the important and surprising discovery is that they alone seem to lead us inexorably to crucial features of physics as we know it.
Where does this physics “come from”? It comes partly from the formal structure of the ruliad, and formal features of the multicomputational processes it involves. And it comes partly from the nature of us as observers. So if we ask “Why is the physics of our universe the way it is?”, an important part of the answer is “Because we observe the universe the way we do”. 
One might imagine that in some sense physics would give us no choice about how we observe the universe. But that’s not the case. Because in the end our “observation” of the universe is about the “abstract conceptual model” we build up for the universe. And, yes, that’s certainly informed by the particular sensory apparatus we have, and so on. But it’s something we can certainly imagine being different.
We can think of ourselves as using some particular description language for the universe. The structure of that language is constrained by the assertions we gave above. But within such a description language, the laws of physics necessarily work out the way they do. But if we chose a different description language, we’d end up with different laws of physics.
Much of our perception of the universe is based on our raw biological structure—the way our sensory organs (like our eyes) work, as well as the way our brains integrate the inputs we get. But that’s not all there is to it. There’s also a certain base of knowledge in our civilization that informs how we parse our “raw perception”—and in effect what description language we use. Once we have the idea of periodic behavior, say, we can use it to describe things that we’d previously have to talk about in a less “economical” way.
But what if our knowledge changed? Or we had different sensory capabilities? Or we used technology to integrate our sensory input in different ways? Then we’d be able to perceive and describe the universe in different ways.
One’s first impression might be that the ruliad effectively contains many possible “parallel universes”, and that we have selected ourselves into one of these, perhaps as a result of our particular characteristics. But in fact the ruliad isn’t about “parallel universes”, it’s about universes that are entangled at the finest possible level. And an important consequence of this is that it means we’re not “stuck in a particular parallel universe”. Instead, we can expect that by somehow “changing our point of view”, we can effectively find ourselves in a “different universe”.
Put another way, a given description of the universe is roughly represented by being at a certain location in rulial space. But it’s possible to move in rulial space—and end up with a different description, and different effective laws for the universe.
But how difficult is motion in rulial space? It could be that some impressive future technology would allow us to “move far enough” to end up with significantly different laws of physics. But it seems more likely that we’d be able to move only comparatively little—and never be able to “escape the box” of things like computational boundedness, and coherence of the observer.
Of course, even changing a little might lead us to different detailed laws of physics—say attributing a different mass to the electron, or a different value of the electromagnetic coupling constant α. But actually, even in traditional physics, this is already something that happens. When viewed at different energy scales—or in a sense with different technology—these quantities have different effective values (as characterized by the renormalization group). 
At first it might seem a little strange to say that as our knowledge or technology change, the laws of physics change. But the whole point is that it’s really our perceived laws of physics. At the level of the raw ruliad there aren’t definite laws of physics. It’s only when we “sample our slice” of the ruliad that we perceive definite laws.
What does all this mean operationally for the search for a fundamental theory of physics? At some level we could just point to the ruliad and declare victory. But this certainly wouldn’t give us specific predictions about the particulars of our perceived universe. To get that we have to go further—and we have to be able to say something about what “slice of the ruliad” we’re dealing with. But the good news is that we don’t seem to have to make many assumptions about ourselves as observers to be able to identify many physical laws that observers like us should perceive. 
So can we ever expect to nail down a single, specific rule for the universe, say one a particular observer would attribute to it? Given our characteristics as observers, the answer is undoubtedly no. We’re simply not that small in rulial space. But we’re not that big, either. And, importantly, we’re small enough that we can expect to “do science” and consider the universe to “behave in definite ways”. But just as in physical space we’re vastly larger than the scale associated with the atoms of space, so similarly we’re also undoubtedly vastly larger in rulial space than the individual components of the ruliad—so we can’t expect our experience to all be “concentrated in one thread” of the ruliad, following one particular rule. 
As we discussed above, by doing experiments we can use scientific inference to attempt to localize ourselves in rulial space. But we won’t be able to do enough to say “from our point of view, the universe is operating according to this one specific rule, and not another”. Instead, there’ll be a whole collection of rules that are “good enough”, in the sense that they’ll be sufficient to predict the results of experiments we can realistically do.
People have often imagined that, try as we might, we’d never be able to “get to the bottom of physics” and find a specific rule for our universe. And in a sense our inability to localize ourselves in rulial space supports this intuition. But what our Physics Project seems to rather dramatically suggest is that we can “get close enough” in rulial space to have vast predictive power about how our universe must work, or at least how observers like us must perceive it to work. 
Alien Views of the Ruliad
We’ve discussed how “observers like us” will necessarily “parse the ruliad” in ways that make us perceive the universe to follow the laws of physics as we know them. But how different could things get? We have a definite sense of what constitutes a “reasonable observer” based on our 21st-century human experience—and in particular our biology, our technology and our ways of thinking. 
But what other kinds of observers can we imagine? What about, for example, animals other than humans—in particular say ones whose sensory experience emphasizes olfaction or echolocation or fluid motion? We can think of such animals as operating in a different rulial reference frame or at a different place in rulial space. But how far away in rulial space will they be? How similar or not will their “world views” (and perceived laws of physics) be to ours? It’s hard to know. Presumably our basic assertions about computational boundedness and coherence still apply. But just how the specifics of something like sequentialization in time play out, say, for an ant colony, seems quite unclear.
Maybe one day we’ll be able to systematically “think like other animals”. But as of now we haven’t been able to “travel that far” in rulial space. We’ve quite thoroughly explored physical space, say on the surface of our planet, but we haven’t explored very far at all in rulial space. We don’t have a way to translate our thinking into some kind of “thinking differently”—and we don’t, for example, have a common language to get there.
There’s often an assumption (a kind of “human exceptionalism”) that if it wasn’t for details of the human experience—like brains and words—then we’d necessarily be dealing with something fundamentally simpler, that could not, for example, show features that we might identify as intelligence. But the Principle of Computational Equivalence tells us this isn’t correct. Because it says that there’s a certain maximal computational sophistication that’s achieved not just by us humans but also by a vast range of other systems. The restrictions of what we’ve chosen to study (in science and elsewhere) have often made us miss this, but in fact computational sophistication—and the direct generalization of our notion of intelligence that’s associated with it—seems quite ubiquitous across many different kinds of systems.
So can those other kinds of systems act as “observers like us”? To do so, they need not just computational sophistication, but also a certain alignment with the features we have that lead to our coherent thread of “conscious experience”. And even given that, to actually “connect with” such systems, we need to be able to reach far enough in rulial space to sufficiently make a translation. 
Imagine the weather (sometimes said to “have a mind of its own”). It’s got plenty of computational sophistication. But is there any sense in which it sequentializes time like we do? Or can one only think of all those different parts of our atmosphere “running in their own time”? To know things like this, we effectively have to have a way to “translate” from the operation of the weather to our (current) way of thinking.
And in some sense we can consider the whole enterprise of natural science as being an effort to find a method of translation—or a common language—between nature and our way of thinking. 
We as observers in effect trace out particular trajectories in rulial space; the challenge of natural science is to “reach out” in rulial space and “pull in” more of the ruliad; to be able to define a way to translate more parts of the ruliad to our processes of thinking. Every time we do an experiment, we can think of this as representing a moment of “connection” or “communication” between us and some aspect of nature. The experiment in effect defines a small piece of “common history” between us and nature—which helps “knit together” the parts of rulial space associated with us and with nature. 
One of the great mysteries of science has been why—in the vastness of physical space—we’ve never detected something we identify as “alien intelligence”, or an “alien civilization”. We might have thought that it was because we humans have either achieved a unique pinnacle of intelligence or computational ability—or have fundamentally not gotten far enough. But the Principle of Computational Equivalence explodes the idea of this kind of cosmic computational pecking order.
So what could actually be going on? Thinking in terms of the ruliad suggests an answer. Our radio telescopes might be able to detect signals from far away in physical space. But our putative aliens might not only live far away in physical space, but also in rulial space.
Put another way, the “alien civilization” might be sampling aspects of the ruliad—and in effect the universe—that are utterly different from those we’re used to. That different sampling might be happening right down at the level of atoms of space, or it might be that the rulial distance from us to the aliens is small enough that there’s enough “shared description language” that the alien civilization might rise to the level of seeming like some kind of “noise” relative to our view of “what’s important in the universe”.
We might wonder how far apart what we could consider “alien civilizations” would be in physical space. But what we now realize is that we also have to consider how far apart they might be in rulial space. And just like in exploring physical space we can imagine building better spacecraft or better telescopes, so also we can imagine building better ways to reach across rulial space. 
We’re so used to physical space that it seems to us very concrete to reach across it. Of course, in our Physics Project, things like motion in physical space end up—like everything else—being pure computational processes. And from this point of view, reaching across rulial space is ultimately no more abstract—even though today we would describe it in terms of “doing (abstract) computations” rather than “moving in space”.
Relative to our own physical size, the universe already seems like a vast place. But the full ruliad is even incredibly more vast. And we are likely much tinier in rulial space relative to the whole universe than we are in physical space. From the Principle of Computational Equivalence we can expect that there’s ultimately no lack of raw computational sophistication out there—but thinking in terms of the ruliad, the issue is whether what’s going on is close enough to us in rulial space that we can successfully see it as an “alien civilization”.
One test of rulial distance might be to ask whether our putative aliens perceive the same laws of physics for the universe that we do. We know that at least the general forms of those laws depend only on what seem to us rather loose conditions. But to get good alignment presumably requires at the very least that we and the aliens are somehow “comparable in size” not only in physical space (and branchial space), but also in rulial space. 
It’s humbling how difficult it is to imagine the universe from the point of view of an alien at a different place in rulial space. But for example if the alien is big compared to us in rulial space, we can say that they’ll inevitably have a version of science that seems to us much “vaguer” than ours. Because if they maintain a coherent thread of experience, they’ll have to conflate more distant paths in rulial space, on which the universe will do things that are “more different” than what we’re used to. (And, yes, there should be rulial analogs of quantum phenomena, associated for example with conflated paths that diverge far in rulial space.)
What would it mean operationally for there to be an alien civilization perhaps nearby in physical space but at a distance in rulial space? Basically the alien civilization will be “operating” in features of the universe that our parsing of the universe just doesn’t pick up. As a simple analogy, our view of, for example, a box of gas might be that it’s something with a certain temperature and pressure. But a different “parsing” of that system might identify a whole world of detailed motions of molecules that with respect to that parsing can be viewed as a vast “alien civilization”. Of course, the situation is much more extreme when it comes to the whole ruliad, and all the paths of history and configurations of atoms of space that it represents.
Relative to the whole ruliad, our civilization and our experience have carved out an extremely tiny piece. And what we’re thinking of as “alien civilizations” might also have carved out their own tiny pieces. And while we’re all “living in the same ruliad”, we might no more be able to detect each other or communicate (and likely very much less) than we can across vast distances in physical space. 
What of the future? The future of our civilization might well be a story of mapping out more of rulial space. If we continue to invent new technology, explore new ideas and generally broaden our ways of thinking and perceiving, we will gradually—albeit in tiny steps—map out more of rulial space. How far can we get? The ultimate limit is determined by the maximum rulial speed. But if we expect to maintain our character as “observers like us”, we’ll no doubt be limited to something much less.
Among other issues, moving in rulial space involves doing computation. (The ultimate scale is set by the “processing power” of the universe—which defines the maximum rulial speed.) But “density of computation” effectively corresponds to a generalized version of mass—and is for example a source of “generalized gravity”. And it could be that to “move any significant distance” in rulial space, we’d have to “experience enough generalized gravity” that we could never maintain things like the kind of coherence we need to be an “observer like us”.
Put another way: yes, it might in principle be possible to “reach out in rulial space” and “contact the rulial aliens”. But it might be that doing so would require us to be so different from the way we currently are that we wouldn’t recognize anything like consciousness or anything that really makes us “identifiably us”. And if this is so, we are in a sense limited to experiencing the ruliad “on our own” from our particular place in rulial space, forever isolated from “alien civilizations” elsewhere in rulial space.
Conceptual Implications of the Ruliad
What does the concept of the ruliad mean for the fundamental way we think about things like science? The typical conception of “what science does” is that it’s about us figuring out—as “objectively” as we can—how the world happens to be. But the concept of the ruliad in a sense turns this on its head. 
Because it says that at some ultimate level, everything is a matter of abstract necessity. And it’s just our “parsing” of it that defines the subject matter of what we call science. We might have thought that the science of the universe was just something that’s “out there”. But what we’re realizing is that instead in some fundamental sense, it’s all “on us”.
But does that mean that there’s no “objective truth”, and nothing that can robustly be said about the universe without “passing it through us”? Well, no. Because what we’ve discovered through our Physics Project is that actually there are quite global things that can (“objectively”) be said about our universe and the laws it follows, as perceived by observers like us. 
We don’t have to know in detail about us humans and the particular ways we perceive things. All we need are some general features—particularly that we are computationally bounded, and that we have a certain persistence and coherence. And this is all it takes to deduce some quite specific statements about how our universe operates, at least as we perceive it.
So in a sense what this means is that there is a large “zone of objectivity”; a large set of choices for how we could be that will still lead us to the same “objective truth” about our universe. But if we go far enough away in our mechanism for “parsing the ruliad”, this will no longer be the case. From our current vantage point, we’d no doubt then be hard-pressed to recognize how we’re “doing the parsing”, but the results we’d get would no longer give us the same laws of physics or general perception of the universe that we’re used to.
This view of things has all sorts of implications for various long-discussed philosophical issues. But it’s also a view that has precise scientific consequences. And these don’t just relate to physics. Because the ruliad is really a general object that represents the entangled behavior of all possible abstract rules. When we think of ourselves as observers embedded within this object, it means that for us things are actualized, and we have what we call physics. But we can also imagine sampling the ruliad in different ways. 
Some of those ways correspond to mathematics (or metamathematics). Some correspond to theoretical computer science. The ruliad is the single object that underlies all of them. And which of them we’re talking about just depends on how we imagine we’re sampling or parsing the ruliad, and how we’re describing what we’re observing.
With this degree of generality and universality, it’s inevitable that the ruliad must be a complicated object; in fact, in a sense it must encapsulate all possible achievable complexity. But what’s important is that we now have a definite concept of the ruliad, as something we can study and analyze.
It’s not simple to do this. The ruliad is at some level an object of great and perhaps supremely elegant abstract regularity. But for us to get any concrete handle on it and its structure, we need to break it down into some kind of “digestible slices” which inevitably lose much of its abstract regularity. 
And we’re just at the beginning of seeing how best to “unpack” and “pick through” the ruliad. With explicit computations, we can only chip away at the very simplest approximations to the ruliad. In a sense it’s a tribute to the naturalness and inevitability of the ruliad that it’s so closely related to some of the most advanced abstract mathematical methods we know so far. But again, even with these methods we’re barely scratching the surface of the ruliad and what it contains.
The theoretical exploration of the ruliad will be a long and difficult journey. But the incredible generality and universality of the ruliad means that every piece of progress is likely to have exceptionally powerful consequences. In some sense the exploration of the ruliad can be seen as the encapsulated expression of everything it means to do theoretical investigation: a kind of ultimately abstract limit of theoretical science and more.
For me, the ruliad in a sense builds on a tower of ideas, that include the computational paradigm in general, the exploration of the computational universe of simple programs, the Principle of Computational Equivalence, our Physics Project and the notion of multicomputation. But even with all of these it’s still a significant further jump in abstraction. And one whose consequences will take considerable time to unfold.
But for now it’s exciting to have at least been able to define this thing I call the ruliad, and to start seeing some of its unprecedentedly broad and deep implications. 
Appendix: The Case of the “Multiplicad”
As a very simple example of something like the ruliad, we can consider what we might call the “multiplicad”: a rulial multiway system based on integers, in which the rules simply multiply by successive integers: 
	
(Note that this kind of pure multiplication is presumably not computation universal, so the limiting object here will not be a coordinatization of the actual full ruliad.)
Just like with the full ruliad, there are many different “directions” in which to build up the multiplicad. We could allow as many multipliers and steps as we want, but limit the total size of numbers generated, here say to 30:
	
As an alternative, we can limit the number of multipliers s, say to . Then the multiplicad would build up like this:
	
In the pictures we’ve drawn so far, we’re effectively always deduplicating different occurrences of the same integer. So, for example, the integer 12 can be generated as 1 × 3 × 4 or 1 × 6 × 2 or 1 × 3 × 2 × 2, etc. And in principle we could show each of these “different 12s” separately. But in our deduplicated graph, only a single 12 appears—with the different possible decompositions of 12 being reflected in the presence of multiple paths that lead to the 12.
Sometimes the structure we get is richer—if much bigger—when we don’t immediately do deduplication. For example, if we allow any number of multipliers (i.e. take ) then after just 1 step we will get all integers—and if we do deduplication, then this will be the end of our graph, because we “already have all the integers”. But if we don’t do deduplication, we’ll get a slightly more complicated picture, that begins like this:
	
The “topological” structure of this graph is now straightforward, but its “labeling” with numbers is less so—and if we ask, for example, where a particular number appears after t steps, this can be more complicated. 
Imagine that we are looking only at the subtrees associated with up to s multipliers at the first step—or, equivalently, that we are looking at the rulial multiway system “truncated” with only s rules. Which numbers will appear after steps? The answer is that it will be precisely those numbers that show up in an s × s multiplication table where we start from :
	
Clearly no primes appear here, but some numbers can appear multiple times (e.g. 12 appears 4 times). In general, the number of times that the number will show up is the number of proper divisors it has, or DivisorSigma[0, n]–2: 
	
	
We can continue this, to ask how many times a given number n will occur at a particular step t: 
	
	
	
We can think of these results as being determined by the number of times that n appears in an s × s × s… (t times) multiplication array. Alternatively, to know the results for a given number n, we can look at all the ways n can be decomposed into factors. For , for example, we would have:
	
And from this we can deduce that 12 appears once at (i.e. with 1 factor), 4 times at (i.e. with 2 factors) and 3 times at (i.e. with 3 factors). 
The full multiplicad is formed by taking the limits and (as well as what is essentially the limit for an infinite set of possible initial conditions). As we can see, our “finite perception” of the multiplicad will be different depending on how we sample it in s and t. 
As an example, let’s consider what happens for given s as a function of t. For , we simply have powers of 2:
	
For , where can multiply by both 2 and 3, we get:
	
In studying multiway systems, it is often of interest to ask about the growth rates of the number of states reached over the course of t steps (i.e. the growth rates of volumes of geodesic balls). In the case , the number of states reached by step t is just t. For , it’s the triangular numbers t (t – 1)/2:
	
Here are some results for larger s:
	
Each of these sequences is generated by a linear recurrence relation with a kernel given by a sequence of signed binomial coefficients. The values for successive t can be represented by polynomials:
	
The leading term in the growth of number of states is then determined by the orders of these polynomials, which turn out to be just PrimePi[s]:
	
	
For the case , we saw above that the multiway graph essentially forms a simple 2D grid. For larger s, the graph becomes more complicated, though it still approximates a grid—but in dimension PrimePi[s]. (The reason PrimePi[s] appears is that in a sense the combining of primes less than s are the largest “drivers” of structure in the multiway graph.)
In our general analysis of multiway graphs, it is common to consider branchial graphs—or for a rulial multiway system what we can call rulial graphs—obtained by looking at a slice of the multiway graph, effectively for a given t, and asking what states are connected by having a common ancestor. The results for are rather trivial (here shown for , 2, 3):
	
For we get:
	
And for we have:
	
In a sense these pictures show how numbers in the multiplicad can be “laid out in rulial space”. For , the “large-t graph” has a very linear form
	
and the numbers that appear “from left to right” are arranged more or less in numerical order:
	
For , the result is a 2D-like structure:
	
And again the numbers that appear are roughly arranged in a kind of “numerical sequence”:
	
We can then think of this as suggesting that the multiplicad “generates a rulial space” that can be approximately coordinatized purely by the values of the numbers that appear in it. How all this works in the limit is not clear, though somehow the “1D” sequence of numerical values presumably “snakes through” PrimePi[s]-dimensional space as some kind of approximation to a space-filling curve. 
It should be noted that we’ve only considered one particular way of sampling the rulial multiway graph as a function of t. In general there are many different possible foliations that could be used, all of them giving us in effect a different view of the multiplicad, from a different “reference frame”.
As mentioned at the beginning, the multiplicad is presumably not on its own capable of giving us the full ruliad. But if we change the underlying rules—probably even just inserting addition as well as multiplication—we’ll potentially get a system that is capable of universal computation, and which can therefore generate the full ruliad. Needless to say, the particular representation of the ruliad obtained by the kind of “numerical processes” that we’ve used here may be utterly different from any representation that we would recognize from our perception of the physical universe.
Thanks &amp; Note
Thanks for discussions of various aspects of the ruliad to Xerxes Arsiwalla, James Boyd, Elise Cawley, Hatem Elshatlawy, Jonathan Gorard and Nik Murzin. Thanks also to Ed Pegg and Joseph Stocke for input about the multiplicad. A new paper by Xerxes Arsiwalla and Jonathan Gorard discusses in a more technical way some ideas and results related to the ruliad.
The Concept of the Ruliad—Stephen Wolfram Writings
https://writings.stephenwolfram.com/2021/11/the-concept-of-the-ruliad/
https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/
Exploring Rulial Space: The Case of Turing Machines—Wolfram Physics Bulletins
2020-06-09
Stephen Wolfram
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<li><a href="#2020-06-09-related-livestreams">Related livestreams</a></li>
<li><a href="#2020-06-09-related-notebooks">Related notebooks</a></li>
<li><a href="#2020-06-09-video-worklogs">Video work logs</a></li>
<li><a href="#2020-06-09-related-tweets">Related tweets</a></li>
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<h2>Generalized Physics and the Theory of Computation</h2>
<p>Let’s say we find a rule that reproduces physics. A big question would then be: “Why this rule, and not another?” I think there’s a <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/#why-this-universe-the-relativity-of-rules">very elegant potential answer</a> to this question, that uses what we’re calling <a href="https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/multiway-systems-in-the-space-of-all-possible-rules/">rule space relativity</a>—and that essentially says that there isn’t just one rule: actually all possible rules are being used, but we’re basically picking a reference frame that makes us attribute what we see to some particular rule. In other words, our description of the universe is a sense of our making, and there can be many other—potentially utterly incoherent—descriptions, etc.</p>
<p>But so how does this work at a more formal level? This bulletin is going to explore one very simple case. And in doing so we’ll discover that what we’re exploring is potentially relevant not only for questions of “generalized physics”, but also for fundamental questions in the theory of computation. In essence, what we’ll be doing is to study the structure of spaces created by applying all possible rules, potentially, for example, allowing us to “geometrize” spaces of possible algorithms and their applications.</p>
<p>In our models of physics, we begin by considering <a href="https://www.wolframphysics.org/technical-introduction/additional-material/appendix-graph-types/">spatial hypergraphs</a> that describe relations between “atoms of space”. Then, looking at all possible ways a given rule can update these hypergraphs, we form what we call a <a href="https://www.wolframphysics.org/technical-introduction/additional-material/appendix-graph-types/">multiway graph</a>. The transversals of this graph define what we call <a href="https://www.wolframphysics.org/technical-introduction/additional-material/appendix-graph-types/">branchial space</a>, in which we can see the pattern of entanglements between quantum states. </p>
<p>But there’s also a third level we can consider. Instead of just forming a multiway graph in which we do all possible updates with a given rule, we form a rulial multiway (or “ultramultiway”) graph in which we follow not only all possible updates, but also all possible rules. The transversals to this rulial multiway graph define what we call rulial space. <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-phenomenon-of-causal-invariance/">Causal invariance</a> in the rulial multiway graph then implies “rule space relativity” which is what allows us to use different possible reference frames to describe the universe. </p>
<p>In the end, the <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> implies a certain invariance to the limiting structure of the rulial multiway graph, independent of the particular parametrization of the set of possible rules. But for the sake of understanding rulial space and the rulial multiway graph—and getting intuition about these—this bulletin is going to look at a specific set of possible rules, defined by <a href="https://www.wolframscience.com/nks/p78--turing-machines/">simple Turing machines</a>.</p>
<p>The rules we’ll use aren’t a good fit for describing our universe. But the comparative simplicity of their structure will help us in trying to elucidate some of the complexities of rulial space. In addition, using Turing machines will make it easier for us to make contact with the theory of computation, where Turing machines are standard models.</p>
<h2>Turing Machines</h2>
<p>Here’s a representation of the rule for a particular <a href="https://www.wolframscience.com/nks/p78--turing-machines/">2-state (<em>s</em> = 2), 2-color (<em>k</em> = 2) Turing machine</a>:</p>
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<p>The pointer represents the “head” of the Turing machine and its orientation represents the state of the head. Here’s what this particular Turing machine does over the course of a few steps starting from a “blank tape” (i.e. all squares white):</p>
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<p>There are (2 <em>s k</em>)<sup><em>sk</em></sup> possible <em>s</em>-state <em>k</em>-color Turing machines, or 4096 <em>s</em> = 2, <em>k</em> = 2 machines. Here are all the distinct behaviors that occur (up to left-right reflection) in these 4096 machines, starting from a blank tape:</p>
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</p>
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<p>But note that all these are deterministic Turing machines, in the sense that, for a given machine, it is always the same rule that is applied at every step. But what about <a href="https://www.wolframscience.com/nks/p767--undecidability-and-intractability/">non-deterministic Turing machines</a>? The idea here is to allow several different rules at any given step. </p>
<p>As a simple example, consider the pair of rules:</p>
<div>
<p>
</p>
</div>
<p>To show the possible evolution histories, we can construct a multiway system:</p>
<div>
<p>
</p>
</div>
<p>Each arrow represents an application of one of the two possible rules. A non-deterministic Turing machine is usually considered to follow a particular path, corresponding to a particular evolution history. Then a typical question is whether there exists some path that leads to a particular final state (which might be viewed as the solution to some particular problem). (In a <a href="https://writings.stephenwolfram.com/2020/04/the-wolfram-physics-project-the-first-two-weeks/#quantumcomputing">quantum Turing machine</a>, one considers collections of multiple states, viewed as being in a quantum superposition.)</p>
<p>In what we’re going to be doing here, we want to study the “extreme case” of non-determinism—where at every step, every possible Turing machine rule (at least with a given <em>s</em>, <em>k</em>) is applied. Then we’ll be interested in the full rulial multiway graph that’s created.</p>
<h2>The Rulial Multiway Graph</h2>
<p>Start with a blank tape. Then apply all possible 2,2 Turing machine rules. This is the rulial multiway graph that’s formed after 1 step:</p>
<div>
<p>
</p>
</div>
<p>Each individual arrow here represents many possible rules from all the 4096 discussed above. Here’s what happens after 2 steps:</p>
<div>
<p>
</p>
</div>
<p>In layered form, this becomes:</p>
<div>
<p>
</p>
</div>
<p>In creating these rulial multiway graphs, there’s an important simplification we’re able to make. We don’t need to separately apply all 4096 2,2 Turing machine rules to each state at each step—because all that ever matters is the one case of the rule that’s relevant to the particular state one has. There are 4 possible individual cases—and for each of these there are 8 possible outcomes, leading to 32 “micro-rules”:</p>
<div>
<p>
</p>
</div>
<p>After 3 steps, the rulial multiway graph has the form:</p>
<div>
<p>
</p>
</div>
<p>In 3D this becomes:</p>
<div>
<p>
</p>
</div>
<p>In layered form it is:</p>
<div>
<p>
</p>
</div>
<p>After 5 steps, the rulial multiway graph is:</p>
<div>
<p>
</p>
</div>
<p>What about other numbers of states and colors? Here are the rulial multiway graphs after 3 steps for Turing machines with various numbers of states and colors :</p>
<div>
<p>
</p>
</div>
<p>Notice the presence of <em>s</em> = 1 examples. Even with a single possible state, it is already possible to form a nontrivial rulial multiway system. Here are the results for <em>k</em> = 2 after 1 step:</p>
<div>
<p>
</p>
</div>
<p>2 steps:</p>
<div>
<p>
</p>
</div>
<p>6 steps:</p>
<div>
<p>
</p>
</div>
<p>In principle one can also consider the even simpler case of <em>s</em> = <em>k</em> = 1. After 1 step one gets:</p>
<div>
<p>
</p>
</div>
<p>After 2 steps this becomes:</p>
<div>
<p>
</p>
</div>
<p>And after 3 steps:</p>
<div>
<p>
</p>
</div>
<p>As a layered graph rendering, this becomes:</p>
<div>
<p>
</p>
</div>
<p>With <em>s</em> = 2, <em>k</em> = 1 one gets after 1 step:</p>
<div>
<p>
</p>
</div>
<p>After 2 steps this becomes:</p>
<div>
<p>
</p>
</div>
<p>And after 3 steps:</p>
<div>
<p>
</p>
</div>
<p>After 6 steps this becomes:</p>
<div>
<p>
</p>
</div>
<p>At least in the central part of this graph, there are edges going in both directions. Combining these, and treating the graph as undirected, one gets:</p>
<div>
<p>
</p>
</div>
<p>With 3 states (<em>s</em> = 3) there are 3 tracks:</p>
<div>
<p>
</p>
</div>
<p>One can also consider Turing machines in which the head can not only move left or right, but can also stay still. In this case, with <em>s</em> = 2, <em>k</em> = 1 one gets after 1 step:</p>
<div>
<p>
</p>
</div>
<p>After 2 steps this becomes:</p>
<div>
<p>
</p>
</div>
<p>After 4 steps, removing repeated edges, etc. this gives:</p>
<div>
<p>
</p>
</div>
<p>As another generalization, one can consider Turing machines that can move not just one, but also two squares:</p>
<div>
<p>
</p>
</div>
<p>After 2 steps one gets:</p>
<div>
<p>
</p>
</div>
<p>After 3 steps, removing repeated edges, etc. this gives:</p>
<div>
<p>
</p>
</div>
<h2>The Limit of the Rulial Multiway Graph</h2>
<p>What is the limiting structure of the rulial multiway graph after an infinite number of steps? The first crucial observation is that it’s in a sense homogeneous: the structure of the graph around any given node is always the same (i.e. it’s a <a href="https://www.wolframphysics.org/technical-introduction/limiting-behavior-and-emergent-geometry/manifolds-and-model-spaces">vertex-transitive graph</a>). To see why this is true, recall that each node in the graph corresponds to a particular distinct configuration of the Turing machine. This node will lead to all nodes that can be obtained from it by one step of Turing machine evolution. But (assuming the head always moves ±1 square) there are always exactly 2 <em>s k</em> of these. And because we are following all possible “micro-rules”, we can think of ourselves as just “blindly overwriting” whatever configuration we had, so that the structure of the graph is the same independent of what configuration we’re at. </p>
<p>As a simple example, here’s what happens in the <em>s</em> = 2, <em>k</em> = 1 case, after 3 steps:</p>
<div>
<p>
</p>
</div>
<p>There’s some trickiness at the ends, but in the central region we see as expected that each node has exactly 4 successors. If we pick out the subgraph around the blank-tape starting node it has the form:</p>
<div>
<p>
</p>
</div>
<p>In general, the limiting neighborhood of every node is just:</p>
<div>
<p>
</p>
</div>
<p>In the full graph, these neighborhoods are knitted together, giving pieces like:</p>
<div>
<p>
</p>
</div>
<p>Let’s now consider the case <em>s</em> = 1, <em>k</em> = 2. After 4 steps the full graph is:</p>
<div>
<p>
</p>
</div>
<p>Here the neighborhood of the start node is:</p>
<div>
<p>
</p>
</div>
<p>And in the limiting graph, every node will have a local neighborhood with this same structure. If we look at successively larger neighborhoods, the limiting form of these for every node will be:</p>
<div>
<p>
</p>
</div>
<p>Here’s a table of the number of distinct nodes in these successively larger neighborhoods for Turing machines with various values of <em>s</em> and <em>k</em>:</p>
<div>
<p>
</p>
</div>
<p>In the limit <em>t</em>   the number of nodes reached in all cases goes like:</p>
<p></p>
<p>For <em>k</em> = 1, one finds (for ):</p>
<p></p>
<p>For <em>t</em> = 1 one has:</p>
<p></p>
<p>If one ignores directedness in the graph, and just counts the total number of neighbors out to distance <em>t</em>, the results for <em>k</em> = 1 are the same as before, but in other cases they are different:</p>
<div>
<p>
</p>
</div>
<p>(These results asymptotically seem larger by a factor .)</p>
<p>So for <em>k</em> = 1, the limiting rulial multiway graph behaves like a 1-dimensional space, but for all <em>k</em>  2, it behaves like an infinite-dimensional space, in which the volumes of geodesic balls grow exponentially with volume.</p>
<h2>Finite Tapes</h2>
<p>So far we’ve always assumed that our Turing machine tape is unbounded. But in some ways it’s easier to see what’s going on if we instead limit the tape, so we have a finite total number of states (). One thing we can do is to consider a <a href="https://www.wolframscience.com/nks/p255--systems-of-limited-size-and-class-2-behavior/">cyclic tape</a>. </p>
<p>For with successively longer tapes we then get:</p>
<div>
<p>
</p>
</div>
<p>For we get:</p>
<div>
<p>
</p>
</div>
<p>For tape size 10, the results for successive values of <em>s</em> are:</p>
<div>
<p>
</p>
</div>
<p>If instead of having a cyclic tape, we just have a finite tape, and insist that the head never goes off either end, then for we get: </p>
<div>
<p>
</p>
</div>
<p>For we get</p>
<div>
<p>
</p>
</div>
<p>and the overall behavior for successive <em>s</em> is exactly as we saw above for unbounded tapes:</p>
<div>
<p>
</p>
</div>
<p>For , this is what happens in the cyclic case for length 3:</p>
<div>
<p>
</p>
</div>
<p>The results for lengths 1 through 6 are:</p>
<div>
<p>
</p>
</div>
<p>Rendering these in 3D makes the connection with progressively high-dimensional hypercubes slightly clearer:</p>
<div>
<p>
</p>
</div>
<p>The non-cyclic case for lengths 2 through 6 (length 1 is trivial):</p>
<div>
<p>
</p>
</div>
<p>Rendered in 3D these become:</p>
<div>
<p>
</p>
</div>
<p>In the case for a length-3 cyclic tape we get:</p>
<div>
<p>
</p>
</div>
<p>Rendering the results for lengths 1 through 6 in 3D gives:</p>
<div>
<p>
</p>
</div>
<p>In the non-cyclic case the results for lengths 2 through 6 in 3D are (the length-1 case is trivial):</p>
<div>
<p>
</p>
</div>
<h2>The Turing Machine Group</h2>
<p>It turns out that there’s a nice mathematical characterization of rulial multiway graphs for Turing machines: they’re just Cayley graphs of groups that we can call “Turing machine groups”. Why is this? Basically it’s because the possible configurations of a Turing machine have a direct correspondence with transformations that can act on these configurations. And in particular, one can pick out certain transformations that correspond to individual transitions in a non-deterministic Turing machine, and use these as generators in a presentation of the group. </p>
<p>Let’s start by considering the case of Turing machines with finite cyclic tapes. If the tape has length <em>n</em>, the total number of possible configurations of the machine is <em>n s k</em><sup><em>n</em></sup>. (Note that if the tape is finite but not cyclic then one doesn’t get a group.)</p>
<p>Assume for now <em>s</em> = 1, <em>k</em> = 2. Then for <em>n</em> = 3, there are 24 possible configurations, and the rulial multiway graph is:</p>
<div>
<p>
</p>
</div>
<p>But this graph turns out to be a <a href="https://reference.wolfram.com/language/ref/CayleyGraph.html">Cayley graph</a> for the finite group <a href="http://reference.wolfram.com/language/ref/AlternatingGroup.html"><em>A</em><sub>4</sub></a>  <a href="http://reference.wolfram.com/language/ref/CyclicGroup.html"><sub>2</sub></a>:</p>
<div>
<p>
</p>
</div>
<p>To construct this Cayley graph let’s represent the configurations of the Turing machine by pairs of integers , where 0  <em>i</em>  <em>n</em> – 1 gives the position of the head, and the bits of <em>u</em> (with 0  <em>u</em>  2<sup><em>n</em></sup> – 1) give the values on the tape. (Since <em>s</em> = 1, we don’t have to worry about the state of the head.) With this representation of the configurations, consider the “multiplication” operation:</p>
<div>
<p>
</p>
</div>
<p>When this operation acts on the configurations it defines a group. Here’s the multiplication table for <em>n</em> = 3:</p>
<div>
<p>
</p>
</div>
<p>Or equivalently:</p>
<div>
<p>
</p>
</div>
<p>But now consider the four elements , (position –1 wraps around on the cyclic tape): </p>
<div>
<p>
</p>
</div>
<p>We can consider these as the result of applying the 4 possible Turing machine transitions</p>
<div>
<p>
</p>
</div>
<p>to the configuration (corresponding to ):</p>
<div>
<p>
</p>
</div>
<p>And now by treating these elements as generators (and effectively applying them not just to the initial configuration, but to any configuration), we get as the Cayley graph of the group exactly the rulial multiway graph above.</p>
<p>It’s worth noting that the 4 elements we’ve used don’t correspond to the minimal set of generators for the group. Two elements suffice. And for example, we can use and , which can be thought of respectively as moving the head right, and flipping the color of one cell (again relative to the “identity” configuration ):</p>
<div>
<p>
</p>
</div>
<p>With these generators, we get the Cayley graph:</p>
<div>
<p>
</p>
</div>
<p>How else can we characterize the group? For any tape size <em>n</em> we can write it in terms of explicit permutations:</p>
<div>
<p>
</p>
</div>
<p>(For , the group can be generated by the permutations .)</p>
<p>We can also represent it symbolically in terms of generators and relations. Calling our “move-right” generator <em>R</em>, and “bit-flip” generator <em>F</em>, the group then satisfies at least the relations:</p>
<p></p>
<p>OK, so does the group have a name? For <em>n</em> = 2, it’s the 8-element dihedral group <em>D</em><sub>4</sub> and for <em>n</em> = 3, it’s the 24-element group <em>A</em><sub>4</sub>  <sub>2</sub>. For larger <em>n</em>, there doesn’t seem to be a standard name. But given our derivation we can just call it . And we can express it as a <a href="https://mathworld.wolfram.com/SemidirectProduct.html">semidirect product</a> (or <a href="https://ncatlab.org/nlab/show/wreath+product+of+groups">wreath product</a>):</p>
<p></p>
<p>The normal subgroup (<sub>2</sub>)<sup><em>n</em></sup> here represents states of the tape, and corresponds to a Boolean <em>n</em>-cube. The cyclic group <sub><em>n</em></sub> represents the position of the head, and acts on the Boolean <em>n</em>-cube by rotating its coordinates.</p>
<p>For any <em>n</em>, we can use just two generators, producing the sequence of Cayley graphs:</p>
<div>
<p>
</p>
</div>
<p>Undirected versions of these are exactly the <a href="https://mathworld.wolfram.com/Cube-ConnectedCycleGraph.html">cube-connected cycle graphs</a> that have arisen in studying communications networks: </p>
<div>
<p>
</p>
</div>
<p>So now what about the limit <em>n</em>  ? Now, the group is no longer finite, but we’ve still got the relations </p>
<p></p>
<p>and in the end, we can see that the group can be described as a semidirect product:</p>
<p></p>
<p>In a sense, the group—and its Cayley graph—is dominated by the <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/typical-forms-of-branchial-graphs/#p-271">infinite-dimensional Boolean hypercube</a>. But there’s more going on. And perhaps there’s a useful characterization of the limit that can be derived by the methods of <a href="https://en.wikipedia.org/wiki/Geometric_group_theory">modern geometric theory</a>. </p>
<p>For arbitrary <em>s</em> and <em>k</em>, we can potentially generalize to get:</p>
<p></p>
<h2>Causal Graphs for Deterministic Turing Machines</h2>
<p>In a deterministic Turing machine, every step involves one updating event—and the <a href="https://www.wolframscience.com/nks/p489--time-and-causal-networks/">causal graph</a> can be drawn by just joining successive locations of the head, and successive points where the head returns to a square where it has been before:</p>
<div>
<p>
</p>
</div>
<p>Continuing for a few more steps, the causal graph for this particular Turing machine becomes:</p>
<div>
<p>
</p>
</div>
<p>Continuing for more steps, and redrawing the graph, we see that we get a simple grid:</p>
<div>
<p>
</p>
</div>
<p>Of <em>s</em> = 2, <em>k</em> = 2 Turing machines, the one with the most exotic causal graph is the <a href="https://www.wolframscience.com/nks/notes-3-4--counter-turing-machine/">“binary counter” machine 1953</a>:</p>
<div>
<p>
</p>
</div>
<div>
<p>
</p>
</div>
<p>The causal graph gives a good “overall map” of the behavior of a Turing machine. <a href="https://www.wolframscience.com/nks/p80--turing-machines/">Here are a few Turing machines</a> (with <em>s</em> = 3, <em>k</em> = 2 and <em>s</em> = 4, <em>k</em> = 2) from <em><a href="https://www.wolframscience.com/nks">A New Kind of Science</a></em> (compare also the <em>s</em> = 2, <em>k</em> = 3 <a href="https://www.wolframscience.com/prizes/tm23/gallery/">universal Turing machine</a>):</p>
<div>
<p>
</p>
</div>
<p>And here are their respective causal graphs:</p>
<div>
<p>
</p>
</div>
<h2>Rulial Multiway Causal Graphs</h2>
<p>What can we say about the causal graph associated with a rulial multiway system? The first important observation is that rulial multiway graphs always exhibit causal invariance, since by including transitions associated with all possible rules one is inevitably including both rules and their inverses, with the result that every branching of edges in the rulial multiway graph is always associated with a corresponding merging.</p>
<p>It’s fairly easy to see this for two steps of <em>s</em> = 2, <em>k</em> = 2 Turing machines. The rulial multiway states graph here is:</p>
<div>
<p>
</p>
</div>
<p>This can also be rendered:</p>
<div>
<p>
</p>
</div>
<p>Explicitly showing events we get:</p>
<div>
<p>
</p>
</div>
<p>Including causal connections we get</p>
<div>
<p>
</p>
</div>
<p>or after 3 steps:</p>
<div>
<p>
</p>
</div>
<p>The pure rulial multiway causal graph in this case is then:</p>
<div>
<p>
</p>
</div>
<p>In layered form this becomes:</p>
<div>
<p>
</p>
</div>
<p>After 5 steps the growth in the number of nodes reached going from the root grows like:</p>
<div>
<p>
</p>
</div>
<p>Causal invariance implies that this multiway causal graph is ultimately composed of a large number of interwoven copies of a single causal graph. After any given number of steps of evolution, the various copies of this causal graph will have “reached different stages”. Here are the results after 3 steps: </p>
<div>
<p>
</p>
</div>
<p>And after 4 steps:</p>
<div>
<p>
</p>
</div>
<p>One can also look at rulial multiway causal graphs for other sets of Turing machines. For <em>s</em> = 1, <em>k</em> = 1 one has (here after 5 steps)</p>
<div>
<p>
</p>
</div>
<p>which is equivalent to:</p>
<div>
<p>
</p>
</div>
<p>The individual causal graphs in this case are immediately all the same, and are just:</p>
<div>
<p>
</p>
</div>
<p>For <em>s</em> = 2, <em>k</em> = 1 one already has after 3 steps</p>
<div>
<p>
</p>
</div>
<p>or in layered form:</p>
<div>
<p>
</p>
</div>
<p>After 5 steps this becomes</p>
<div>
<p>
</p>
</div>
<p>or in layered form:</p>
<div>
<p>
</p>
</div>
<p>In this case, the individual causal graphs after 5 steps are:</p>
<div>
<p>
</p>
</div>
<p>For <em>s</em> = 1, <em>k</em> = 2 one has very similar results to the case <em>s</em> = 2, <em>k</em> = 2. After 3 steps the causal graph is: </p>
<div>
<p>
</p>
</div>
<p>Or after 5 steps:</p>
<div>
<p>
</p>
</div>
<p>Removing multiple edges this is:</p>
<div>
<p>
</p>
</div>
<p>The individual causal graphs in this case are:</p>
<div>
<p>
</p>
</div>
<h2>Rulial Graphs</h2>
<p>Just as for ordinary multiway graphs one can study <a href="https://www.wolframphysics.org/technical-introduction/the-updating-process-for-string-substitution-systems/the-concept-of-branchial-graphs/">branchial graphs</a> which represent their transversals, so similarly for rulial multiway graphs one can study <a href="https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/multiway-systems-in-the-space-of-all-possible-rules/">rulial graphs</a> which represent their transversals. The layered way we have drawn multiway graphs corresponds to a particular choice of foliation—and with this choice, we can immediately generate rulial graphs.</p>
<p>For <em>s</em> = 2, <em>k</em> = 1 one has after 2 steps</p>
<div>
<p>
</p>
</div>
<p>while after 3 steps one gets:</p>
<div>
<p>
</p>
</div>
<p>For <em>s</em> = 1, <em>k</em> = 2 one gets after 2 steps:</p>
<div>
<p>
</p>
</div>
<p>while after 3 steps one has:</p>
<div>
<p>
</p>
</div>
<p>The sequence of results for steps 1 through 4 is:</p>
<div>
<p>
</p>
</div>
<p>For <em>s</em> = 2, <em>k</em> = 2 the corresponding results are:</p>
<div>
<p>
</p>
</div>
<p>What do these pictures mean? Just as branchial graphs in ordinary multiway systems can be thought of as “entanglement maps” for states (interpreted in our models as quantum states) in ordinary “multiway space”, so here these rulial graphs can be thought of as “entanglement maps” for states in rulial space. In other words, they are a kind of map of which Turing machine configurations are “evolutionarily close” to which other ones, in the sense that they can be reached with only a few different choices of rules.</p>
<h2>Deterministic Turing Machine Paths in Rulial Space</h2>
<p>The rulial multiway graph defines all paths that can be followed by all non-deterministic Turing machines. At each node in this graph there is therefore an outgoing edge corresponding to any possible Turing machine transition from the configuration corresponding to that node. But what if one considers just a single deterministic Turing machine? Its evolution is then a single path within the rulial multiway graph.</p>
<p>Consider for example the <em>s</em> = 2, <em>k</em> = 2 Turing machine:</p>
<div>
<p>
</p>
</div>
<p>This machine evolves from a blank tape according to:</p>
<div>
<p>
</p>
</div>
<p>This corresponds to a path in the rulial multiway graph:</p>
<div>
<p>
</p>
</div>
<p>With a different initial condition</p>
<div>
<p>
</p>
</div>
<p>the path can be very different:</p>
<div>
<p>
</p>
</div>
<p>This pictures might make it seem that the paths corresponding to the evolution of deterministic Turing machines are geodesics in the rulial multiway graph. But in general they are definitely not. For example, starting from a blank tape, the rule</p>
<div>
<p>
</p>
</div>
<p>follows this path on the rulial multiway graph:</p>
<div>
<p>
</p>
</div>
<p>If one allows any possible Turing machine transition at every step, then one can follow a geodesic path from a node corresponding to an initial condition to a node corresponding to any other configuration. But if one restricts oneself to the transitions in a particular (deterministic) Turing machine, then there will in general be many configurations one will never reach, and even those that one can reach, one may reach by a circuitous route in the rulial multiway graph.</p>
<p>Given a particular configuration (corresponding to a node in the rulial multiway graph), there may be many deterministic Turing machines that can reach it from a given initial state. One can consider each of these Turing machines to be implementing a certain algorithm. So then the “optimal algorithm” will be the one which is shortest among deterministic Turing machine paths. As I just mentioned, this won’t typically be the shortest possible path: that will usually be achieved by a non-deterministic Turing machine with a particular sequence of transitions. But there is still an optimal case (i.e. an “optimal algorithm”) among deterministic Turing machines.</p>
<p>But now we can ask across all possible deterministic Turing machines where they can reach in the rulial multiway graph. Here is the result for 4 steps for the <em>s</em> = 2, <em>k</em> = 2 Turing machines we have been considering:</p>
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<p>These are the results for 1, 2 and 3 steps:</p>
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<p>What is the significance of this? Essentially what we’re seeing is a comparison of what can be achieved with deterministic computation versus non-deterministic. Taking the 4-step case as an example, the “background” gray rulial multiway graph shows what can be achieved with arbitrary non-deterministic computation in 4 steps. The red region is what deterministic computation can achieve in the same number of steps.</p>
<p>In a sense this is a very simple empirical analog of the <a href="https://www.wolframscience.com/nks/p765--undecidability-and-intractability/">P vs. NP problem</a>. Unlike the real P vs. NP case, we’re not allowing arbitrary polynomial-time algorithms here; we’re just looking at possible <em>s</em> = 2, <em>k</em> = 2 Turing machine algorithms running specifically for 4 steps. But if we were to generalize this appropriately, P = NP would imply that the “red region” must in some limit in effect “reach anywhere in the graph”. </p>
<p>A little more precisely, the official definition of P and NP is for decision problems: you start from some initial condition which defines an instance of a problem (“Is this <a href="https://www.wolframscience.com/nks/p768--undecidability-and-intractability/">Boolean formula satisfiable</a>?” or whatever), and the system must eventually evolve to a state representing either “yes” or “no”. We can imagine setting things up so that the outcomes correspond to particular configurations of the Turing machine. The inputs are then also encoded as initial configurations of the Turing machine, and we want to know what happens as we consider inputs of progressively larger sizes. We can imagine drawing the rulial multiway graph so that progressively larger inputs are shown “progressively further from the center”. If we consider non-deterministic Turing machines (associated with the class of NP computations), then the shortest computation will be a geodesic path in the rulial multiway graph from the input configuration to the outcome configuration. But for deterministic Turing machines (associated with the class P) it will in general be some much more circuitous path.</p>
<p>The standard P vs. NP problem asks about limiting behavior as one increases the size of the input computation—and one might imagine that the question could be “geometrized” in terms of some continuum limit of the rulial multiway graph. Of course, there is no guarantee that any reasonable limit exists. And it could perfectly well be that the question of whether P  NP is <a href="https://www.wolframscience.com/nks/notes-12-8--p-versus-np-questions/">actually undecidable</a>, or in other words, that no finite proof of it can be given within a standard axiom system (such as Peano arithmetic or ZFC set theory).</p>
<p>One could imagine empirically testing more and more Turing machines, and seeing how they perform on an NP-complete problem. One might think of plotting their running times as a function of <em>n</em> for increasingly large <em>n</em>. For a while a particular Turing machine might be the winner. But then another one might take over. And there might be no end to how many “surprises” would occur as one increases <em>n</em>. (Somehow this is reminiscent of the <a href="https://www.wolframscience.com/nks/notes-4-4--numbers-of-primes/">story of the Skewes number</a> and whether .)</p>
<p>In computational complexity theory one usually thinks about explicitly constructing optimal algorithms by standard “engineering-like” methods. But I think there’s a lot to be learned from a more empirical approach—in which one doesn’t try to construct optimal algorithms, but just finds them “in the wild” by searching all possible programs. In the past, it might not have seemed that “just searching for programs” would ever produce anything terribly interesting. But one of the big consequences of what I discussed in <em>A New Kind of Science</em> is that even among tiny programs—small enough that one can, for example, enumerate all of them—there’s often very complex and potentially “useful” behavior. And that makes it seem much more reasonable to try to do “<a href="https://www.wolframscience.com/nks/p761--undecidability-and-intractability/">empirical computational complexity theory</a>”—enumerating possible programs to find optimal ones, and so on.</p>
<p>Small programs can be thought of as ones with <a href="https://www.wolframscience.com/nks/notes-10-3--algorithmic-information-theory/">low algorithmic complexity</a>. So searching for “fast programs” among these can be thought of as searching for programs with low time complexity, and low algorithmic complexity. We don’t know exactly how strong the constraint of low algorithmic complexity is, but from what I’ve seen in the computational universe (and what’s embodied in things like the Principle of Computational Equivalence), it seems as if it’s not such a big constraint. </p>
<p>I studied “empirical computational complexity theory” a bit in <em>A New Kind of Science</em>, notably for Turing machines. And one of the interesting observations was that the <a href="https://www.wolframscience.com/nks/p763--undecidability-and-intractability/">optimal algorithm for things was often not something</a> that could readily be constructed in a step-by-step engineering way. Instead, it was something that one basically could only find pretty much by doing a search of possible programs—and where there usually didn’t seem to be a “general form” that would “apply for all <em>n</em>”, and let one readily deduce the properties of the <em>n</em>   limit. In other words, it didn’t seem like what we’d now think of as the sequence of paths in rulial space would show any sign of converging to a “continuum limit”.</p>
<p>A decent example of all this occurs in <a href="https://www.wolframscience.com/nks/notes-12-8--sorting-networks/">sorting networks</a>. Imagine that you are given a collection of <em>n</em> numbers to sort. You can straightforwardly do this by making about <em>n</em><sup>2</sup> pairwise comparisons, and it’s pretty easy to optimize this a bit.</p>
<p>But explicit searches have revealed that the actual optimal networks for successive <em>n</em> are:</p>
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<p>What’s notable is how complicated and “random” they look; there doesn’t seem to be any obvious pattern to them (and my guess is that there fundamentally isn’t). Here’s a plot of the sizes of these networks (divided by <em>n</em><sup>2</sup>):</p>
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<p>It’s worth noting that these are deterministic networks. One could also imagine non-deterministic networks (and indeed one could construct a rulial multiway graph by considering all possible successive placements of pairwise comparisons)—and in a non-deterministic network it’s always possible to sort <em>n</em> numbers in at most <em>n</em> – 1 steps. </p>
<h2>The Space of Deterministic Turing Machine Computations</h2>
<p>We’ve just seen how the results of deterministic Turing machine computations lay out in the rulial multiway space of all possible non-deterministic Turing machine computations. But what happens if we just look at the graph of deterministic Turing machine computations on their own?</p>
<p>Here are the full rulial multiway graphs for 2 and 3 steps with the graphs of deterministic Turing machine computations superimposed, as before: </p>
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<p>But now let’s “pull out just the red subgraphs”—in other words, include as nodes only those configurations that at least one of the 4096 <em>s</em> = 2, <em>k</em> = 2 Turing machines can reach after 2 or 3 steps: </p>
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<p>Notice that after 2 steps, deterministic Turing machines can still reach all 36 configurations that non-deterministic ones can reach (though not through quite as many paths). But after 3 steps, among all the deterministic Turing machines, they can only reach 68 possible configurations, while non-deterministic ones can reach 100 configurations. </p>
<p>For all possible non-deterministic Turing machines with <em>s</em> = 2, <em>k</em> = 2 the total number of configurations that can be reached eventually roughly doubles on successive steps:</p>
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<p>For deterministic Turing machines, however, the number of possible configurations that can be reached soon increases much more slowly:</p>
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<p>And in fact there’s an obvious bound here: at any given step, the most that can happen is that each of the 4096 <em>s</em> = 2, <em>k</em> = 2 Turing machines leads to a new configuration—or, in other words, the maximum number of configurations reached increases by 4096.</p>
<p>Looking at the differences on successive steps, we find:</p>
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<p>In other words, among all 4096 Turing machines, about 100 “novel configurations” are reached at each successive step. (The actual sequence here looks surprisingly random; it’s not clear whether there’s any particular regularity.)</p>
<p>Now let’s look at the actual graphs formed. With 4 through 7 steps we get:</p>
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<p>After 10 and 20 steps the results are:</p>
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<p>Here is the result after 50 steps:</p>
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<p>There’s a surprising amount of structure in these graphs. There’s a “central region” near the initial blank-tape configuration (shown highlighted below) in which many different Turing machines end up visiting the same configurations:</p>
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<p>Here’s a 3D rendering of this region:</p>
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<p>But away from this region there end up being “spokes” (about 100 of them) corresponding to Turing machines that “independently explore new territory” in the space of configurations.</p>
<p>What are those “configurations on the edge” like? Here are sorted collections of them for the first few steps:</p>
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<p>For comparison, here is the result for all configurations that can be reached by non-deterministic Turing machines after just 2 steps:</p>
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<p>Here are the results for deterministic Turing machines after more steps:</p>
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<p>We can also ask which machines are the ones that typically “explore new territory”. Here’s the result for 30 steps:</p>
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<p>As we go to more steps, the graph of configurations that can be reached by deterministic Turing machines grows. But does at least the core of it reach some kind of limit after sufficiently many steps? We can get a sense of this by looking—as we have done so many times before—at the growth rate of the geodesic ball in the graph starting from the initial blank-tape configuration. The total number of new configurations that can be reached on each new layer of the geodesic ball is at most 4096—and in reality it’s much smaller. Here are the numbers of new configurations added on successive layers at steps 100, 200, …, 500 in the overall evolution:</p>
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<p>The cause of the “steps down” becomes clearer if one examines more closely the “spokes” in the graph above. Here’s one example:</p>
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<p>And basically what’s happening is that multiple Turing machines are tracing out roughly the same sequences of configurations, but some do it “efficiently”, while others “waste time”, for example having the head flip around on alternate steps. The “inner parts” of the spokes—that are closer to the initial node—involve both “inefficient” and more efficient Turing machines. But the “inefficient” Turing machines will simply not get as far, so they do not contribute to the outer layers of the geodesic ball. The final “step down” in the plot above—at basically half the total number of steps used for the Turing machines—involves the “petering out” of the roughly half the Turing machines that effectively “waste half their steps”.</p>
<p>For the “spoke” shown, here are the actual Turing machine histories involved (there are 8 machines that made identical copies of the first history):</p>
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<h2>The Cellular Automaton Analog</h2>
<p>The kind of graphs we’ve just made for deterministic Turing machines can be made for any family of deterministic computational systems. And in particular they can be made for the (at least for me, much more familiar) case of the <a href="https://www.wolframscience.com/nks/p55--more-cellular-automata/">256 <em>k</em> = 2, <em>r</em> = 1 cellular automata</a>. (And, yes, it’s somewhat amazing that in all these years I’ve never made such pictures before—though there’s a <a href="https://www.wolframscience.com/nks/notes-6-6--uniqueness-of-patterns-in-cellular-automata/">note on page 956</a> of <em>A New Kind of Science</em> that gets close.) </p>
<p>Here are the results for 5 and 10 steps, starting from an initial condition containing a single black cell:</p>
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<p>And here is the result for 50 steps:</p>
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<p>Here is the result for 10 steps, annotated with the actual cellular automaton evolution for the rules that “reach furthest”:</p>
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<p>It’s slightly easier to see what’s going on if we include only even-numbered rules (which leave a blank state blank):</p>
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<p>It’s an interesting map of “cellular automaton space”. (Note the presence of rule 30 on the lower left, and rule 110 on the lower right.)</p>
<p>The total number of new configurations explored by all rules on successive steps has a more regular form than for Turing machines:</p>
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<p>The result mostly alternates with period 4 between 72 and 84, though with dips at steps of the form 2<sup><em>m</em></sup>.</p>
<p>If we go for a certain number of steps (say 200), and then look at the geodesic ball centered on the initial condition, the number of configurations in successive layers is just:</p>
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<p>These results are all for ordinary, deterministic cellular automata. So are there non-deterministic cellular automata? Typically, cellular automata are defined to consistently update every cell at every step. But one can also consider <a href="https://www.wolframscience.com/nks/notes-9-10--sequential-cellular-automata/">sequential cellular automata</a>, where specific cells are updated at each step—and in this case it is straightforward to define a non-deterministic version, for which things like multiway systems can be constructed. (Note that for a full rulial multiway system, such non-deterministic cellular automata are basically the same as non-deterministic <a href="https://www.wolframscience.com/nks/p71--mobile-automata/">mobile automata</a>, which are close to Turing machines.)</p>
<h2>Computation Capabilities and the Structure of Rulial Space</h2>
<p>What do the computational capabilities of Turing machines mean for their rulial space? Let’s start with computation universality. </p>
<p>An important fact about deterministic Turing machines is that there are universal ones. Among the 4096 <em>s</em> = 2, <em>k</em> = 2 rules there aren’t any. But as soon one goes to <em>s</em> = 2, <em>k</em> = 3 rules (of which there are 2,985,984) it’s known (thanks to my <a href="https://www.wolframscience.com/nks/p709--universality-in-turing-machines-and-other-systems/">results in <em>A New Kind of Science</em></a>, and <a href="https://www.wolframscience.com/prizes/tm23/">Alex Smith’s 2007 proof</a>) that there is a universal machine. The rule is:</p>
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<p>Starting from a blank tape, <a href="https://www.wolframscience.com/prizes/tm23/gallery/">this machine gives</a></p>
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<p>and the <a href="https://www.wolframscience.com/prizes/tm23/gallery/">corresponding causal graph</a> is:</p>
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<p>But what does the existence of a universal machine mean in the rulial multiway system? From any given initial condition, any deterministic Turing machine will trace out some trajectory in the rulial multiway graph. And if a machine is universal it means that by appropriately picking its initial conditions it can be “programmed” to “emulate” any other Turing machine, in the sense that its trajectory will track the trajectory of whatever Turing machine it’s emulating. What does “track” mean? Basically, that there’s some fixed scheme that allows one to go from the states of the universal Turing machine to the states of the machine it’s emulating. The “scheme” will correspond to some translation in the rulial multiway graph, and the requirement is that this translation is somehow always limited. In other words, the trajectory of a universal machine will “flail around” as its starting point (i.e. initial condition) moves in the rulial multiway graph, and this “flailing” will be diverse enough that the trajectory can get close to the trajectory of any given other machine.</p>
<p>If, on the other hand, the machine one’s dealing with isn’t universal, then its trajectory won’t “flail around” enough for this work; the trajectory will in a sense be too constrained to successfully “track” all possible other deterministic Turing machine trajectories. </p>
<p>What about non-deterministic Turing machines? What universality can be interpreted to mean in this case is that given a particular initial condition, the output one wants occurs somewhere on the different paths followed by the non-deterministic Turing machine. (If one’s trying to do a decision problem—as in the <a href="https://www.wolframscience.com/nks/p765--undecidability-and-intractability/">computational complexity class NP</a>—then one can arrange to “signal that one’s got an answer” through some feature of the Turing machine state.) In the case of “extreme non-determinism”—as used to construct the rulial multiway graph—computation universality in a sense becomes a statement purely about the structure of the rulial multiway graph. And basically it just requires that the rulial multiway graph is sufficiently connected—which is guaranteed if there’s causal invariance (so there’s nothing like an “event horizon” anywhere). </p>
<p>But does one have to allow “extreme non-determinism” to get universality? With <em>s</em> = 2, <em>k</em> = 3 <a href="https://www.wolframscience.com/prizes/tm23/">we know that there’s a purely deterministic Turing machine that achieves it</a>. And my guess is that among <em>s</em> = 2, <em>k</em> = 2 there are “slightly multiway” rules that also do. In a standard deterministic <em>s</em> = 2, <em>k</em> = 2 Turing machine, there are 4 cases of the rule that are each specified to have a unique outcome. But what if even just one of those cases in the rule has two outcomes? The rule is non-deterministic, but it can be thought of as just being the result of specifying 5 defining cases for the rule. And it’s my guess that even this will be sufficient to get universality in the system. </p>
<p>A non-deterministic rule like this will not trace out a single path in the rulial multiway graph. Instead, it’ll give a bundle of paths, with different paths corresponding to different non-deterministic choices. But the story of universality is very similar to the deterministic case: one simply has to ask whether anything in the bundle successfully manages to track the trajectory of the machine one’s emulating.</p>
<p>It’s worth remembering that any given rule won’t typically be following geodesics in rulial space. It’ll be following some more circuitous path (or bundle of paths). But let’s say one has some rule that traces out some trajectory—corresponding to performing some computation. The <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> implies that across different possible rules, there’s a standard “maximum computational sophistication” for these computations, and many rules achieve it. But then the principle also implies that there’s equivalence between these maximally sophisticated computations, in the sense that there’s always a limited computation that translates between them.</p>
<p>Let’s think about this in rulial space. Let’s say we have two rules, starting from the same initial condition. Well, then, at the beginning it’s trivial to translate between the rules. But after <em>t</em> steps, the states reached by these rules can have diverged—making it potentially progressively more difficult to translate between them. A key idea of the rulial multiway graph—and rulial space—is that it lets one talk about both computations and translations between them in uniform ways. Let’s say that the trajectories of two rules have gone a certain distance in rulial space. Then one can look at their divergence, and see how long a “translation computation” one has to do to get from one to the other. </p>
<p>In ordinary spacetime, let’s say a certain time <em>t</em> has elapsed. Then we know that the maximum spatial distance that can have been traversed is <em>c t</em>, where <em>c</em> is the speed of light. In rulial space, there’s something directly analogous: in time <em>t</em>, there’s a maximum rulial distance that can be traversed, which we can call  <em>t</em>. But here the Principle of Computational Equivalence makes it a crucial contribution: it implies that throughout rulial space, and in all situations, is fixed. It can take an irreducible amount of computational work to successfully translate from the outcome of one rule to another. But this always scales the same way. There’s in effect one scale of computational irreducibility, and it’s characterized by . Like the constancy of the speed of light uniformly limits physical motion, the constancy of uniformly limits rulial motion.</p>
<p>But let’s say you’re trying to get from one point in rulial space to another. If from your starting point you can follow the path of an irreducible—and effectively universal—computation, then you’ll successfully be able to reach the other point. But if from your starting point you can only follow a reducible computation this won’t generally be true. And what this means is that “pockets of computational reducibility” in rulial space act a bit like black holes in physical space. You can get into them from regions of irreducibility, but you can’t get out of them.</p>
<p>There are probably signs of phenomena like this even in the rulial space for simple Turing machines that we’ve explored here. But there’s considerably more that needs to be worked out to be able to make all the necessary connections.</p>
<h2>The Emerging Picture of Rulial Space</h2>
<p>There are lots of analogies between physical space, the branchial space, and rulial space. For example, in physical space, there are light cones that govern the maximum rate at which effects can propagate between different parts of physical space. In branchial space, there are entanglement cones that govern the maximum rate of quantum entanglement. And in rulial space, one can think of “emulation cones”, which govern the maximum rate at which one description of behavior can be translated into another. </p>
<p>And when it comes to applying these things to modeling the physical universe, a crucial point is that the observer is necessarily part of the system, governed by the same rules as everything else. And that means that the observer can only be sensitive to certain “appropriately modded-out” aspects of the system. But in actually imagining how an observer “perceives” a system it’s almost always convenient to think about coordinatizing the system—by defining some appropriate foliation. In physical space, this involves foliating the causal graph using reference frames like in relativity. In branchial space, it involves our concept of quantum observation frames. And in rulial space, we can invent another such concept: a rulial description frame, or just a rulial frame. </p>
<p>Different rulial frames in effect correspond to describing the evolution of the universe as operating according to different rules. Causal invariance implies that in the end different rulial frames must give equivalent results. But the specific way one describes the time evolution of the universe will depend on what rulial frame one’s using. In one frame one would be describing the universe in one way; in another frame, another way. And the “story one tells” about how the universe evolves will be different in the different frames.</p>
<p>Much like with superposition in quantum mechanics, there’s probably some notion of regions in rulial space, in which one’s somehow viewing the universe as operating according to “rulially entangled” collections of rules. </p>
<p>But while our original motivation was understanding physics, a lot of what we’re studying about rulial space also applies to purely computational systems. For example, we can think of rulial space even without having any notion of physical space. And we can in effect imagine that rulial space is some kind of map of a space of possible rules for computational systems. (Notice that because of computation universality and the <a href="https://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> it ultimately doesn’t matter what particular type of rule—<a href="https://www.wolframscience.com/nks/p78--turing-machines/">Turing machine</a>, <a href="https://www.wolframscience.com/nks/p53--more-cellular-automata/">cellular automaton</a>, <a href="https://www.wolframscience.com/nks/p102--symbolic-systems/">combinator</a>, whatever—is used to “parametrize” rulial space.)</p>
<p>Different places in rulial space in some sense correspond to different rules. Paths at different places in rulial space correspond to evolution according to different rules. So what is the analog of motion in rulial space? In effect it’s having a frame which progressively changes the rule one’s using. If one’s trying to find out what happens in one’s system, it’s fundamentally most efficient to “stick with one rule”. If one progressively changes the rule, one’s going to have to keep “translating back” to the original rule, by somehow emulating this rule with whatever rule one’s reached. And the result of this is there’ll be exactly the analog of relativistic time dilation. Faster motion in rulial space leads to slower progression in time. Of course, to discuss this properly, we really have to talk about the rulial multiway causal graph, etc.</p>
<p>But one thing is clear: motion faster than some maximum speed is impossible. From within the system, you simply can’t keep the correct rulial causal connections if you’re changing your rulial location faster than . </p>
<p>In an abstract study of Turing machines, is just an arbitrary parameter. But in the actual physical universe, it must have a definite value, like the speed of light <em>c</em>, or our maximum entanglement speed . It’s difficult even to estimate what might be. And presumably estimating will be even harder. But it’s interesting to discuss at least how we might start to think about estimating . </p>
<p>The first key observation about rulial space is that in our models, it’s discrete. In other words, there’s a discrete space of possible rules. Or, put another way, theories are quantized, and there’s somehow an elementary distance between theories—or a minimum distance in “theory space” between neighboring theories. </p>
<p>But what units is this distance in? Basically it’s in units of rule—or program—size. Given any program—or rule—we can imagine writing that program out in some language (say in <a href="https://www.wolfram.com/language/">Wolfram Language</a>, or as a program for a particular universal Turing machine, or whatever) And now we can characterize the size of the program by just looking at how many tokens it takes to write the program out.</p>
<p>Of course, with different languages, that number will be different—at the simplest level just like the number of decimal digits necessary to represent a number is different from the number of binary digits, or the length of its representation in terms of primes. But it’s just like measuring a length in feet or meters: even though the numerical value is different, we’re still describing the same length.</p>
<p>It’s important to point out that it’s not enough to just measure things in terms of “raw information content”, or ordinary bits, as discussed in <a href="https://www.wolframscience.com/nks/notes-12-10--theories-of-communication/">information theory</a>. Rather, we want some kind of measure of “semantic information content”: information content that directly tells us what computation to do.</p>
<p>It’s also important to point out that what we need is different from what’s discussed in <a href="https://www.wolframscience.com/nks/notes-10-3--algorithmic-information-theory/">algorithmic information theory</a>. Once one has a computation universal system, one can always use it to <a href="https://www.wolframscience.com/nks/chap-11--the-notion-of-computation/">translate from any one language to any other</a>. And in algorithmic information theory the concept is that one can measure the length of a program up to an additive constant by just expecting to include an “emulation program” that adapts to whatever language one’s measuring the length in. But in the usual formalism of algorithmic information theory one doesn’t worry about how long it’s going to take for the emulation to be done; it’s just a question of whether there’s ultimately enough information to do it. </p>
<p>In our setup, however, it does matter how long the emulation takes, because that process of emulation is actually part of our system. And basically we need the number of steps needed for the emulation to be in some sense bounded by a constant.</p>
<p>So, OK, what does this mean for the value of ? Its units are presumably program size per unit time. And so to define its value, we’ll have to say how we’re measuring program size. Perhaps we could imagine we write our rules in the <a href="https://www.wolfram.com/language/">Wolfram Language</a>. Then there should be a definite value of for our universe, measured in Wolfram-Language-tokens per second. If we chose to use <a href="https://www.wolframscience.com/prizes/tm23/">(2,3)-Turing-machine</a>-tape-values per year then we’d get a different numerical value. But assuming we used the correct conversions, the value would be the same. (And, yes, there’s all sorts of subtlety about constant-time or not emulation, etc.)</p>
<p>In some sense, we may be able to think of as the ultimate “processor speed for the universe”: how fast tokens in whatever language we’re using are being “interpreted” and actually “executed” to determine the behavior of the universe. </p>
<p>Can we estimate the value of ? If our units are Wolfram-Language-tokens per second we could start by imagining just computing in the Wolfram Language some piece of the rulial multiway graph for our models and seeing how many operations it takes. To allow “all possible rules” we’d have to increase the possible left- (and right-) hand sides of our rules to reflect the size of the hypergraph representing the universe at each step. But now we’d need to divide by the “number of parallel threads” in the rulial multiway graph. So we can argue that all we’d be left with is something like (size of spatial hypergraph represented in Wolfram Language) / (elementary time). </p>
<p>So, based on our <a href="https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/units-and-scales/">previous estimates</a> (which I don’t consider anything more than vaguely indicative yet) we might conclude that perhaps:</p>
<p> ~ 10<sup>450</sup> Wolfram-Language-tokens/second</p>
<p>The number of “parallel threads” in the rulial multiway graph (the rulial analog of Ξ) might then be related to the number of possible hypergraphs that contain about the number of nodes in the universe, or <a href="https://www.wolframphysics.org/technical-introduction/typical-behaviors/the-number-of-possible-rules/">very roughly</a> (10<sup>350</sup>)^(10<sup>350</sup>) ≈ . If we ask the total number of Wolfram Language tokens processed by the universe, there’ll be another factor ~10<sup>467</sup>, but this “parallelism” will completely dominate, and the result will be about:</p>
<p> Wolfram-Language-tokens</p>
<p>OK, so given a value of , how might we conceivably observe it? Presumably there’s an analog of quantum uncertainty in rulial space, that’s somehow proportional to the value of . It’s not completely clear how this would show up, but one possibility is that it would lead to intrinsic limits on inductive inference. For example, given only a limited observation time, it might be fundamentally impossible to determine beyond a certain (“rulial”) accuracy what rule the universe is following in your description language. There’d be a minimum rate of divergence of behaviors from different rules, associated with the minimum distance between theories—and it would take a certain time to distinguish theories at this rate of divergence.</p>
<p>In our models, just like every causal edge in physical and branchial space is associated with energy, so should every causal edge in rulial space be. In other words, the more processing that happens in a particular part of rulial space, the more physical energy one should consider exists there. And just as with the Einstein equations in physical space, or the Feynman path integral in branchial space, we should expect that the presence of energy in a particular region of rulial space should reflect in a deflection of geodesics there.</p>
<p>Geodesics in rulial space are the shortest paths from one configuration of the universe to another, using whatever sequences of rules are needed. But although that’s somewhat like what’s considered at a theoretical level in non-deterministic computation, it’s not something we’re usually familiar with: we’re used to picking a particular description language and sticking with it. So exactly what the interpretation of deflections of geodesics in rulial space should be isn’t clear.</p>
<p>But there are a few other things we can consider. For example, presumably the universe is expanding in rulial space, perhaps implying that in some sense more descriptions of it are becoming possible over time. What about rulial black holes? As mentioned above, parts of rulial space that correspond to computational reducibility should behave like black holes, where in effect “time stops”. Or, in other words, while in most of the universe time is progressing, and irreducible computation is going on, computational reducibility will cause that process to stop in a rulial black hole.</p>
<p>Presumably geodesics near the rulial black hole will be pulled towards it. Somehow when there’s a description language that leads to computational reducibility, languages near it will tend to also stop being able to successfully describe computationally irreducible processes.</p>
<p>Can we estimate the density of rulial black holes? Let’s consider the Turing machine case. In effect we’re going to want to know what the density of non-universality is among all possible non-deterministic Turing machines. Imagine we emulate all Turing machines using a single universal machine. Then this is effectively equivalent to asking what fraction of initial conditions for that machine lead to reducible behaviors—or, in essence, in the traditional characterization of Turing machines, halt. But the probability across all possible inputs that a universal Turing machine will halt is exactly <a href="https://www.wolframscience.com/nks/notes-10-3--algorithmic-information-theory/">Greg Chaitin’s </a>. In other words, the density of rulial black holes in rulial space is governed by .</p>
<p>But <a href="https://mathworld.wolfram.com/ChaitinsConstant.html"> isn’t just a number like </a> that we can compute as accurately as we want; it’s noncomputable, in the sense that it can’t be computed to arbitrary accuracy in any finite time by a Turing machine. Now, in a sense it’s not too surprising that the density of rulial black holes is noncomputable—because, given computational irreducibility, to determine for certain whether something is truly a rulial black hole from which nothing can escape one might have to watch it for an unbounded amount of time.</p>
<p>But for me there’s something personally interesting about Greg Chaitin’s showing up in any way in a potential description of anything to do with the universe. You see, I’ve had a nearly 40-year-long debate with Greg about whether the universe is “like ” or “like ”. In other words, is it possible to have a rule that will let us compute what the universe does just like we can compute (say, in principle, with a Turing machine) the digits of ? Or will we have to go beyond a Turing machine to describe our universe? I’ve always thought that our universe is “like ”; Greg has thought that it might be “like ”. But now it looks as if we might both be right!</p>
<p>In our models, we’re saying that we can compute what the universe does, in principle with a Turing machine. But what we’re now finding out is that in the full rulial space, general limiting statements pull in . In a particular rulial observation frame, we’re able to analyze the universe “like ”. But if we want to know about all possible rulial observation frames—or in a sense the space of all possible descriptions of the universe—we’ll be confronted with .</p>
<p>In our models, the actual operation of the universe, traced in a particular rulial observation frame, is assumed never to correspond to anything computationally more powerful than a Turing machine. But let’s say there was a <a href="https://www.wolframscience.com/nks/p720--the-content-of-the-principle/">hypercomputer</a> in our universe. What would it look like? It’d be a place where the effective is infinite—a place where rulial geodesics infinitely diverge—a kind of white hole in rulial space (or perhaps a cosmic event horizon). (We can also think about the hypercomputer as introducing infinite-shortcut paths in the rulial multiway graph which ordinary Turing-machine paths can never “catch up to” and therefore causally affect.)</p>
<p>But given a universe that does hypercomputation, we can then imagine defining a rulial multiway graph for it. And then our universe will show up as a black hole in this higher-level rulial space.</p>
<p>But OK, so if there’s one level of hypercomputer, why not consider all possible levels? In other words, why not define a hyperrulial multiway graph in which the possible rules that are used include both ordinary computational ones, but also hypercomputational ones? </p>
<p>And once we’re dealing with hypercomputational systems, we can just keep going, adding in effect more and more levels of <a href="https://www.wolframscience.com/nks/notes-12-3--oracles-for-universal-systems/">oracles</a>—and progressively ascending the arithmetic hierarchy. (The concept of “<a href="https://www.wolframscience.com/nks/notes-12-4--intermediate-degrees/">intermediate degrees</a>” might be thought to lead to something that isn’t a perfect hierarchy—but I suspect that it’s not robust enough to apply to complete rulial spaces.) Within the hyperrulial multiway graph at a particular level, levels below it will be presumably appear as rulial black holes, while ones above will appear as rulial white holes. </p>
<p>And there’s nothing to keep us only considering finite levels of the arithmetic hierarchy; we can also imagine ascending to <a href="https://www.wolframscience.com/nks/notes-12-9--transfinite-numbers/">transfinite</a> levels, and then just keeping going to higher and higher levels of infinity. Of course, according to our models, none of this is relevant to our particular physical universe. But at a theoretical level, we can still at least to some extent “symbolically describe it”, even in our universe.</p>
<h2>Thanks</h2>
<p>Thanks to Jonathan Gorard for a variety of discussions, as well as to <a href="https://www.wolframcloud.com/obj/wolframphysics/WorkingMaterial/2020/TuringMachineGroup.nb">Tali Beynon</a>, Todd Rowland, Ed Pegg, Jose Martin-Garcia and Christopher Wolfram.</p>
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 Wolfram Physics Project Bulletins

 Contents
 * Top
 * Generalized Physics and the Theory of Computation
 * Turing Machines
 * The Rulial Multiway Graph
 * The Limit of the Rulial Multiway Graph
 * Finite Tapes
 * The Turing Machine Group
 * Causal Graphs for Deterministic Turing Machines
 * Rulial Multiway Causal Graphs
 * Rulial Graphs
 * Deterministic Turing Machine Paths in Rulial Space
 * The Space of Deterministic Turing Machine Computations
 * The Cellular Automaton Analog
 * Computation Capabilities and the Structure of Rulial Space
 * The Emerging Picture of Rulial Space
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 Exploring Rulial Space: The Case of Turing Machines

 Exploring Rulial Space: The Case of Turing Machines

 Stephen Wolfram

 June 9, 2020
 * Related livestreams
 * Related notebooks
 * Video work logs
 * Related tweets

 Related livestreams

 May 26, 2020 (with Jonathan Gorard, Ed Pegg and Max Piskunov)
 June 4, 2020 (with Jonathan Gorard)

 Related notebooks

 Rulial-01-NDTM (by Stephen Wolfram)
 Rulial-02-SS (by Stephen Wolfram)
 Rulial-03-NDTM (by Stephen Wolfram)
 Rulial-04-continuum (by Stephen Wolfram)
 Rulial-05-NDTM (by Stephen Wolfram)
 Rulial-06-NDTM (by Stephen Wolfram)
 Rulial-07-NDTM (by Stephen Wolfram)
 Rulial-08-trees (by Stephen Wolfram)
 Rulial-09-NDTM (by Stephen Wolfram)
 Rulial-10-NDTM (by Stephen Wolfram)
 Rulial-11-NDTM (by Stephen Wolfram)
 Rulial-12-NDTM (by Stephen Wolfram)
 Rulial-13-NDTM (by Stephen Wolfram)
 TuringMachineGroup (by Tali Beynon)

 Video work logs

 May 24, 2020
 May 25, 2020 [Part 1]
 May 25, 2020 [Part 2]
 May 27, 2020
 May 29, 2020
 May 30, 2020
 May 31, 2020
 June 2, 2020

 Related tweets

 Just finished a 3-hour #WolframPhysicsLive update and Q&A. Lots of emerging new ideas in physical, branchial and rulial space…https://t.co/FHyXhfYvBa

 — WolframPhysics (@wolframphysics) June 4, 2020

 New in the Wolfram Function Repository: MultiwayTuringMachine 🙂 https://t.co/697tXBdDWr pic.twitter.com/zwSjN47LNL

 — WolframPhysics (@wolframphysics) June 3, 2020

 #WolframPhysicsLive: The shape of "Turing space" … What can all possible Turing machines do? pic.twitter.com/QvN5tgMuQm

 — WolframPhysics (@wolframphysics) May 26, 2020

 #WolframPhysicsLive: The space of deterministic vs. non-deterministic computations… (Working towards understanding relations between different descriptions of the universe) pic.twitter.com/PCJq2QUMLK

 — WolframPhysics (@wolframphysics) May 26, 2020

 Today's #WolframPhysicsLive: Continuing to explore rulial space … The structure of extreme non-determinism in Turing machines (every possible transition at every step)https://t.co/yNVC1fJE16 pic.twitter.com/nkiNc2lI9P

 — WolframPhysics (@wolframphysics) May 26, 2020

 From #WolframPhysicsLive: Meet the rulial space of all possible Turing machines. The beginning of turning computability theory into topology and computational complexity theory into geometry—in rulial space…https://t.co/QEjOsusalT pic.twitter.com/pSqcarRwkK

 — WolframPhysics (@wolframphysics) May 22, 2020

 From #WolframPhysicsLive: What's the analog of the Einstein equations / path integral in rulial space? Presumably it tells us about bundles of nearby algorithms. Geometrization of computational complexity theory?

 — WolframPhysics (@wolframphysics) April 30, 2020

 Generalized Physics and the Theory of Computation

 Let’s say we find a rule that reproduces physics. A big question would then be: “Why this rule, and not another?” I think there’s a very elegant potential answer to this question, that uses what we’re calling rule space relativity—and that essentially says that there isn’t just one rule: actually all possible rules are being used, but we’re basically picking a reference frame that makes us attribute what we see to some particular rule. In other words, our description of the universe is a sense of our making, and there can be many other—potentially utterly incoherent—descriptions, etc.

 But so how does this work at a more formal level? This bulletin is going to explore one very simple case. And in doing so we’ll discover that what we’re exploring is potentially relevant not only for questions of “generalized physics”, but also for fundamental questions in the theory of computation. In essence, what we’ll be doing is to study the structure of spaces created by applying all possible rules, potentially, for example, allowing us to “geometrize” spaces of possible algorithms and their applications.

 In our models of physics, we begin by considering spatial hypergraphs that describe relations between “atoms of space”. Then, looking at all possible ways a given rule can update these hypergraphs, we form what we call a multiway graph. The transversals of this graph define what we call branchial space, in which we can see the pattern of entanglements between quantum states.

 But there’s also a third level we can consider. Instead of just forming a multiway graph in which we do all possible updates with a given rule, we form a rulial multiway (or “ultramultiway”) graph in which we follow not only all possible updates, but also all possible rules. The transversals to this rulial multiway graph define what we call rulial space. Causal invariance in the rulial multiway graph then implies “rule space relativity” which is what allows us to use different possible reference frames to describe the universe.

 In the end, the Principle of Computational Equivalence implies a certain invariance to the limiting structure of the rulial multiway graph, independent of the particular parametrization of the set of possible rules. But for the sake of understanding rulial space and the rulial multiway graph—and getting intuition about these—this bulletin is going to look at a specific set of possible rules, defined by simple Turing machines.

 The rules we’ll use aren’t a good fit for describing our universe. But the comparative simplicity of their structure will help us in trying to elucidate some of the complexities of rulial space. In addition, using Turing machines will make it easier for us to make contact with the theory of computation, where Turing machines are standard models.

 Turing Machines

 Here’s a representation of the rule for a particular 2-state (s = 2), 2-color (k = 2) Turing machine:

 ✕
 

 The pointer represents the “head” of the Turing machine and its orientation represents the state of the head. Here’s what this particular Turing machine does over the course of a few steps starting from a “blank tape” (i.e. all squares white):

 ✕
 

 There are (2 s k)sk possible s-state k-color Turing machines, or 4096 s = 2, k = 2 machines. Here are all the distinct behaviors that occur (up to left-right reflection) in these 4096 machines, starting from a blank tape:

 ✕
 

 But note that all these are deterministic Turing machines, in the sense that, for a given machine, it is always the same rule that is applied at every step. But what about non-deterministic Turing machines? The idea here is to allow several different rules at any given step.

 As a simple example, consider the pair of rules:

 ✕
 

 To show the possible evolution histories, we can construct a multiway system:

 ✕
 

 Each arrow represents an application of one of the two possible rules. A non-deterministic Turing machine is usually considered to follow a particular path, corresponding to a particular evolution history. Then a typical question is whether there exists some path that leads to a particular final state (which might be viewed as the solution to some particular problem). (In a quantum Turing machine, one considers collections of multiple states, viewed as being in a quantum superposition.)

 In what we’re going to be doing here, we want to study the “extreme case” of non-determinism—where at every step, every possible Turing machine rule (at least with a given s, k) is applied. Then we’ll be interested in the full rulial multiway graph that’s created.

 The Rulial Multiway Graph

 Start with a blank tape. Then apply all possible 2,2 Turing machine rules. This is the rulial multiway graph that’s formed after 1 step:

 ✕
 

 Each individual arrow here represents many possible rules from all the 4096 discussed above. Here’s what happens after 2 steps:

 ✕
 

 In layered form, this becomes:

 ✕
 

 In creating these rulial multiway graphs, there’s an important simplification we’re able to make. We don’t need to separately apply all 4096 2,2 Turing machine rules to each state at each step—because all that ever matters is the one case of the rule that’s relevant to the particular state one has. There are 4 possible individual cases—and for each of these there are 8 possible outcomes, leading to 32 “micro-rules”:

 ✕
 

 After 3 steps, the rulial multiway graph has the form:

 ✕
 

 In 3D this becomes:

 ✕
 

 In layered form it is:

 ✕
 

 After 5 steps, the rulial multiway graph is:

 ✕
 

 What about other numbers of states and colors? Here are the rulial multiway graphs after 3 steps for Turing machines with various numbers of states and colors :

 ✕
 

 Notice the presence of s = 1 examples. Even with a single possible state, it is already possible to form a nontrivial rulial multiway system. Here are the results for k = 2 after 1 step:

 ✕
 

 2 steps:

 ✕
 

 6 steps:

 ✕
 

 In principle one can also consider the even simpler case of s = k = 1. After 1 step one gets:

 ✕
 

 After 2 steps this becomes:

 ✕
 

 And after 3 steps:

 ✕
 

 As a layered graph rendering, this becomes:

 ✕
 

 With s = 2, k = 1 one gets after 1 step:

 ✕
 

 After 2 steps this becomes:

 ✕
 

 And after 3 steps:

 ✕
 

 After 6 steps this becomes:

 ✕
 

 At least in the central part of this graph, there are edges going in both directions. Combining these, and treating the graph as undirected, one gets:

 ✕
 

 With 3 states (s = 3) there are 3 tracks:

 ✕
 

 One can also consider Turing machines in which the head can not only move left or right, but can also stay still. In this case, with s = 2, k = 1 one gets after 1 step:

 ✕
 

 After 2 steps this becomes:

  ✕
 

 After 4 steps, removing repeated edges, etc. this gives:

  ✕
  

  As another generalization, one can consider Turing machines that can move not just one, but also two squares:

  ✕
  

  After 2 steps one gets:

  ✕
  

  After 3 steps, removing repeated edges, etc. this gives:

  ✕
  

  The Limit of the Rulial Multiway Graph

  What is the limiting structure of the rulial multiway graph after an infinite number of steps? The first crucial observation is that it’s in a sense homogeneous: the structure of the graph around any given node is always the same (i.e. it’s a vertex-transitive graph). To see why this is true, recall that each node in the graph corresponds to a particular distinct configuration of the Turing machine. This node will lead to all nodes that can be obtained from it by one step of Turing machine evolution. But (assuming the head always moves ±1 square) there are always exactly 2 s k of these. And because we are following all possible “micro-rules”, we can think of ourselves as just “blindly overwriting” whatever configuration we had, so that the structure of the graph is the same independent of what configuration we’re at.

  As a simple example, here’s what happens in the s = 2, k = 1 case, after 3 steps:

  ✕
  

  There’s some trickiness at the ends, but in the central region we see as expected that each node has exactly 4 successors. If we pick out the subgraph around the blank-tape starting node it has the form:

  ✕
  

  In general, the limiting neighborhood of every node is just:

  ✕
  

  In the full graph, these neighborhoods are knitted together, giving pieces like:

  ✕
  

  Let’s now consider the case s = 1, k = 2. After 4 steps the full graph is:

  ✕
  

  Here the neighborhood of the start node is:

  ✕
  

  And in the limiting graph, every node will have a local neighborhood with this same structure. If we look at successively larger neighborhoods, the limiting form of these for every node will be:

  ✕
  

  Here’s a table of the number of distinct nodes in these successively larger neighborhoods for Turing machines with various values of s and k:

  ✕
  

  In the limit t → ∞ the number of nodes reached in all cases goes like:

  For k = 1, one finds (for ):

  For t = 1 one has:

  If one ignores directedness in the graph, and just counts the total number of neighbors out to distance t, the results for k = 1 are the same as before, but in other cases they are different:

  ✕
  

  (These results asymptotically seem larger by a factor .)

  So for k = 1, the limiting rulial multiway graph behaves like a 1-dimensional space, but for all k ≥ 2, it behaves like an infinite-dimensional space, in which the volumes of geodesic balls grow exponentially with volume.

  Finite Tapes

  So far we’ve always assumed that our Turing machine tape is unbounded. But in some ways it’s easier to see what’s going on if we instead limit the tape, so we have a finite total number of states (n s kn). One thing we can do is to consider a cyclic tape.

  For s = k = 1 with successively longer tapes we then get:

  ✕
  

  For s = 2, k = 1 we get:

  ✕
  

  For tape size 10, the results for successive values of s are:

  ✕
  

  If instead of having a cyclic tape, we just have a finite tape, and insist that the head never goes off either end, then for s = k = 1 we get:

  ✕
  

  For s = 2, k = 1 we get

  ✕
  

  and the overall behavior for successive s is exactly as we saw above for unbounded tapes:

  ✕
  

  For s = 1, k = 2 , this is what happens in the cyclic case for length 3:

  ✕
  

  The results for lengths 1 through 6 are:

  ✕
  

  Rendering these in 3D makes the connection with progressively high-dimensional hypercubes slightly clearer:

  ✕
  

  The non-cyclic case for lengths 2 through 6 (length 1 is trivial):

  ✕
  

  Rendered in 3D these become:

  ✕
  

  In the case s = 2, k = 2 for a length-3 cyclic tape we get:

  ✕
  

  Rendering the results for lengths 1 through 6 in 3D gives:

  ✕
  

  In the non-cyclic case the results for lengths 2 through 6 in 3D are (the length-1 case is trivial):

  ✕
  

  The Turing Machine Group

  It turns out that there’s a nice mathematical characterization of rulial multiway graphs for Turing machines: they’re just Cayley graphs of groups that we can call “Turing machine groups”. Why is this? Basically it’s because the possible configurations of a Turing machine have a direct correspondence with transformations that can act on these configurations. And in particular, one can pick out certain transformations that correspond to individual transitions in a non-deterministic Turing machine, and use these as generators in a presentation of the group.

  Let’s start by considering the case of Turing machines with finite cyclic tapes. If the tape has length n, the total number of possible configurations of the machine is n s kn. (Note that if the tape is finite but not cyclic then one doesn’t get a group.)

  Assume for now s = 1, k = 2. Then for n = 3, there are 24 possible configurations, and the rulial multiway graph is:

  ✕
  

  But this graph turns out to be a Cayley graph for the finite group A4 ×  2:

  ✕
  

  To construct this Cayley graph let’s represent the configurations of the Turing machine by pairs of integers , where 0 ≤ i ≤ n – 1 gives the position of the head, and the bits of u (with 0 ≤ u ≤ 2n – 1) give the values on the tape. (Since s = 1, we don’t have to worry about the state of the head.) With this representation of the configurations, consider the “multiplication” operation:

  ✕
  

  When this operation acts on the configurations it defines a group. Here’s the multiplication table for n = 3:

  ✕
  

  Or equivalently:

   ✕
  

  But now consider the four elements , (position –1 wraps around on the cyclic tape):

   ✕
   

   We can consider these as the result of applying the 4 possible Turing machine transitions

   ✕
   

   to the configuration (corresponding to ):

   ✕
   

   And now by treating these elements as generators (and effectively applying them not just to the initial configuration, but to any configuration), we get as the Cayley graph of the group exactly the rulial multiway graph above.

   It’s worth noting that the 4 elements we’ve used don’t correspond to the minimal set of generators for the group. Two elements suffice. And for example, we can use and , which can be thought of respectively as moving the head right, and flipping the color of one cell (again relative to the “identity” configuration ):

   ✕
   

   With these generators, we get the Cayley graph:

   ✕
   

   How else can we characterize the group? For any tape size n we can write it in terms of explicit permutations:

   ✕
   

   (For , the group can be generated by the permutations .)

   We can also represent it symbolically in terms of generators and relations. Calling our “move-right” generator R, and “bit-flip” generator F, the group then satisfies at least the relations:

   OK, so does the group have a name? For n = 2, it’s the 8-element dihedral group D4 and for n = 3, it’s the 24-element group A4 ×  2. For larger n, there doesn’t seem to be a standard name. But given our derivation we can just call it . And we can express it as a semidirect product (or wreath product):

   The normal subgroup (  2)n here represents states of the tape, and corresponds to a Boolean n-cube. The cyclic group  n represents the position of the head, and acts on the Boolean n-cube by rotating its coordinates.

   For any n, we can use just two generators, producing the sequence of Cayley graphs:

   ✕
   

   Undirected versions of these are exactly the cube-connected cycle graphs that have arisen in studying communications networks:

   ✕
   

   So now what about the limit n → ∞ ? Now, the group is no longer finite, but we’ve still got the relations

   and in the end, we can see that the group can be described as a semidirect product:

   In a sense, the group—and its Cayley graph—is dominated by the infinite-dimensional Boolean hypercube. But there’s more going on. And perhaps there’s a useful characterization of the limit that can be derived by the methods of modern geometric theory.

   For arbitrary s and k, we can potentially generalize to get:

   Causal Graphs for Deterministic Turing Machines

   In a deterministic Turing machine, every step involves one updating event—and the causal graph can be drawn by just joining successive locations of the head, and successive points where the head returns to a square where it has been before:

   ✕
   

   Continuing for a few more steps, the causal graph for this particular Turing machine becomes:

   ✕
   

   Continuing for more steps, and redrawing the graph, we see that we get a simple grid:

   ✕
   

   Of s = 2, k = 2 Turing machines, the one with the most exotic causal graph is the “binary counter” machine 1953:

   ✕
   
   ✕
   

   The causal graph gives a good “overall map” of the behavior of a Turing machine. Here are a few Turing machines (with s = 3, k = 2 and s = 4, k = 2) from A New Kind of Science (compare also the s = 2, k = 3 universal Turing machine):

   ✕
   

   And here are their respective causal graphs:

   ✕
   

   Rulial Multiway Causal Graphs

   What can we say about the causal graph associated with a rulial multiway system? The first important observation is that rulial multiway graphs always exhibit causal invariance, since by including transitions associated with all possible rules one is inevitably including both rules and their inverses, with the result that every branching of edges in the rulial multiway graph is always associated with a corresponding merging.

   It’s fairly easy to see this for two steps of s = 2, k = 2 Turing machines. The rulial multiway states graph here is:

   ✕
   

   This can also be rendered:

   ✕
   

   Explicitly showing events we get:

   ✕
   

   Including causal connections we get

   ✕
   

   or after 3 steps:

   ✕
   

   The pure rulial multiway causal graph in this case is then:

   ✕
   

   In layered form this becomes:

   ✕
   

   After 5 steps the growth in the number of nodes reached going from the root grows like:

   ✕
   

   Causal invariance implies that this multiway causal graph is ultimately composed of a large number of interwoven copies of a single causal graph. After any given number of steps of evolution, the various copies of this causal graph will have “reached different stages”. Here are the results after 3 steps:

   ✕
   

   And after 4 steps:

   ✕
   

   One can also look at rulial multiway causal graphs for other sets of Turing machines. For s = 1, k = 1 one has (here after 5 steps)

   ✕
   

   which is equivalent to:

   ✕
   

   The individual causal graphs in this case are immediately all the same, and are just:

   ✕
   

   For s = 2, k = 1 one already has after 3 steps

   ✕
   

   or in layered form:

   ✕
   

   After 5 steps this becomes

   ✕
   

   or in layered form:

    ✕
   

   In this case, the individual causal graphs after 5 steps are:

    ✕
    

    For s = 1, k = 2 one has very similar results to the case s = 2, k = 2. After 3 steps the causal graph is:

    ✕
    

    Or after 5 steps:

    ✕
    

    Removing multiple edges this is:

    ✕
    

    The individual causal graphs in this case are:

    ✕
    

    Rulial Graphs

    Just as for ordinary multiway graphs one can study branchial graphs which represent their transversals, so similarly for rulial multiway graphs one can study rulial graphs which represent their transversals. The layered way we have drawn multiway graphs corresponds to a particular choice of foliation—and with this choice, we can immediately generate rulial graphs.

    For s = 2, k = 1 one has after 2 steps

    ✕
    

    while after 3 steps one gets:

    ✕
    

    For s = 1, k = 2 one gets after 2 steps:

    ✕
    

    while after 3 steps one has:

    ✕
    

    The sequence of results for steps 1 through 4 is:

    ✕
    

    For s = 2, k = 2 the corresponding results are:

    ✕
    

    What do these pictures mean? Just as branchial graphs in ordinary multiway systems can be thought of as “entanglement maps” for states (interpreted in our models as quantum states) in ordinary “multiway space”, so here these rulial graphs can be thought of as “entanglement maps” for states in rulial space. In other words, they are a kind of map of which Turing machine configurations are “evolutionarily close” to which other ones, in the sense that they can be reached with only a few different choices of rules.

    Deterministic Turing Machine Paths in Rulial Space

    The rulial multiway graph defines all paths that can be followed by all non-deterministic Turing machines. At each node in this graph there is therefore an outgoing edge corresponding to any possible Turing machine transition from the configuration corresponding to that node. But what if one considers just a single deterministic Turing machine? Its evolution is then a single path within the rulial multiway graph.

    Consider for example the s = 2, k = 2 Turing machine:

    ✕
    

    This machine evolves from a blank tape according to:

    ✕
    

    This corresponds to a path in the rulial multiway graph:

    ✕
    

    With a different initial condition

    ✕
    

    the path can be very different:

    ✕
    

    This pictures might make it seem that the paths corresponding to the evolution of deterministic Turing machines are geodesics in the rulial multiway graph. But in general they are definitely not. For example, starting from a blank tape, the rule

    ✕
    

    follows this path on the rulial multiway graph:

    ✕
    

    If one allows any possible Turing machine transition at every step, then one can follow a geodesic path from a node corresponding to an initial condition to a node corresponding to any other configuration. But if one restricts oneself to the transitions in a particular (deterministic) Turing machine, then there will in general be many configurations one will never reach, and even those that one can reach, one may reach by a circuitous route in the rulial multiway graph.

    Given a particular configuration (corresponding to a node in the rulial multiway graph), there may be many deterministic Turing machines that can reach it from a given initial state. One can consider each of these Turing machines to be implementing a certain algorithm. So then the “optimal algorithm” will be the one which is shortest among deterministic Turing machine paths. As I just mentioned, this won’t typically be the shortest possible path: that will usually be achieved by a non-deterministic Turing machine with a particular sequence of transitions. But there is still an optimal case (i.e. an “optimal algorithm”) among deterministic Turing machines.

    But now we can ask across all possible deterministic Turing machines where they can reach in the rulial multiway graph. Here is the result for 4 steps for the s = 2, k = 2 Turing machines we have been considering:

    ✕
    

    These are the results for 1, 2 and 3 steps:

    ✕
    

    What is the significance of this? Essentially what we’re seeing is a comparison of what can be achieved with deterministic computation versus non-deterministic. Taking the 4-step case as an example, the “background” gray rulial multiway graph shows what can be achieved with arbitrary non-deterministic computation in 4 steps. The red region is what deterministic computation can achieve in the same number of steps.

    In a sense this is a very simple empirical analog of the P vs. NP problem. Unlike the real P vs. NP case, we’re not allowing arbitrary polynomial-time algorithms here; we’re just looking at possible s = 2, k = 2 Turing machine algorithms running specifically for 4 steps. But if we were to generalize this appropriately, P = NP would imply that the “red region” must in some limit in effect “reach anywhere in the graph”.

    A little more precisely, the official definition of P and NP is for decision problems: you start from some initial condition which defines an instance of a problem (“Is this Boolean formula satisfiable?” or whatever), and the system must eventually evolve to a state representing either “yes” or “no”. We can imagine setting things up so that the outcomes correspond to particular configurations of the Turing machine. The inputs are then also encoded as initial configurations of the Turing machine, and we want to know what happens as we consider inputs of progressively larger sizes. We can imagine drawing the rulial multiway graph so that progressively larger inputs are shown “progressively further from the center”. If we consider non-deterministic Turing machines (associated with the class of NP computations), then the shortest computation will be a geodesic path in the rulial multiway graph from the input configuration to the outcome configuration. But for deterministic Turing machines (associated with the class P) it will in general be some much more circuitous path.

    The standard P vs. NP problem asks about limiting behavior as one increases the size of the input computation—and one might imagine that the question could be “geometrized” in terms of some continuum limit of the rulial multiway graph. Of course, there is no guarantee that any reasonable limit exists. And it could perfectly well be that the question of whether P ⊂ NP is actually undecidable, or in other words, that no finite proof of it can be given within a standard axiom system (such as Peano arithmetic or ZFC set theory).

    One could imagine empirically testing more and more Turing machines, and seeing how they perform on an NP-complete problem. One might think of plotting their running times as a function of n for increasingly large n. For a while a particular Turing machine might be the winner. But then another one might take over. And there might be no end to how many “surprises” would occur as one increases n. (Somehow this is reminiscent of the story of the Skewes number and whether LogIntegral[n] > Prime[n].)

    In computational complexity theory one usually thinks about explicitly constructing optimal algorithms by standard “engineering-like” methods. But I think there’s a lot to be learned from a more empirical approach—in which one doesn’t try to construct optimal algorithms, but just finds them “in the wild” by searching all possible programs. In the past, it might not have seemed that “just searching for programs” would ever produce anything terribly interesting. But one of the big consequences of what I discussed in A New Kind of Science is that even among tiny programs—small enough that one can, for example, enumerate all of them—there’s often very complex and potentially “useful” behavior. And that makes it seem much more reasonable to try to do “empirical computational complexity theory”—enumerating possible programs to find optimal ones, and so on.

    Small programs can be thought of as ones with low algorithmic complexity. So searching for “fast programs” among these can be thought of as searching for programs with low time complexity, and low algorithmic complexity. We don’t know exactly how strong the constraint of low algorithmic complexity is, but from what I’ve seen in the computational universe (and what’s embodied in things like the Principle of Computational Equivalence), it seems as if it’s not such a big constraint.

    I studied “empirical computational complexity theory” a bit in A New Kind of Science, notably for Turing machines. And one of the interesting observations was that the optimal algorithm for things was often not something that could readily be constructed in a step-by-step engineering way. Instead, it was something that one basically could only find pretty much by doing a search of possible programs—and where there usually didn’t seem to be a “general form” that would “apply for all n”, and let one readily deduce the properties of the n → ∞ limit. In other words, it didn’t seem like what we’d now think of as the sequence of paths in rulial space would show any sign of converging to a “continuum limit”.

    A decent example of all this occurs in sorting networks. Imagine that you are given a collection of n numbers to sort. You can straightforwardly do this by making about n2 pairwise comparisons, and it’s pretty easy to optimize this a bit.

    But explicit searches have revealed that the actual optimal networks for successive n are:

    ✕
    

    What’s notable is how complicated and “random” they look; there doesn’t seem to be any obvious pattern to them (and my guess is that there fundamentally isn’t). Here’s a plot of the sizes of these networks (divided by n2):

    ✕
    

    It’s worth noting that these are deterministic networks. One could also imagine non-deterministic networks (and indeed one could construct a rulial multiway graph by considering all possible successive placements of pairwise comparisons)—and in a non-deterministic network it’s always possible to sort n numbers in at most n – 1 steps.

    The Space of Deterministic Turing Machine Computations

    We’ve just seen how the results of deterministic Turing machine computations lay out in the rulial multiway space of all possible non-deterministic Turing machine computations. But what happens if we just look at the graph of deterministic Turing machine computations on their own?

    Here are the full rulial multiway graphs for 2 and 3 steps with the graphs of deterministic Turing machine computations superimposed, as before:

    ✕
    

    But now let’s “pull out just the red subgraphs”—in other words, include as nodes only those configurations that at least one of the 4096 s = 2, k = 2 Turing machines can reach after 2 or 3 steps:

    ✕
    

    Notice that after 2 steps, deterministic Turing machines can still reach all 36 configurations that non-deterministic ones can reach (though not through quite as many paths). But after 3 steps, among all the deterministic Turing machines, they can only reach 68 possible configurations, while non-deterministic ones can reach 100 configurations.

    For all possible non-deterministic Turing machines with s = 2, k = 2 the total number of configurations that can be reached eventually roughly doubles on successive steps:

    ✕
    

    For deterministic Turing machines, however, the number of possible configurations that can be reached soon increases much more slowly:

    ✕
    

    And in fact there’s an obvious bound here: at any given step, the most that can happen is that each of the 4096 s = 2, k = 2 Turing machines leads to a new configuration—or, in other words, the maximum number of configurations reached increases by 4096.

    Looking at the differences on successive steps, we find:

    ✕
    

    In other words, among all 4096 Turing machines, about 100 “novel configurations” are reached at each successive step. (The actual sequence here looks surprisingly random; it’s not clear whether there’s any particular regularity.)

    Now let’s look at the actual graphs formed. With 4 through 7 steps we get:

    ✕
    

    After 10 and 20 steps the results are:

    ✕
    

    Here is the result after 50 steps:

    ✕
    

    There’s a surprising amount of structure in these graphs. There’s a “central region” near the initial blank-tape configuration (shown highlighted below) in which many different Turing machines end up visiting the same configurations:

    ✕
    

    Here’s a 3D rendering of this region:

     ✕
    

    But away from this region there end up being “spokes” (about 100 of them) corresponding to Turing machines that “independently explore new territory” in the space of configurations.

    What are those “configurations on the edge” like? Here are sorted collections of them for the first few steps:

     ✕
     

     For comparison, here is the result for all configurations that can be reached by non-deterministic Turing machines after just 2 steps:

     ✕
     

     Here are the results for deterministic Turing machines after more steps:

     ✕
     

     We can also ask which machines are the ones that typically “explore new territory”. Here’s the result for 30 steps:

     ✕
     

     As we go to more steps, the graph of configurations that can be reached by deterministic Turing machines grows. But does at least the core of it reach some kind of limit after sufficiently many steps? We can get a sense of this by looking—as we have done so many times before—at the growth rate of the geodesic ball in the graph starting from the initial blank-tape configuration. The total number of new configurations that can be reached on each new layer of the geodesic ball is at most 4096—and in reality it’s much smaller. Here are the numbers of new configurations added on successive layers at steps 100, 200, …, 500 in the overall evolution:

     ✕
     

     The cause of the “steps down” becomes clearer if one examines more closely the “spokes” in the graph above. Here’s one example:

     ✕
     

     And basically what’s happening is that multiple Turing machines are tracing out roughly the same sequences of configurations, but some do it “efficiently”, while others “waste time”, for example having the head flip around on alternate steps. The “inner parts” of the spokes—that are closer to the initial node—involve both “inefficient” and more efficient Turing machines. But the “inefficient” Turing machines will simply not get as far, so they do not contribute to the outer layers of the geodesic ball. The final “step down” in the plot above—at basically half the total number of steps used for the Turing machines—involves the “petering out” of the roughly half the Turing machines that effectively “waste half their steps”.

     For the “spoke” shown, here are the actual Turing machine histories involved (there are 8 machines that made identical copies of the first history):

     ✕
     

     The Cellular Automaton Analog

     The kind of graphs we’ve just made for deterministic Turing machines can be made for any family of deterministic computational systems. And in particular they can be made for the (at least for me, much more familiar) case of the 256 k = 2, r = 1 cellular automata. (And, yes, it’s somewhat amazing that in all these years I’ve never made such pictures before—though there’s a note on page 956 of A New Kind of Science that gets close.)

     Here are the results for 5 and 10 steps, starting from an initial condition containing a single black cell:

     ✕
     

     And here is the result for 50 steps:

     ✕
     

     Here is the result for 10 steps, annotated with the actual cellular automaton evolution for the rules that “reach furthest”:

     ✕
     

     It’s slightly easier to see what’s going on if we include only even-numbered rules (which leave a blank state blank):

     ✕
     

     It’s an interesting map of “cellular automaton space”. (Note the presence of rule 30 on the lower left, and rule 110 on the lower right.)

     The total number of new configurations explored by all rules on successive steps has a more regular form than for Turing machines:

     ✕
     

     The result mostly alternates with period 4 between 72 and 84, though with dips at steps of the form 2m.

     If we go for a certain number of steps (say 200), and then look at the geodesic ball centered on the initial condition, the number of configurations in successive layers is just:

     ✕
     

     These results are all for ordinary, deterministic cellular automata. So are there non-deterministic cellular automata? Typically, cellular automata are defined to consistently update every cell at every step. But one can also consider sequential cellular automata, where specific cells are updated at each step—and in this case it is straightforward to define a non-deterministic version, for which things like multiway systems can be constructed. (Note that for a full rulial multiway system, such non-deterministic cellular automata are basically the same as non-deterministic mobile automata, which are close to Turing machines.)

     Computation Capabilities and the Structure of Rulial Space

     What do the computational capabilities of Turing machines mean for their rulial space? Let’s start with computation universality.

     An important fact about deterministic Turing machines is that there are universal ones. Among the 4096 s = 2, k = 2 rules there aren’t any. But as soon one goes to s = 2, k = 3 rules (of which there are 2,985,984) it’s known (thanks to my results in A New Kind of Science, and Alex Smith’s 2007 proof) that there is a universal machine. The rule is:

     ✕
     

     Starting from a blank tape, this machine gives

     ✕
     

     and the corresponding causal graph is:

     ✕
     

     But what does the existence of a universal machine mean in the rulial multiway system? From any given initial condition, any deterministic Turing machine will trace out some trajectory in the rulial multiway graph. And if a machine is universal it means that by appropriately picking its initial conditions it can be “programmed” to “emulate” any other Turing machine, in the sense that its trajectory will track the trajectory of whatever Turing machine it’s emulating. What does “track” mean? Basically, that there’s some fixed scheme that allows one to go from the states of the universal Turing machine to the states of the machine it’s emulating. The “scheme” will correspond to some translation in the rulial multiway graph, and the requirement is that this translation is somehow always limited. In other words, the trajectory of a universal machine will “flail around” as its starting point (i.e. initial condition) moves in the rulial multiway graph, and this “flailing” will be diverse enough that the trajectory can get close to the trajectory of any given other machine.

     If, on the other hand, the machine one’s dealing with isn’t universal, then its trajectory won’t “flail around” enough for this work; the trajectory will in a sense be too constrained to successfully “track” all possible other deterministic Turing machine trajectories.

     What about non-deterministic Turing machines? What universality can be interpreted to mean in this case is that given a particular initial condition, the output one wants occurs somewhere on the different paths followed by the non-deterministic Turing machine. (If one’s trying to do a decision problem—as in the computational complexity class NP—then one can arrange to “signal that one’s got an answer” through some feature of the Turing machine state.) In the case of “extreme non-determinism”—as used to construct the rulial multiway graph—computation universality in a sense becomes a statement purely about the structure of the rulial multiway graph. And basically it just requires that the rulial multiway graph is sufficiently connected—which is guaranteed if there’s causal invariance (so there’s nothing like an “event horizon” anywhere).

     But does one have to allow “extreme non-determinism” to get universality? With s = 2, k = 3 we know that there’s a purely deterministic Turing machine that achieves it. And my guess is that among s = 2, k = 2 there are “slightly multiway” rules that also do. In a standard deterministic s = 2, k = 2 Turing machine, there are 4 cases of the rule that are each specified to have a unique outcome. But what if even just one of those cases in the rule has two outcomes? The rule is non-deterministic, but it can be thought of as just being the result of specifying 5 defining cases for the rule. And it’s my guess that even this will be sufficient to get universality in the system.

     A non-deterministic rule like this will not trace out a single path in the rulial multiway graph. Instead, it’ll give a bundle of paths, with different paths corresponding to different non-deterministic choices. But the story of universality is very similar to the deterministic case: one simply has to ask whether anything in the bundle successfully manages to track the trajectory of the machine one’s emulating.

     It’s worth remembering that any given rule won’t typically be following geodesics in rulial space. It’ll be following some more circuitous path (or bundle of paths). But let’s say one has some rule that traces out some trajectory—corresponding to performing some computation. The Principle of Computational Equivalence implies that across different possible rules, there’s a standard “maximum computational sophistication” for these computations, and many rules achieve it. But then the principle also implies that there’s equivalence between these maximally sophisticated computations, in the sense that there’s always a limited computation that translates between them.

     Let’s think about this in rulial space. Let’s say we have two rules, starting from the same initial condition. Well, then, at the beginning it’s trivial to translate between the rules. But after t steps, the states reached by these rules can have diverged—making it potentially progressively more difficult to translate between them. A key idea of the rulial multiway graph—and rulial space—is that it lets one talk about both computations and translations between them in uniform ways. Let’s say that the trajectories of two rules have gone a certain distance in rulial space. Then one can look at their divergence, and see how long a “translation computation” one has to do to get from one to the other.

     In ordinary spacetime, let’s say a certain time t has elapsed. Then we know that the maximum spatial distance that can have been traversed is c t, where c is the speed of light. In rulial space, there’s something directly analogous: in time t, there’s a maximum rulial distance that can be traversed, which we can call ρ t. But here the Principle of Computational Equivalence makes it a crucial contribution: it implies that throughout rulial space, and in all situations, ρ is fixed. It can take an irreducible amount of computational work to successfully translate from the outcome of one rule to another. But this always scales the same way. There’s in effect one scale of computational irreducibility, and it’s characterized by ρ . Like the constancy of the speed of light uniformly limits physical motion, the constancy of ρ uniformly limits rulial motion.

     But let’s say you’re trying to get from one point in rulial space to another. If from your starting point you can follow the path of an irreducible—and effectively universal—computation, then you’ll successfully be able to reach the other point. But if from your starting point you can only follow a reducible computation this won’t generally be true. And what this means is that “pockets of computational reducibility” in rulial space act a bit like black holes in physical space. You can get into them from regions of irreducibility, but you can’t get out of them.

     There are probably signs of phenomena like this even in the rulial space for simple Turing machines that we’ve explored here. But there’s considerably more that needs to be worked out to be able to make all the necessary connections.

     The Emerging Picture of Rulial Space

     There are lots of analogies between physical space, the branchial space, and rulial space. For example, in physical space, there are light cones that govern the maximum rate at which effects can propagate between different parts of physical space. In branchial space, there are entanglement cones that govern the maximum rate of quantum entanglement. And in rulial space, one can think of “emulation cones”, which govern the maximum rate at which one description of behavior can be translated into another.

     And when it comes to applying these things to modeling the physical universe, a crucial point is that the observer is necessarily part of the system, governed by the same rules as everything else. And that means that the observer can only be sensitive to certain “appropriately modded-out” aspects of the system. But in actually imagining how an observer “perceives” a system it’s almost always convenient to think about coordinatizing the system—by defining some appropriate foliation. In physical space, this involves foliating the causal graph using reference frames like in relativity. In branchial space, it involves our concept of quantum observation frames. And in rulial space, we can invent another such concept: a rulial description frame, or just a rulial frame.

     Different rulial frames in effect correspond to describing the evolution of the universe as operating according to different rules. Causal invariance implies that in the end different rulial frames must give equivalent results. But the specific way one describes the time evolution of the universe will depend on what rulial frame one’s using. In one frame one would be describing the universe in one way; in another frame, another way. And the “story one tells” about how the universe evolves will be different in the different frames.

     Much like with superposition in quantum mechanics, there’s probably some notion of regions in rulial space, in which one’s somehow viewing the universe as operating according to “rulially entangled” collections of rules.

     But while our original motivation was understanding physics, a lot of what we’re studying about rulial space also applies to purely computational systems. For example, we can think of rulial space even without having any notion of physical space. And we can in effect imagine that rulial space is some kind of map of a space of possible rules for computational systems. (Notice that because of computation universality and the Principle of Computational Equivalence it ultimately doesn’t matter what particular type of rule—Turing machine, cellular automaton, combinator, whatever—is used to “parametrize” rulial space.)

     Different places in rulial space in some sense correspond to different rules. Paths at different places in rulial space correspond to evolution according to different rules. So what is the analog of motion in rulial space? In effect it’s having a frame which progressively changes the rule one’s using. If one’s trying to find out what happens in one’s system, it’s fundamentally most efficient to “stick with one rule”. If one progressively changes the rule, one’s going to have to keep “translating back” to the original rule, by somehow emulating this rule with whatever rule one’s reached. And the result of this is there’ll be exactly the analog of relativistic time dilation. Faster motion in rulial space leads to slower progression in time. Of course, to discuss this properly, we really have to talk about the rulial multiway causal graph, etc.

     But one thing is clear: motion faster than some maximum speed ρ is impossible. From within the system, you simply can’t keep the correct rulial causal connections if you’re changing your rulial location faster than ρ .

     In an abstract study of Turing machines, ρ is just an arbitrary parameter. But in the actual physical universe, it must have a definite value, like the speed of light c, or our maximum entanglement speed ζ . It’s difficult even to estimate what ζ might be. And presumably estimating ρ will be even harder. But it’s interesting to discuss at least how we might start to think about estimating ρ .

     The first key observation about rulial space is that in our models, it’s discrete. In other words, there’s a discrete space of possible rules. Or, put another way, theories are quantized, and there’s somehow an elementary distance between theories—or a minimum distance in “theory space” between neighboring theories.

     But what units is this distance in? Basically it’s in units of rule—or program—size. Given any program—or rule—we can imagine writing that program out in some language (say in Wolfram Language, or as a program for a particular universal Turing machine, or whatever) And now we can characterize the size of the program by just looking at how many tokens it takes to write the program out.

     Of course, with different languages, that number will be different—at the simplest level just like the number of decimal digits necessary to represent a number is different from the number of binary digits, or the length of its representation in terms of primes. But it’s just like measuring a length in feet or meters: even though the numerical value is different, we’re still describing the same length.

     It’s important to point out that it’s not enough to just measure things in terms of “raw information content”, or ordinary bits, as discussed in information theory. Rather, we want some kind of measure of “semantic information content”: information content that directly tells us what computation to do.

     It’s also important to point out that what we need is different from what’s discussed in algorithmic information theory. Once one has a computation universal system, one can always use it to translate from any one language to any other. And in algorithmic information theory the concept is that one can measure the length of a program up to an additive constant by just expecting to include an “emulation program” that adapts to whatever language one’s measuring the length in. But in the usual formalism of algorithmic information theory one doesn’t worry about how long it’s going to take for the emulation to be done; it’s just a question of whether there’s ultimately enough information to do it.

     In our setup, however, it does matter how long the emulation takes, because that process of emulation is actually part of our system. And basically we need the number of steps needed for the emulation to be in some sense bounded by a constant.

     So, OK, what does this mean for the value of ρ ? Its units are presumably program size per unit time. And so to define its value, we’ll have to say how we’re measuring program size. Perhaps we could imagine we write our rules in the Wolfram Language. Then there should be a definite value of ρ for our universe, measured in Wolfram-Language-tokens per second. If we chose to use (2,3)-Turing-machine-tape-values per year then we’d get a different numerical value. But assuming we used the correct conversions, the value would be the same. (And, yes, there’s all sorts of subtlety about constant-time or not emulation, etc.)

     In some sense, we may be able to think of ρ as the ultimate “processor speed for the universe”: how fast tokens in whatever language we’re using are being “interpreted” and actually “executed” to determine the behavior of the universe.

     Can we estimate the value of ρ ? If our units are Wolfram-Language-tokens per second we could start by imagining just computing in the Wolfram Language some piece of the rulial multiway graph for our models and seeing how many operations it takes. To allow “all possible rules” we’d have to increase the possible left- (and right-) hand sides of our rules to reflect the size of the hypergraph representing the universe at each step. But now we’d need to divide by the “number of parallel threads” in the rulial multiway graph. So we can argue that all we’d be left with is something like (size of spatial hypergraph represented in Wolfram Language) / (elementary time).

     So, based on our previous estimates (which I don’t consider anything more than vaguely indicative yet) we might conclude that perhaps:

     ρ ~ 10450 Wolfram-Language-tokens/second

     The number of “parallel threads” in the rulial multiway graph (the rulial analog of Ξ) might then be related to the number of possible hypergraphs that contain about the number of nodes in the universe, or very roughly (10350)^(10350) ≈ . If we ask the total number of Wolfram Language tokens processed by the universe, there’ll be another factor ~10467, but this “parallelism” will completely dominate, and the result will be about:

     Wolfram-Language-tokens

     OK, so given a value of ρ , how might we conceivably observe it? Presumably there’s an analog of quantum uncertainty in rulial space, that’s somehow proportional to the value of ρ . It’s not completely clear how this would show up, but one possibility is that it would lead to intrinsic limits on inductive inference. For example, given only a limited observation time, it might be fundamentally impossible to determine beyond a certain (“rulial”) accuracy what rule the universe is following in your description language. There’d be a minimum rate of divergence of behaviors from different rules, associated with the minimum distance between theories—and it would take a certain time to distinguish theories at this rate of divergence.

     In our models, just like every causal edge in physical and branchial space is associated with energy, so should every causal edge in rulial space be. In other words, the more processing that happens in a particular part of rulial space, the more physical energy one should consider exists there. And just as with the Einstein equations in physical space, or the Feynman path integral in branchial space, we should expect that the presence of energy in a particular region of rulial space should reflect in a deflection of geodesics there.

     Geodesics in rulial space are the shortest paths from one configuration of the universe to another, using whatever sequences of rules are needed. But although that’s somewhat like what’s considered at a theoretical level in non-deterministic computation, it’s not something we’re usually familiar with: we’re used to picking a particular description language and sticking with it. So exactly what the interpretation of deflections of geodesics in rulial space should be isn’t clear.

     But there are a few other things we can consider. For example, presumably the universe is expanding in rulial space, perhaps implying that in some sense more descriptions of it are becoming possible over time. What about rulial black holes? As mentioned above, parts of rulial space that correspond to computational reducibility should behave like black holes, where in effect “time stops”. Or, in other words, while in most of the universe time is progressing, and irreducible computation is going on, computational reducibility will cause that process to stop in a rulial black hole.

     Presumably geodesics near the rulial black hole will be pulled towards it. Somehow when there’s a description language that leads to computational reducibility, languages near it will tend to also stop being able to successfully describe computationally irreducible processes.

     Can we estimate the density of rulial black holes? Let’s consider the Turing machine case. In effect we’re going to want to know what the density of non-universality is among all possible non-deterministic Turing machines. Imagine we emulate all Turing machines using a single universal machine. Then this is effectively equivalent to asking what fraction of initial conditions for that machine lead to reducible behaviors—or, in essence, in the traditional characterization of Turing machines, halt. But the probability across all possible inputs that a universal Turing machine will halt is exactly Greg Chaitin’s Ω . In other words, the density of rulial black holes in rulial space is governed by Ω .

     But Ω isn’t just a number like π that we can compute as accurately as we want; it’s noncomputable, in the sense that it can’t be computed to arbitrary accuracy in any finite time by a Turing machine. Now, in a sense it’s not too surprising that the density of rulial black holes is noncomputable—because, given computational irreducibility, to determine for certain whether something is truly a rulial black hole from which nothing can escape one might have to watch it for an unbounded amount of time.

     But for me there’s something personally interesting about Greg Chaitin’s Ω showing up in any way in a potential description of anything to do with the universe. You see, I’ve had a nearly 40-year-long debate with Greg about whether the universe is “like π ” or “like Ω ”. In other words, is it possible to have a rule that will let us compute what the universe does just like we can compute (say, in principle, with a Turing machine) the digits of π ? Or will we have to go beyond a Turing machine to describe our universe? I’ve always thought that our universe is “like π ”; Greg has thought that it might be “like Ω ”. But now it looks as if we might both be right!

     In our models, we’re saying that we can compute what the universe does, in principle with a Turing machine. But what we’re now finding out is that in the full rulial space, general limiting statements pull in Ω . In a particular rulial observation frame, we’re able to analyze the universe “like π ”. But if we want to know about all possible rulial observation frames—or in a sense the space of all possible descriptions of the universe—we’ll be confronted with Ω .

     In our models, the actual operation of the universe, traced in a particular rulial observation frame, is assumed never to correspond to anything computationally more powerful than a Turing machine. But let’s say there was a hypercomputer in our universe. What would it look like? It’d be a place where the effective ρ is infinite—a place where rulial geodesics infinitely diverge—a kind of white hole in rulial space (or perhaps a cosmic event horizon). (We can also think about the hypercomputer as introducing infinite-shortcut paths in the rulial multiway graph which ordinary Turing-machine paths can never “catch up to” and therefore causally affect.)

     But given a universe that does hypercomputation, we can then imagine defining a rulial multiway graph for it. And then our universe will show up as a black hole in this higher-level rulial space.

     But OK, so if there’s one level of hypercomputer, why not consider all possible levels? In other words, why not define a hyperrulial multiway graph in which the possible rules that are used include both ordinary computational ones, but also hypercomputational ones?

     And once we’re dealing with hypercomputational systems, we can just keep going, adding in effect more and more levels of oracles—and progressively ascending the arithmetic hierarchy. (The concept of “intermediate degrees” might be thought to lead to something that isn’t a perfect hierarchy—but I suspect that it’s not robust enough to apply to complete rulial spaces.) Within the hyperrulial multiway graph at a particular level, levels below it will be presumably appear as rulial black holes, while ones above will appear as rulial white holes.

     And there’s nothing to keep us only considering finite levels of the arithmetic hierarchy; we can also imagine ascending to transfinite levels, and then just keeping going to higher and higher levels of infinity. Of course, according to our models, none of this is relevant to our particular physical universe. But at a theoretical level, we can still at least to some extent “symbolically describe it”, even in our universe.

     Thanks

     Thanks to Jonathan Gorard for a variety of discussions, as well as to Tali Beynon, Todd Rowland, Ed Pegg, Jose Martin-Garcia and Christopher Wolfram.

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Generalized Physics and the Theory of Computation
Let’s say we find a rule that reproduces physics. A big question would then be: “Why this rule, and not another?” I think there’s a very elegant potential answer to this question, that uses what we’re calling rule space relativity—and that essentially says that there isn’t just one rule: actually all possible rules are being used, but we’re basically picking a reference frame that makes us attribute what we see to some particular rule. In other words, our description of the universe is a sense of our making, and there can be many other—potentially utterly incoherent—descriptions, etc.
But so how does this work at a more formal level? This bulletin is going to explore one very simple case. And in doing so we’ll discover that what we’re exploring is potentially relevant not only for questions of “generalized physics”, but also for fundamental questions in the theory of computation. In essence, what we’ll be doing is to study the structure of spaces created by applying all possible rules, potentially, for example, allowing us to “geometrize” spaces of possible algorithms and their applications.
In our models of physics, we begin by considering spatial hypergraphs that describe relations between “atoms of space”. Then, looking at all possible ways a given rule can update these hypergraphs, we form what we call a multiway graph. The transversals of this graph define what we call branchial space, in which we can see the pattern of entanglements between quantum states. 
But there’s also a third level we can consider. Instead of just forming a multiway graph in which we do all possible updates with a given rule, we form a rulial multiway (or “ultramultiway”) graph in which we follow not only all possible updates, but also all possible rules. The transversals to this rulial multiway graph define what we call rulial space. Causal invariance in the rulial multiway graph then implies “rule space relativity” which is what allows us to use different possible reference frames to describe the universe. 
In the end, the Principle of Computational Equivalence implies a certain invariance to the limiting structure of the rulial multiway graph, independent of the particular parametrization of the set of possible rules. But for the sake of understanding rulial space and the rulial multiway graph—and getting intuition about these—this bulletin is going to look at a specific set of possible rules, defined by simple Turing machines.
The rules we’ll use aren’t a good fit for describing our universe. But the comparative simplicity of their structure will help us in trying to elucidate some of the complexities of rulial space. In addition, using Turing machines will make it easier for us to make contact with the theory of computation, where Turing machines are standard models.
Turing Machines
Here’s a representation of the rule for a particular 2-state (s = 2), 2-color (k = 2) Turing machine:
The pointer represents the “head” of the Turing machine and its orientation represents the state of the head. Here’s what this particular Turing machine does over the course of a few steps starting from a “blank tape” (i.e. all squares white):
There are (2 s k) sk  possible s-state k-color Turing machines, or 4096 s = 2, k = 2 machines. Here are all the distinct behaviors that occur (up to left-right reflection) in these 4096 machines, starting from a blank tape:
But note that all these are deterministic Turing machines, in the sense that, for a given machine, it is always the same rule that is applied at every step. But what about non-deterministic Turing machines? The idea here is to allow several different rules at any given step. 
As a simple example, consider the pair of rules:
To show the possible evolution histories, we can construct a multiway system:
Each arrow represents an application of one of the two possible rules. A non-deterministic Turing machine is usually considered to follow a particular path, corresponding to a particular evolution history. Then a typical question is whether there exists some path that leads to a particular final state (which might be viewed as the solution to some particular problem). (In a quantum Turing machine, one considers collections of multiple states, viewed as being in a quantum superposition.)
In what we’re going to be doing here, we want to study the “extreme case” of non-determinism—where at every step, every possible Turing machine rule (at least with a given s, k) is applied. Then we’ll be interested in the full rulial multiway graph that’s created.
The Rulial Multiway Graph
Start with a blank tape. Then apply all possible 2,2 Turing machine rules. This is the rulial multiway graph that’s formed after 1 step:
Each individual arrow here represents many possible rules from all the 4096 discussed above. Here’s what happens after 2 steps:
In layered form, this becomes:
In creating these rulial multiway graphs, there’s an important simplification we’re able to make. We don’t need to separately apply all 4096 2,2 Turing machine rules to each state at each step—because all that ever matters is the one case of the rule that’s relevant to the particular state one has. There are 4 possible individual cases—and for each of these there are 8 possible outcomes, leading to 32 “micro-rules”:
After 3 steps, the rulial multiway graph has the form:
In 3D this becomes:
In layered form it is:
After 5 steps, the rulial multiway graph is:
What about other numbers of states and colors? Here are the rulial multiway graphs after 3 steps for Turing machines with various numbers of states and colors :
Notice the presence of s = 1 examples. Even with a single possible state, it is already possible to form a nontrivial rulial multiway system. Here are the results for k = 2 after 1 step:
2 steps:
6 steps:
In principle one can also consider the even simpler case of s = k = 1. After 1 step one gets:
After 2 steps this becomes:
And after 3 steps:
As a layered graph rendering, this becomes:
With s = 2, k = 1 one gets after 1 step:
After 2 steps this becomes:
And after 3 steps:
After 6 steps this becomes:
At least in the central part of this graph, there are edges going in both directions. Combining these, and treating the graph as undirected, one gets:
With 3 states (s = 3) there are 3 tracks:
One can also consider Turing machines in which the head can not only move left or right, but can also stay still. In this case, with s = 2, k = 1 one gets after 1 step:
After 2 steps this becomes:
After 4 steps, removing repeated edges, etc. this gives:
As another generalization, one can consider Turing machines that can move not just one, but also two squares:
After 2 steps one gets:
After 3 steps, removing repeated edges, etc. this gives:
The Limit of the Rulial Multiway Graph
What is the limiting structure of the rulial multiway graph after an infinite number of steps? The first crucial observation is that it’s in a sense homogeneous: the structure of the graph around any given node is always the same (i.e. it’s a vertex-transitive graph). To see why this is true, recall that each node in the graph corresponds to a particular distinct configuration of the Turing machine. This node will lead to all nodes that can be obtained from it by one step of Turing machine evolution. But (assuming the head always moves ±1 square) there are always exactly 2 s k of these. And because we are following all possible “micro-rules”, we can think of ourselves as just “blindly overwriting” whatever configuration we had, so that the structure of the graph is the same independent of what configuration we’re at. 
As a simple example, here’s what happens in the s = 2, k = 1 case, after 3 steps:
There’s some trickiness at the ends, but in the central region we see as expected that each node has exactly 4 successors. If we pick out the subgraph around the blank-tape starting node it has the form:
In general, the limiting neighborhood of every node is just:
In the full graph, these neighborhoods are knitted together, giving pieces like:
Let’s now consider the case s = 1, k = 2. After 4 steps the full graph is:
Here the neighborhood of the start node is:
And in the limiting graph, every node will have a local neighborhood with this same structure. If we look at successively larger neighborhoods, the limiting form of these for every node will be:
Here’s a table of the number of distinct nodes in these successively larger neighborhoods for Turing machines with various values of s and k:
In the limit t   the number of nodes reached in all cases goes like:
 
For k = 1, one finds (for ):
 
For t = 1 one has:
 
If one ignores directedness in the graph, and just counts the total number of neighbors out to distance t, the results for k = 1 are the same as before, but in other cases they are different:
(These results asymptotically seem larger by a factor .)
So for k = 1, the limiting rulial multiway graph behaves like a 1-dimensional space, but for all k  2, it behaves like an infinite-dimensional space, in which the volumes of geodesic balls grow exponentially with volume.
Finite Tapes
So far we’ve always assumed that our Turing machine tape is unbounded. But in some ways it’s easier to see what’s going on if we instead limit the tape, so we have a finite total number of states (). One thing we can do is to consider a cyclic tape. 
For with successively longer tapes we then get:
For we get:
For tape size 10, the results for successive values of s are:
If instead of having a cyclic tape, we just have a finite tape, and insist that the head never goes off either end, then for we get: 
For we get
and the overall behavior for successive s is exactly as we saw above for unbounded tapes:
For , this is what happens in the cyclic case for length 3:
The results for lengths 1 through 6 are:
Rendering these in 3D makes the connection with progressively high-dimensional hypercubes slightly clearer:
The non-cyclic case for lengths 2 through 6 (length 1 is trivial):
Rendered in 3D these become:
In the case for a length-3 cyclic tape we get:
Rendering the results for lengths 1 through 6 in 3D gives:
In the non-cyclic case the results for lengths 2 through 6 in 3D are (the length-1 case is trivial):
The Turing Machine Group
It turns out that there’s a nice mathematical characterization of rulial multiway graphs for Turing machines: they’re just Cayley graphs of groups that we can call “Turing machine groups”. Why is this? Basically it’s because the possible configurations of a Turing machine have a direct correspondence with transformations that can act on these configurations. And in particular, one can pick out certain transformations that correspond to individual transitions in a non-deterministic Turing machine, and use these as generators in a presentation of the group. 
Let’s start by considering the case of Turing machines with finite cyclic tapes. If the tape has length n, the total number of possible configurations of the machine is n s k  n . (Note that if the tape is finite but not cyclic then one doesn’t get a group.)
Assume for now s = 1, k = 2. Then for n = 3, there are 24 possible configurations, and the rulial multiway graph is:
But this graph turns out to be a Cayley graph for the finite group  A 4    2 :
To construct this Cayley graph let’s represent the configurations of the Turing machine by pairs of integers , where 0  i  n – 1 gives the position of the head, and the bits of u (with 0  u  2 n  – 1) give the values on the tape. (Since s = 1, we don’t have to worry about the state of the head.) With this representation of the configurations, consider the “multiplication” operation:
When this operation acts on the configurations it defines a group. Here’s the multiplication table for n = 3:
Or equivalently:
But now consider the four elements , (position –1 wraps around on the cyclic tape): 
We can consider these as the result of applying the 4 possible Turing machine transitions
to the configuration (corresponding to ):
And now by treating these elements as generators (and effectively applying them not just to the initial configuration, but to any configuration), we get as the Cayley graph of the group exactly the rulial multiway graph above.
It’s worth noting that the 4 elements we’ve used don’t correspond to the minimal set of generators for the group. Two elements suffice. And for example, we can use and , which can be thought of respectively as moving the head right, and flipping the color of one cell (again relative to the “identity” configuration ):
With these generators, we get the Cayley graph:
How else can we characterize the group? For any tape size n we can write it in terms of explicit permutations:
(For , the group can be generated by the permutations .)
We can also represent it symbolically in terms of generators and relations. Calling our “move-right” generator R, and “bit-flip” generator F, the group then satisfies at least the relations:
 
OK, so does the group have a name? For n = 2, it’s the 8-element dihedral group D 4 and for n = 3, it’s the 24-element group A 4  2. For larger n, there doesn’t seem to be a standard name. But given our derivation we can just call it . And we can express it as a semidirect product (or wreath product):
 
The normal subgroup (2) n  here represents states of the tape, and corresponds to a Boolean n-cube. The cyclic group  n  represents the position of the head, and acts on the Boolean n-cube by rotating its coordinates.
For any n, we can use just two generators, producing the sequence of Cayley graphs:
Undirected versions of these are exactly the cube-connected cycle graphs that have arisen in studying communications networks: 
So now what about the limit n  ? Now, the group is no longer finite, but we’ve still got the relations 
 
and in the end, we can see that the group can be described as a semidirect product:
 
In a sense, the group—and its Cayley graph—is dominated by the infinite-dimensional Boolean hypercube. But there’s more going on. And perhaps there’s a useful characterization of the limit that can be derived by the methods of modern geometric theory. 
For arbitrary s and k, we can potentially generalize to get:
 
Causal Graphs for Deterministic Turing Machines
In a deterministic Turing machine, every step involves one updating event—and the causal graph can be drawn by just joining successive locations of the head, and successive points where the head returns to a square where it has been before:
Continuing for a few more steps, the causal graph for this particular Turing machine becomes:
Continuing for more steps, and redrawing the graph, we see that we get a simple grid:
Of s = 2, k = 2 Turing machines, the one with the most exotic causal graph is the “binary counter” machine 1953:
The causal graph gives a good “overall map” of the behavior of a Turing machine. Here are a few Turing machines (with s = 3, k = 2 and s = 4, k = 2) from  A New Kind of Science  (compare also the s = 2, k = 3 universal Turing machine):
And here are their respective causal graphs:
Rulial Multiway Causal Graphs
What can we say about the causal graph associated with a rulial multiway system? The first important observation is that rulial multiway graphs always exhibit causal invariance, since by including transitions associated with all possible rules one is inevitably including both rules and their inverses, with the result that every branching of edges in the rulial multiway graph is always associated with a corresponding merging.
It’s fairly easy to see this for two steps of s = 2, k = 2 Turing machines. The rulial multiway states graph here is:
This can also be rendered:
Explicitly showing events we get:
Including causal connections we get
or after 3 steps:
The pure rulial multiway causal graph in this case is then:
In layered form this becomes:
After 5 steps the growth in the number of nodes reached going from the root grows like:
Causal invariance implies that this multiway causal graph is ultimately composed of a large number of interwoven copies of a single causal graph. After any given number of steps of evolution, the various copies of this causal graph will have “reached different stages”. Here are the results after 3 steps: 
And after 4 steps:
One can also look at rulial multiway causal graphs for other sets of Turing machines. For s = 1, k = 1 one has (here after 5 steps)
which is equivalent to:
The individual causal graphs in this case are immediately all the same, and are just:
For s = 2, k = 1 one already has after 3 steps
or in layered form:
After 5 steps this becomes
or in layered form:
In this case, the individual causal graphs after 5 steps are:
For s = 1, k = 2 one has very similar results to the case s = 2, k = 2. After 3 steps the causal graph is: 
Or after 5 steps:
Removing multiple edges this is:
The individual causal graphs in this case are:
Rulial Graphs
Just as for ordinary multiway graphs one can study branchial graphs which represent their transversals, so similarly for rulial multiway graphs one can study rulial graphs which represent their transversals. The layered way we have drawn multiway graphs corresponds to a particular choice of foliation—and with this choice, we can immediately generate rulial graphs.
For s = 2, k = 1 one has after 2 steps
while after 3 steps one gets:
For s = 1, k = 2 one gets after 2 steps:
while after 3 steps one has:
The sequence of results for steps 1 through 4 is:
For s = 2, k = 2 the corresponding results are:
What do these pictures mean? Just as branchial graphs in ordinary multiway systems can be thought of as “entanglement maps” for states (interpreted in our models as quantum states) in ordinary “multiway space”, so here these rulial graphs can be thought of as “entanglement maps” for states in rulial space. In other words, they are a kind of map of which Turing machine configurations are “evolutionarily close” to which other ones, in the sense that they can be reached with only a few different choices of rules.
Deterministic Turing Machine Paths in Rulial Space
The rulial multiway graph defines all paths that can be followed by all non-deterministic Turing machines. At each node in this graph there is therefore an outgoing edge corresponding to any possible Turing machine transition from the configuration corresponding to that node. But what if one considers just a single deterministic Turing machine? Its evolution is then a single path within the rulial multiway graph.
Consider for example the s = 2, k = 2 Turing machine:
This machine evolves from a blank tape according to:
This corresponds to a path in the rulial multiway graph:
With a different initial condition
the path can be very different:
This pictures might make it seem that the paths corresponding to the evolution of deterministic Turing machines are geodesics in the rulial multiway graph. But in general they are definitely not. For example, starting from a blank tape, the rule
follows this path on the rulial multiway graph:
If one allows any possible Turing machine transition at every step, then one can follow a geodesic path from a node corresponding to an initial condition to a node corresponding to any other configuration. But if one restricts oneself to the transitions in a particular (deterministic) Turing machine, then there will in general be many configurations one will never reach, and even those that one can reach, one may reach by a circuitous route in the rulial multiway graph.
Given a particular configuration (corresponding to a node in the rulial multiway graph), there may be many deterministic Turing machines that can reach it from a given initial state. One can consider each of these Turing machines to be implementing a certain algorithm. So then the “optimal algorithm” will be the one which is shortest among deterministic Turing machine paths. As I just mentioned, this won’t typically be the shortest possible path: that will usually be achieved by a non-deterministic Turing machine with a particular sequence of transitions. But there is still an optimal case (i.e. an “optimal algorithm”) among deterministic Turing machines.
But now we can ask across all possible deterministic Turing machines where they can reach in the rulial multiway graph. Here is the result for 4 steps for the s = 2, k = 2 Turing machines we have been considering:
These are the results for 1, 2 and 3 steps:
What is the significance of this? Essentially what we’re seeing is a comparison of what can be achieved with deterministic computation versus non-deterministic. Taking the 4-step case as an example, the “background” gray rulial multiway graph shows what can be achieved with arbitrary non-deterministic computation in 4 steps. The red region is what deterministic computation can achieve in the same number of steps.
In a sense this is a very simple empirical analog of the P vs. NP problem. Unlike the real P vs. NP case, we’re not allowing arbitrary polynomial-time algorithms here; we’re just looking at possible s = 2, k = 2 Turing machine algorithms running specifically for 4 steps. But if we were to generalize this appropriately, P = NP would imply that the “red region” must in some limit in effect “reach anywhere in the graph”. 
A little more precisely, the official definition of P and NP is for decision problems: you start from some initial condition which defines an instance of a problem (“Is this Boolean formula satisfiable?” or whatever), and the system must eventually evolve to a state representing either “yes” or “no”. We can imagine setting things up so that the outcomes correspond to particular configurations of the Turing machine. The inputs are then also encoded as initial configurations of the Turing machine, and we want to know what happens as we consider inputs of progressively larger sizes. We can imagine drawing the rulial multiway graph so that progressively larger inputs are shown “progressively further from the center”. If we consider non-deterministic Turing machines (associated with the class of NP computations), then the shortest computation will be a geodesic path in the rulial multiway graph from the input configuration to the outcome configuration. But for deterministic Turing machines (associated with the class P) it will in general be some much more circuitous path.
The standard P vs. NP problem asks about limiting behavior as one increases the size of the input computation—and one might imagine that the question could be “geometrized” in terms of some continuum limit of the rulial multiway graph. Of course, there is no guarantee that any reasonable limit exists. And it could perfectly well be that the question of whether P  NP is actually undecidable, or in other words, that no finite proof of it can be given within a standard axiom system (such as Peano arithmetic or ZFC set theory).
One could imagine empirically testing more and more Turing machines, and seeing how they perform on an NP-complete problem. One might think of plotting their running times as a function of n for increasingly large n. For a while a particular Turing machine might be the winner. But then another one might take over. And there might be no end to how many “surprises” would occur as one increases n. (Somehow this is reminiscent of the story of the Skewes number and whether .)
In computational complexity theory one usually thinks about explicitly constructing optimal algorithms by standard “engineering-like” methods. But I think there’s a lot to be learned from a more empirical approach—in which one doesn’t try to construct optimal algorithms, but just finds them “in the wild” by searching all possible programs. In the past, it might not have seemed that “just searching for programs” would ever produce anything terribly interesting. But one of the big consequences of what I discussed in A New Kind of Science is that even among tiny programs—small enough that one can, for example, enumerate all of them—there’s often very complex and potentially “useful” behavior. And that makes it seem much more reasonable to try to do “empirical computational complexity theory”—enumerating possible programs to find optimal ones, and so on.
Small programs can be thought of as ones with low algorithmic complexity. So searching for “fast programs” among these can be thought of as searching for programs with low time complexity, and low algorithmic complexity. We don’t know exactly how strong the constraint of low algorithmic complexity is, but from what I’ve seen in the computational universe (and what’s embodied in things like the Principle of Computational Equivalence), it seems as if it’s not such a big constraint. 
I studied “empirical computational complexity theory” a bit in A New Kind of Science, notably for Turing machines. And one of the interesting observations was that the optimal algorithm for things was often not something that could readily be constructed in a step-by-step engineering way. Instead, it was something that one basically could only find pretty much by doing a search of possible programs—and where there usually didn’t seem to be a “general form” that would “apply for all n”, and let one readily deduce the properties of the n   limit. In other words, it didn’t seem like what we’d now think of as the sequence of paths in rulial space would show any sign of converging to a “continuum limit”.
A decent example of all this occurs in sorting networks. Imagine that you are given a collection of n numbers to sort. You can straightforwardly do this by making about n 2 pairwise comparisons, and it’s pretty easy to optimize this a bit.
But explicit searches have revealed that the actual optimal networks for successive n are:
What’s notable is how complicated and “random” they look; there doesn’t seem to be any obvious pattern to them (and my guess is that there fundamentally isn’t). Here’s a plot of the sizes of these networks (divided by n 2):
It’s worth noting that these are deterministic networks. One could also imagine non-deterministic networks (and indeed one could construct a rulial multiway graph by considering all possible successive placements of pairwise comparisons)—and in a non-deterministic network it’s always possible to sort n numbers in at most n – 1 steps. 
The Space of Deterministic Turing Machine Computations
We’ve just seen how the results of deterministic Turing machine computations lay out in the rulial multiway space of all possible non-deterministic Turing machine computations. But what happens if we just look at the graph of deterministic Turing machine computations on their own?
Here are the full rulial multiway graphs for 2 and 3 steps with the graphs of deterministic Turing machine computations superimposed, as before: 
But now let’s “pull out just the red subgraphs”—in other words, include as nodes only those configurations that at least one of the 4096 s = 2, k = 2 Turing machines can reach after 2 or 3 steps: 
Notice that after 2 steps, deterministic Turing machines can still reach all 36 configurations that non-deterministic ones can reach (though not through quite as many paths). But after 3 steps, among all the deterministic Turing machines, they can only reach 68 possible configurations, while non-deterministic ones can reach 100 configurations. 
For all possible non-deterministic Turing machines with s = 2, k = 2 the total number of configurations that can be reached eventually roughly doubles on successive steps:
For deterministic Turing machines, however, the number of possible configurations that can be reached soon increases much more slowly:
And in fact there’s an obvious bound here: at any given step, the most that can happen is that each of the 4096 s = 2, k = 2 Turing machines leads to a new configuration—or, in other words, the maximum number of configurations reached increases by 4096.
Looking at the differences on successive steps, we find:
In other words, among all 4096 Turing machines, about 100 “novel configurations” are reached at each successive step. (The actual sequence here looks surprisingly random; it’s not clear whether there’s any particular regularity.)
Now let’s look at the actual graphs formed. With 4 through 7 steps we get:
After 10 and 20 steps the results are:
Here is the result after 50 steps:
There’s a surprising amount of structure in these graphs. There’s a “central region” near the initial blank-tape configuration (shown highlighted below) in which many different Turing machines end up visiting the same configurations:
Here’s a 3D rendering of this region:
But away from this region there end up being “spokes” (about 100 of them) corresponding to Turing machines that “independently explore new territory” in the space of configurations.
What are those “configurations on the edge” like? Here are sorted collections of them for the first few steps:
For comparison, here is the result for all configurations that can be reached by non-deterministic Turing machines after just 2 steps:
Here are the results for deterministic Turing machines after more steps:
We can also ask which machines are the ones that typically “explore new territory”. Here’s the result for 30 steps:
As we go to more steps, the graph of configurations that can be reached by deterministic Turing machines grows. But does at least the core of it reach some kind of limit after sufficiently many steps? We can get a sense of this by looking—as we have done so many times before—at the growth rate of the geodesic ball in the graph starting from the initial blank-tape configuration. The total number of new configurations that can be reached on each new layer of the geodesic ball is at most 4096—and in reality it’s much smaller. Here are the numbers of new configurations added on successive layers at steps 100, 200, …, 500 in the overall evolution:
The cause of the “steps down” becomes clearer if one examines more closely the “spokes” in the graph above. Here’s one example:
And basically what’s happening is that multiple Turing machines are tracing out roughly the same sequences of configurations, but some do it “efficiently”, while others “waste time”, for example having the head flip around on alternate steps. The “inner parts” of the spokes—that are closer to the initial node—involve both “inefficient” and more efficient Turing machines. But the “inefficient” Turing machines will simply not get as far, so they do not contribute to the outer layers of the geodesic ball. The final “step down” in the plot above—at basically half the total number of steps used for the Turing machines—involves the “petering out” of the roughly half the Turing machines that effectively “waste half their steps”.
For the “spoke” shown, here are the actual Turing machine histories involved (there are 8 machines that made identical copies of the first history):
The Cellular Automaton Analog
The kind of graphs we’ve just made for deterministic Turing machines can be made for any family of deterministic computational systems. And in particular they can be made for the (at least for me, much more familiar) case of the 256 k = 2, r = 1 cellular automata. (And, yes, it’s somewhat amazing that in all these years I’ve never made such pictures before—though there’s a note on page 956 of A New Kind of Science that gets close.) 
Here are the results for 5 and 10 steps, starting from an initial condition containing a single black cell:
And here is the result for 50 steps:
Here is the result for 10 steps, annotated with the actual cellular automaton evolution for the rules that “reach furthest”:
It’s slightly easier to see what’s going on if we include only even-numbered rules (which leave a blank state blank):
It’s an interesting map of “cellular automaton space”. (Note the presence of rule 30 on the lower left, and rule 110 on the lower right.)
The total number of new configurations explored by all rules on successive steps has a more regular form than for Turing machines:
The result mostly alternates with period 4 between 72 and 84, though with dips at steps of the form 2 m .
If we go for a certain number of steps (say 200), and then look at the geodesic ball centered on the initial condition, the number of configurations in successive layers is just:
These results are all for ordinary, deterministic cellular automata. So are there non-deterministic cellular automata? Typically, cellular automata are defined to consistently update every cell at every step. But one can also consider sequential cellular automata, where specific cells are updated at each step—and in this case it is straightforward to define a non-deterministic version, for which things like multiway systems can be constructed. (Note that for a full rulial multiway system, such non-deterministic cellular automata are basically the same as non-deterministic mobile automata, which are close to Turing machines.)
Computation Capabilities and the Structure of Rulial Space
What do the computational capabilities of Turing machines mean for their rulial space? Let’s start with computation universality. 
An important fact about deterministic Turing machines is that there are universal ones. Among the 4096 s = 2, k = 2 rules there aren’t any. But as soon one goes to s = 2, k = 3 rules (of which there are 2,985,984) it’s known (thanks to my results in A New Kind of Science , and Alex Smith’s 2007 proof) that there is a universal machine. The rule is:
Starting from a blank tape, this machine gives 
and the corresponding causal graph is:
But what does the existence of a universal machine mean in the rulial multiway system? From any given initial condition, any deterministic Turing machine will trace out some trajectory in the rulial multiway graph. And if a machine is universal it means that by appropriately picking its initial conditions it can be “programmed” to “emulate” any other Turing machine, in the sense that its trajectory will track the trajectory of whatever Turing machine it’s emulating. What does “track” mean? Basically, that there’s some fixed scheme that allows one to go from the states of the universal Turing machine to the states of the machine it’s emulating. The “scheme” will correspond to some translation in the rulial multiway graph, and the requirement is that this translation is somehow always limited. In other words, the trajectory of a universal machine will “flail around” as its starting point (i.e. initial condition) moves in the rulial multiway graph, and this “flailing” will be diverse enough that the trajectory can get close to the trajectory of any given other machine.
If, on the other hand, the machine one’s dealing with isn’t universal, then its trajectory won’t “flail around” enough for this work; the trajectory will in a sense be too constrained to successfully “track” all possible other deterministic Turing machine trajectories. 
What about non-deterministic Turing machines? What universality can be interpreted to mean in this case is that given a particular initial condition, the output one wants occurs somewhere on the different paths followed by the non-deterministic Turing machine. (If one’s trying to do a decision problem—as in the computational complexity class NP—then one can arrange to “signal that one’s got an answer” through some feature of the Turing machine state.) In the case of “extreme non-determinism”—as used to construct the rulial multiway graph—computation universality in a sense becomes a statement purely about the structure of the rulial multiway graph. And basically it just requires that the rulial multiway graph is sufficiently connected—which is guaranteed if there’s causal invariance (so there’s nothing like an “event horizon” anywhere). 
But does one have to allow “extreme non-determinism” to get universality? With s = 2, k = 3 we know that there’s a purely deterministic Turing machine that achieves it. And my guess is that among s = 2, k = 2 there are “slightly multiway” rules that also do. In a standard deterministic s = 2, k = 2 Turing machine, there are 4 cases of the rule that are each specified to have a unique outcome. But what if even just one of those cases in the rule has two outcomes? The rule is non-deterministic, but it can be thought of as just being the result of specifying 5 defining cases for the rule. And it’s my guess that even this will be sufficient to get universality in the system. 
A non-deterministic rule like this will not trace out a single path in the rulial multiway graph. Instead, it’ll give a bundle of paths, with different paths corresponding to different non-deterministic choices. But the story of universality is very similar to the deterministic case: one simply has to ask whether anything in the bundle successfully manages to track the trajectory of the machine one’s emulating.
It’s worth remembering that any given rule won’t typically be following geodesics in rulial space. It’ll be following some more circuitous path (or bundle of paths). But let’s say one has some rule that traces out some trajectory—corresponding to performing some computation. The Principle of Computational Equivalence implies that across different possible rules, there’s a standard “maximum computational sophistication” for these computations, and many rules achieve it. But then the principle also implies that there’s equivalence between these maximally sophisticated computations, in the sense that there’s always a limited computation that translates between them.
Let’s think about this in rulial space. Let’s say we have two rules, starting from the same initial condition. Well, then, at the beginning it’s trivial to translate between the rules. But after t steps, the states reached by these rules can have diverged—making it potentially progressively more difficult to translate between them. A key idea of the rulial multiway graph—and rulial space—is that it lets one talk about both computations and translations between them in uniform ways. Let’s say that the trajectories of two rules have gone a certain distance in rulial space. Then one can look at their divergence, and see how long a “translation computation” one has to do to get from one to the other. 
In ordinary spacetime, let’s say a certain time t has elapsed. Then we know that the maximum spatial distance that can have been traversed is c t, where c is the speed of light. In rulial space, there’s something directly analogous: in time t, there’s a maximum rulial distance that can be traversed, which we can call  t. But here the Principle of Computational Equivalence makes it a crucial contribution: it implies that throughout rulial space, and in all situations, is fixed. It can take an irreducible amount of computational work to successfully translate from the outcome of one rule to another. But this always scales the same way. There’s in effect one scale of computational irreducibility, and it’s characterized by . Like the constancy of the speed of light uniformly limits physical motion, the constancy of uniformly limits rulial motion.
But let’s say you’re trying to get from one point in rulial space to another. If from your starting point you can follow the path of an irreducible—and effectively universal—computation, then you’ll successfully be able to reach the other point. But if from your starting point you can only follow a reducible computation this won’t generally be true. And what this means is that “pockets of computational reducibility” in rulial space act a bit like black holes in physical space. You can get into them from regions of irreducibility, but you can’t get out of them.
There are probably signs of phenomena like this even in the rulial space for simple Turing machines that we’ve explored here. But there’s considerably more that needs to be worked out to be able to make all the necessary connections.
The Emerging Picture of Rulial Space
There are lots of analogies between physical space, the branchial space, and rulial space. For example, in physical space, there are light cones that govern the maximum rate at which effects can propagate between different parts of physical space. In branchial space, there are entanglement cones that govern the maximum rate of quantum entanglement. And in rulial space, one can think of “emulation cones”, which govern the maximum rate at which one description of behavior can be translated into another. 
And when it comes to applying these things to modeling the physical universe, a crucial point is that the observer is necessarily part of the system, governed by the same rules as everything else. And that means that the observer can only be sensitive to certain “appropriately modded-out” aspects of the system. But in actually imagining how an observer “perceives” a system it’s almost always convenient to think about coordinatizing the system—by defining some appropriate foliation. In physical space, this involves foliating the causal graph using reference frames like in relativity. In branchial space, it involves our concept of quantum observation frames. And in rulial space, we can invent another such concept: a rulial description frame, or just a rulial frame. 
Different rulial frames in effect correspond to describing the evolution of the universe as operating according to different rules. Causal invariance implies that in the end different rulial frames must give equivalent results. But the specific way one describes the time evolution of the universe will depend on what rulial frame one’s using. In one frame one would be describing the universe in one way; in another frame, another way. And the “story one tells” about how the universe evolves will be different in the different frames.
Much like with superposition in quantum mechanics, there’s probably some notion of regions in rulial space, in which one’s somehow viewing the universe as operating according to “rulially entangled” collections of rules. 
But while our original motivation was understanding physics, a lot of what we’re studying about rulial space also applies to purely computational systems. For example, we can think of rulial space even without having any notion of physical space. And we can in effect imagine that rulial space is some kind of map of a space of possible rules for computational systems. (Notice that because of computation universality and the Principle of Computational Equivalence it ultimately doesn’t matter what particular type of rule—Turing machine, cellular automaton, combinator, whatever—is used to “parametrize” rulial space.)
Different places in rulial space in some sense correspond to different rules. Paths at different places in rulial space correspond to evolution according to different rules. So what is the analog of motion in rulial space? In effect it’s having a frame which progressively changes the rule one’s using. If one’s trying to find out what happens in one’s system, it’s fundamentally most efficient to “stick with one rule”. If one progressively changes the rule, one’s going to have to keep “translating back” to the original rule, by somehow emulating this rule with whatever rule one’s reached. And the result of this is there’ll be exactly the analog of relativistic time dilation. Faster motion in rulial space leads to slower progression in time. Of course, to discuss this properly, we really have to talk about the rulial multiway causal graph, etc.
But one thing is clear: motion faster than some maximum speed is impossible. From within the system, you simply can’t keep the correct rulial causal connections if you’re changing your rulial location faster than . 
In an abstract study of Turing machines, is just an arbitrary parameter. But in the actual physical universe, it must have a definite value, like the speed of light c, or our maximum entanglement speed . It’s difficult even to estimate what might be. And presumably estimating will be even harder. But it’s interesting to discuss at least how we might start to think about estimating . 
The first key observation about rulial space is that in our models, it’s discrete. In other words, there’s a discrete space of possible rules. Or, put another way, theories are quantized, and there’s somehow an elementary distance between theories—or a minimum distance in “theory space” between neighboring theories. 
But what units is this distance in? Basically it’s in units of rule—or program—size. Given any program—or rule—we can imagine writing that program out in some language (say in Wolfram Language, or as a program for a particular universal Turing machine, or whatever) And now we can characterize the size of the program by just looking at how many tokens it takes to write the program out.
Of course, with different languages, that number will be different—at the simplest level just like the number of decimal digits necessary to represent a number is different from the number of binary digits, or the length of its representation in terms of primes. But it’s just like measuring a length in feet or meters: even though the numerical value is different, we’re still describing the same length.
It’s important to point out that it’s not enough to just measure things in terms of “raw information content”, or ordinary bits, as discussed in information theory. Rather, we want some kind of measure of “semantic information content”: information content that directly tells us what computation to do.
It’s also important to point out that what we need is different from what’s discussed in algorithmic information theory. Once one has a computation universal system, one can always use it to translate from any one language to any other. And in algorithmic information theory the concept is that one can measure the length of a program up to an additive constant by just expecting to include an “emulation program” that adapts to whatever language one’s measuring the length in. But in the usual formalism of algorithmic information theory one doesn’t worry about how long it’s going to take for the emulation to be done; it’s just a question of whether there’s ultimately enough information to do it. 
In our setup, however, it does matter how long the emulation takes, because that process of emulation is actually part of our system. And basically we need the number of steps needed for the emulation to be in some sense bounded by a constant.
So, OK, what does this mean for the value of ? Its units are presumably program size per unit time. And so to define its value, we’ll have to say how we’re measuring program size. Perhaps we could imagine we write our rules in the Wolfram Language. Then there should be a definite value of for our universe, measured in Wolfram-Language-tokens per second. If we chose to use (2,3)-Turing-machine-tape-values per year then we’d get a different numerical value. But assuming we used the correct conversions, the value would be the same. (And, yes, there’s all sorts of subtlety about constant-time or not emulation, etc.)
In some sense, we may be able to think of as the ultimate “processor speed for the universe”: how fast tokens in whatever language we’re using are being “interpreted” and actually “executed” to determine the behavior of the universe. 
Can we estimate the value of ? If our units are Wolfram-Language-tokens per second we could start by imagining just computing in the Wolfram Language some piece of the rulial multiway graph for our models and seeing how many operations it takes. To allow “all possible rules” we’d have to increase the possible left- (and right-) hand sides of our rules to reflect the size of the hypergraph representing the universe at each step. But now we’d need to divide by the “number of parallel threads” in the rulial multiway graph. So we can argue that all we’d be left with is something like (size of spatial hypergraph represented in Wolfram Language) / (elementary time). 
So, based on our previous estimates (which I don’t consider anything more than vaguely indicative yet) we might conclude that perhaps:
 ~ 10450 Wolfram-Language-tokens/second
The number of “parallel threads” in the rulial multiway graph (the rulial analog of Ξ) might then be related to the number of possible hypergraphs that contain about the number of nodes in the universe, or very roughly (10350)^(10350) ≈ . If we ask the total number of Wolfram Language tokens processed by the universe, there’ll be another factor ~10467, but this “parallelism” will completely dominate, and the result will be about:
 Wolfram-Language-tokens
OK, so given a value of , how might we conceivably observe it? Presumably there’s an analog of quantum uncertainty in rulial space, that’s somehow proportional to the value of . It’s not completely clear how this would show up, but one possibility is that it would lead to intrinsic limits on inductive inference. For example, given only a limited observation time, it might be fundamentally impossible to determine beyond a certain (“rulial”) accuracy what rule the universe is following in your description language. There’d be a minimum rate of divergence of behaviors from different rules, associated with the minimum distance between theories—and it would take a certain time to distinguish theories at this rate of divergence.
In our models, just like every causal edge in physical and branchial space is associated with energy, so should every causal edge in rulial space be. In other words, the more processing that happens in a particular part of rulial space, the more physical energy one should consider exists there. And just as with the Einstein equations in physical space, or the Feynman path integral in branchial space, we should expect that the presence of energy in a particular region of rulial space should reflect in a deflection of geodesics there.
Geodesics in rulial space are the shortest paths from one configuration of the universe to another, using whatever sequences of rules are needed. But although that’s somewhat like what’s considered at a theoretical level in non-deterministic computation, it’s not something we’re usually familiar with: we’re used to picking a particular description language and sticking with it. So exactly what the interpretation of deflections of geodesics in rulial space should be isn’t clear.
But there are a few other things we can consider. For example, presumably the universe is expanding in rulial space, perhaps implying that in some sense more descriptions of it are becoming possible over time. What about rulial black holes? As mentioned above, parts of rulial space that correspond to computational reducibility should behave like black holes, where in effect “time stops”. Or, in other words, while in most of the universe time is progressing, and irreducible computation is going on, computational reducibility will cause that process to stop in a rulial black hole.
Presumably geodesics near the rulial black hole will be pulled towards it. Somehow when there’s a description language that leads to computational reducibility, languages near it will tend to also stop being able to successfully describe computationally irreducible processes.
Can we estimate the density of rulial black holes? Let’s consider the Turing machine case. In effect we’re going to want to know what the density of non-universality is among all possible non-deterministic Turing machines. Imagine we emulate all Turing machines using a single universal machine. Then this is effectively equivalent to asking what fraction of initial conditions for that machine lead to reducible behaviors—or, in essence, in the traditional characterization of Turing machines, halt. But the probability across all possible inputs that a universal Turing machine will halt is exactly Greg Chaitin’s . In other words, the density of rulial black holes in rulial space is governed by .
But  isn’t just a number like  that we can compute as accurately as we want; it’s noncomputable, in the sense that it can’t be computed to arbitrary accuracy in any finite time by a Turing machine. Now, in a sense it’s not too surprising that the density of rulial black holes is noncomputable—because, given computational irreducibility, to determine for certain whether something is truly a rulial black hole from which nothing can escape one might have to watch it for an unbounded amount of time.
But for me there’s something personally interesting about Greg Chaitin’s showing up in any way in a potential description of anything to do with the universe. You see, I’ve had a nearly 40-year-long debate with Greg about whether the universe is “like ” or “like ”. In other words, is it possible to have a rule that will let us compute what the universe does just like we can compute (say, in principle, with a Turing machine) the digits of ? Or will we have to go beyond a Turing machine to describe our universe? I’ve always thought that our universe is “like ”; Greg has thought that it might be “like ”. But now it looks as if we might both be right!
In our models, we’re saying that we can compute what the universe does, in principle with a Turing machine. But what we’re now finding out is that in the full rulial space, general limiting statements pull in . In a particular rulial observation frame, we’re able to analyze the universe “like ”. But if we want to know about all possible rulial observation frames—or in a sense the space of all possible descriptions of the universe—we’ll be confronted with .
In our models, the actual operation of the universe, traced in a particular rulial observation frame, is assumed never to correspond to anything computationally more powerful than a Turing machine. But let’s say there was a hypercomputer in our universe. What would it look like? It’d be a place where the effective is infinite—a place where rulial geodesics infinitely diverge—a kind of white hole in rulial space (or perhaps a cosmic event horizon). (We can also think about the hypercomputer as introducing infinite-shortcut paths in the rulial multiway graph which ordinary Turing-machine paths can never “catch up to” and therefore causally affect.)
But given a universe that does hypercomputation, we can then imagine defining a rulial multiway graph for it. And then our universe will show up as a black hole in this higher-level rulial space.
But OK, so if there’s one level of hypercomputer, why not consider all possible levels? In other words, why not define a hyperrulial multiway graph in which the possible rules that are used include both ordinary computational ones, but also hypercomputational ones? 
And once we’re dealing with hypercomputational systems, we can just keep going, adding in effect more and more levels of oracles—and progressively ascending the arithmetic hierarchy. (The concept of “intermediate degrees” might be thought to lead to something that isn’t a perfect hierarchy—but I suspect that it’s not robust enough to apply to complete rulial spaces.) Within the hyperrulial multiway graph at a particular level, levels below it will be presumably appear as rulial black holes, while ones above will appear as rulial white holes. 
And there’s nothing to keep us only considering finite levels of the arithmetic hierarchy; we can also imagine ascending to transfinite levels, and then just keeping going to higher and higher levels of infinity. Of course, according to our models, none of this is relevant to our particular physical universe. But at a theoretical level, we can still at least to some extent “symbolically describe it”, even in our universe.
Thanks
Thanks to Jonathan Gorard for a variety of discussions, as well as to Tali Beynon, Todd Rowland, Ed Pegg, Jose Martin-Garcia and Christopher Wolfram.
Exploring Rulial Space: The Case of Turing Machines—Wolfram Physics Bulletins
https://www.wolframphysics.org/bulletins/2020/06/exploring-rulial-space-the-case-of-turing-machines/
https://www.scientificamerican.com/article/physicists-criticize-stephen-wolframs-theory-of-everything/
Physicists Criticize Stephen Wolfram’s ‘Theory of Everything’
2020-05-06
Adam Becker
<div><div><p>Stephen Wolfram blames himself for not changing the face of physics sooner.</p>
<p>“I do fault myself for not having done this 20 years ago,” the physicist turned software entrepreneur says. “To be fair, I also fault some people in the physics community for trying to prevent it happening 20 years ago. They were successful.” Back in 2002, after years of labor, Wolfram self-published <em>A New Kind of Science,</em> a 1,200-page magnum opus detailing the general idea that nature runs on ultrasimple computational rules. The book was an instant best seller and received glowing reviews: the <em>New York Times</em> called it “a first-class intellectual thrill.” But Wolfram’s arguments found few converts among scientists. Their work carried on, and he went back to running his software company Wolfram Research. And that is where things remained—until last month, when, accompanied by breathless press coverage (and <a href="https://arxiv.org/abs/2004.08210">a 448-page preprint paper</a>), Wolfram announced a possible “path to the fundamental theory of physics” based on his unconventional ideas. Once again, physicists are unconvinced—in no small part, they say, because existing theories do a better job than his model.</p>
<p>At its heart, Wolfram’s new approach is a computational picture of the cosmos—one where the fundamental rules that the universe obeys resemble lines of computer code. This code acts on a graph, a network of points with connections between them, that grows and changes as the digital logic of the code clicks forward, one step at a time. According to Wolfram, this graph is the fundamental stuff of the universe. From the humble beginning of a small graph and a short set of rules, fabulously complex structures can rapidly appear. “Even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich and complex,” he wrote in <a href="https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/">a blog post</a> summarizing the idea. “And this got me thinking: Could the universe work this way?” Wolfram and his collaborator Jonathan Gorard, a physics Ph.D. candidate at the University of Cambridge and a consultant at Wolfram Research, found that this kind of model could reproduce some of the aspects of quantum theory and Einstein’s general theory of relativity, the two fundamental<strong> </strong>pillars of modern physics.</p>
<p>But Wolfram’s model’s ability to incorporate currently accepted physics is not necessarily that impressive. “It’s this sort of infinitely flexible philosophy where, regardless of what anyone said was true about physics, they could then assert, ‘Oh, yeah, you could graft something like that onto our model,’” says Scott Aaronson, a quantum computer scientist at the University of Texas at Austin.</p>
<p>When asked about such criticisms, Gorard agrees—to<strong> </strong>a point. “We’re just kind of fitting things,” he says. “But we're only doing that so we can actually go and do a systematized search” for specific rules that fit those of our universe.</p>
<p>Wolfram and Gorard have not yet found any computational rules meeting those requirements, however. And without those rules, they cannot make any definite, concrete new predictions that could be experimentally tested. Indeed, according to critics, Wolfram’s model has yet to even reproduce the most basic quantitative predictions of conventional physics. “The experimental predictions of [quantum physics and general relativity] have been confirmed to many decimal places—in some cases, to a precision of one part in [10 billion],” says Daniel Harlow, a physicist at the Massachusetts Institute of Technology. “So far I see no indication that this could be done using the simple kinds of [computational rules] advocated by Wolfram. The successes he claims are, at best, qualitative.” Further, even that qualitative success is limited: There are crucial features of modern physics missing from the model. And the parts of physics that it can qualitatively reproduce are mostly there because Wolfram and his colleagues put them in to begin with. This arrangement is akin to announcing, “‘If we suppose that a rabbit was coming out of the hat, then remarkably, this rabbit would be coming out of the hat,’” Aaronson says. “And then [going] on and on about how remarkable it is.”</p>
<p>Unsurprisingly, Wolfram disagrees. He claims that his model has replicated most of fundamental physics already. “From an extremely simple model, we’re able to reproduce special relativity, general relativity and the core results of quantum mechanics,” he says, “which, of course, are what have led to so many precise quantitative predictions of physics over the past century.”</p>
<p>Even Wolfram’s critics acknowledge he is right about at least one thing: it is genuinely interesting that simple computational rules can lead to such complex phenomena. But, they hasten to add, that is hardly an original discovery. The idea “goes back long before Wolfram,” Harlow says. He cites the work of computing pioneers Alan Turing in the 1930s and John von Neumann in the 1950s,<strong> </strong>as well as that of mathematician John Conway in the early 1970s. (Conway, a professor at Princeton University, died of COVID-19 last month.) To the contrary, Wolfram insists that he was the first to discover that virtually boundless complexity could arise from simple rules in the 1980s. “John von Neumann, he absolutely didn’t see this,” Wolfram says. “John Conway, same thing.”</p>
<h2>From Prodigy to Prodigal Scientist</h2>
<p>Born in London in 1959, Wolfram was a child prodigy who studied at Eton College and the University of Oxford before earning a Ph.D. in theoretical physics at the California Institute of Technology in 1979—at the age of 20. After his Ph.D., Caltech promptly hired Wolfram to work alongside his mentors, including physicist Richard Feynman. “I don’t know of any others in this field that have the wide range of understanding of Dr. Wolfram,” Feynman wrote in a letter recommending him for the first ever round of MacArthur “genius grants” in 1981. “He seems to have worked on everything and has some original or careful judgement on any topic.” Wolfram won the grant—at age 21, making him among the youngest ever to receive the award—and became a faculty member at Caltech and then a long-term member at the Institute for Advanced Study in Princeton, N.J. While at the latter, he became interested in simple computational systems and then moved to the University of Illinois in 1986 to start a research center to study the emergence of complex phenomena. In 1987 he founded Wolfram Research, and shortly after he left academia altogether. The software company’s flagship product, Mathematica, is a powerful and impressive piece of mathematics software that has sold millions of copies and is today nearly ubiquitous in physics and mathematics departments worldwide.</p>
<p>Then, in the 1990s, Wolfram decided to go back to scientific research—but without the support and input provided by a traditional research environment. By his own account, he sequestered himself for about a decade, putting together what would eventually become <em>A New Kind of Science</em> with the assistance of a small army of his employees.</p>
<p>Upon the release of the book, the media was ensorcelled by the romantic image of the heroic outsider returning from the wilderness to single-handedly change all of science. <em>Wired</em> dubbed Wolfram “the man who cracked the code to everything” on its cover. “Wolfram has earned some bragging rights,” the <em>New York Times</em> proclaimed. “No one has contributed more seminally to this new way of thinking about the world.” Yet then, as now, researchers largely ignored and derided his work. “There’s a tradition of scientists approaching senility to come up with grand, improbable theories,” the late physicist Freeman Dyson told <em>Newsweek</em> back in 2002. “Wolfram is unusual in that he’s doing this in his 40s.”</p><figure>
<p>
</p>
<p>Sign up for <em>Scientific American</em>’s free newsletters.</p>
</figure>
<p>Wolfram’s story is exactly the sort that many people want to hear, because it matches the familiar beats of dramatic tales from science history that they already know: the lone genius (usually white and male), laboring in obscurity and rejected by the establishment, emerges from isolation, triumphantly grasping a piece of the Truth. But that is rarely—if ever—how scientific discovery actually unfolds. There are examples from the history of science that superficially fit this image: Think of Albert Einstein toiling away on relativity as an obscure Swiss patent clerk at the turn of the 20th century. Or, for a more recent example, consider mathematician Andrew Wiles working in his attic for years to prove Fermat’s last theorem before finally announcing his success in 1995. But portraying those discoveries as the work of a solo genius, romantic as it is, belies the real working process of science. Science is a group effort. Einstein was in close contact with researchers of his day, and Wiles’s work followed a path laid out by other mathematicians just a few years before he got started. Both of them were active, regular participants in the wider scientific community. And even so, they remain exceptions to the rule. Most major scientific breakthroughs are <em>far</em> more collaborative—quantum physics, for example, was developed slowly over a quarter-century by dozens of physicists around the world.</p>
<p>“I think the popular notion that physicists are all in search of the eureka moment in which they will discover the theory of everything is an unfortunate one,” says Katie Mack, a cosmologist at North Carolina State University. “We do want to find better, more complete theories. But the way we go about that is to test and refine our models, look for inconsistencies and incrementally work our way toward better, more complete models.”</p>
<p>Most scientists would readily tell you that their discipline is—and always has been—a collaborative, communal process. Nobody can revolutionize a scientific field without first getting the critical appraisal and eventual validation of their peers. Today this requirement is performed through peer review—a process Wolfram’s critics say he has circumvented with his announcement. “Certainly there’s no reason that Wolfram and his colleagues should be able to bypass formal peer review,” Mack says. “And they definitely have a much better chance of getting useful feedback from the physics community if they publish their results in a format we actually have the tools to deal with.”</p>
<p>Mack is not alone in her concerns. “It’s hard to expect physicists to comb through hundreds of pages of a new theory out of the blue, with no buildup in the form of papers, seminars and conference presentations,” says Sean Carroll, a physicist at Caltech. “Personally, I feel it would be more effective to write short papers addressing specific problems with this kind of approach rather than proclaiming a breakthrough without much vetting.”</p>
<p>So why did Wolfram announce his ideas this way? Why not go the traditional route? “I don't really believe in anonymous peer review,” he says. “I think it’s corrupt. It’s all a giant story of somewhat corrupt gaming, I would say. I think it’s sort of inevitable that happens with these very large systems. It’s a pity.”</p>
<p>So what are Wolfram’s goals? He says he wants the attention and feedback of the physics community. But his unconventional approach—soliciting public comments on an exceedingly long paper—almost ensures it shall remain obscure. Wolfram says he wants physicists’ respect. The ones consulted for this story said gaining it would require him to recognize and engage with the prior work of others in the scientific community.</p>
<p>And when provided with some of the responses from other physicists regarding his work, Wolfram is singularly unenthused. “I’m disappointed by the naivete of the questions that you’re communicating,” he grumbles. “I deserve better.”</p></div><section><h3>ABOUT THE AUTHOR(S)</h3><section><p></p><div><ul></ul><p><strong>Adam Becker</strong> is a freelance science journalist and author of <em>What Is Real?,</em> about the sordid untold history of quantum physics. His writing has appeared in the <em>New York Times,</em> the BBC, NPR and elsewhere. He holds a Ph.D. in cosmology from the University of Michigan. Credit: Nick Higgins</p></div></section></section>||||I||||Skip to main content
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 Space & Physics

 Physicists Criticize Stephen Wolfram’s ‘Theory of Everything’

 The iconoclastic researcher and entrepreneur wants more attention for his big ideas. But so far researchers are less than receptive

 * By Adam Becker on May 6, 2020
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 Stephen Wolfram. Credit: Stephen Wolfram and Stephen Faust Wikipedia (CC BY-SA 3.0)

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 Stephen Wolfram blames himself for not changing the face of physics sooner.

 “I do fault myself for not having done this 20 years ago,” the physicist turned software entrepreneur says. “To be fair, I also fault some people in the physics community for trying to prevent it happening 20 years ago. They were successful.” Back in 2002, after years of labor, Wolfram self-published A New Kind of Science, a 1,200-page magnum opus detailing the general idea that nature runs on ultrasimple computational rules. The book was an instant best seller and received glowing reviews: the New York Times called it “a first-class intellectual thrill.” But Wolfram’s arguments found few converts among scientists. Their work carried on, and he went back to running his software company Wolfram Research. And that is where things remained—until last month, when, accompanied by breathless press coverage (and a 448-page preprint paper), Wolfram announced a possible “path to the fundamental theory of physics” based on his unconventional ideas. Once again, physicists are unconvinced—in no small part, they say, because existing theories do a better job than his model.

 At its heart, Wolfram’s new approach is a computational picture of the cosmos—one where the fundamental rules that the universe obeys resemble lines of computer code. This code acts on a graph, a network of points with connections between them, that grows and changes as the digital logic of the code clicks forward, one step at a time. According to Wolfram, this graph is the fundamental stuff of the universe. From the humble beginning of a small graph and a short set of rules, fabulously complex structures can rapidly appear. “Even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich and complex,” he wrote in a blog post summarizing the idea. “And this got me thinking: Could the universe work this way?” Wolfram and his collaborator Jonathan Gorard, a physics Ph.D. candidate at the University of Cambridge and a consultant at Wolfram Research, found that this kind of model could reproduce some of the aspects of quantum theory and Einstein’s general theory of relativity, the two fundamental pillars of modern physics.

 But Wolfram’s model’s ability to incorporate currently accepted physics is not necessarily that impressive. “It’s this sort of infinitely flexible philosophy where, regardless of what anyone said was true about physics, they could then assert, ‘Oh, yeah, you could graft something like that onto our model,’” says Scott Aaronson, a quantum computer scientist at the University of Texas at Austin.

 When asked about such criticisms, Gorard agrees—to a point. “We’re just kind of fitting things,” he says. “But we're only doing that so we can actually go and do a systematized search” for specific rules that fit those of our universe.

 Wolfram and Gorard have not yet found any computational rules meeting those requirements, however. And without those rules, they cannot make any definite, concrete new predictions that could be experimentally tested. Indeed, according to critics, Wolfram’s model has yet to even reproduce the most basic quantitative predictions of conventional physics. “The experimental predictions of [quantum physics and general relativity] have been confirmed to many decimal places—in some cases, to a precision of one part in [10 billion],” says Daniel Harlow, a physicist at the Massachusetts Institute of Technology. “So far I see no indication that this could be done using the simple kinds of [computational rules] advocated by Wolfram. The successes he claims are, at best, qualitative.” Further, even that qualitative success is limited: There are crucial features of modern physics missing from the model. And the parts of physics that it can qualitatively reproduce are mostly there because Wolfram and his colleagues put them in to begin with. This arrangement is akin to announcing, “‘If we suppose that a rabbit was coming out of the hat, then remarkably, this rabbit would be coming out of the hat,’” Aaronson says. “And then [going] on and on about how remarkable it is.”

 Unsurprisingly, Wolfram disagrees. He claims that his model has replicated most of fundamental physics already. “From an extremely simple model, we’re able to reproduce special relativity, general relativity and the core results of quantum mechanics,” he says, “which, of course, are what have led to so many precise quantitative predictions of physics over the past century.”

 Even Wolfram’s critics acknowledge he is right about at least one thing: it is genuinely interesting that simple computational rules can lead to such complex phenomena. But, they hasten to add, that is hardly an original discovery. The idea “goes back long before Wolfram,” Harlow says. He cites the work of computing pioneers Alan Turing in the 1930s and John von Neumann in the 1950s, as well as that of mathematician John Conway in the early 1970s. (Conway, a professor at Princeton University, died of COVID-19 last month.) To the contrary, Wolfram insists that he was the first to discover that virtually boundless complexity could arise from simple rules in the 1980s. “John von Neumann, he absolutely didn’t see this,” Wolfram says. “John Conway, same thing.”

 From Prodigy to Prodigal Scientist

 Born in London in 1959, Wolfram was a child prodigy who studied at Eton College and the University of Oxford before earning a Ph.D. in theoretical physics at the California Institute of Technology in 1979—at the age of 20. After his Ph.D., Caltech promptly hired Wolfram to work alongside his mentors, including physicist Richard Feynman. “I don’t know of any others in this field that have the wide range of understanding of Dr. Wolfram,” Feynman wrote in a letter recommending him for the first ever round of MacArthur “genius grants” in 1981. “He seems to have worked on everything and has some original or careful judgement on any topic.” Wolfram won the grant—at age 21, making him among the youngest ever to receive the award—and became a faculty member at Caltech and then a long-term member at the Institute for Advanced Study in Princeton, N.J. While at the latter, he became interested in simple computational systems and then moved to the University of Illinois in 1986 to start a research center to study the emergence of complex phenomena. In 1987 he founded Wolfram Research, and shortly after he left academia altogether. The software company’s flagship product, Mathematica, is a powerful and impressive piece of mathematics software that has sold millions of copies and is today nearly ubiquitous in physics and mathematics departments worldwide.

 Then, in the 1990s, Wolfram decided to go back to scientific research—but without the support and input provided by a traditional research environment. By his own account, he sequestered himself for about a decade, putting together what would eventually become A New Kind of Science with the assistance of a small army of his employees.

 Upon the release of the book, the media was ensorcelled by the romantic image of the heroic outsider returning from the wilderness to single-handedly change all of science. Wired dubbed Wolfram “the man who cracked the code to everything” on its cover. “Wolfram has earned some bragging rights,” the New York Times proclaimed. “No one has contributed more seminally to this new way of thinking about the world.” Yet then, as now, researchers largely ignored and derided his work. “There’s a tradition of scientists approaching senility to come up with grand, improbable theories,” the late physicist Freeman Dyson told Newsweek back in 2002. “Wolfram is unusual in that he’s doing this in his 40s.”

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 Sign Up

 Wolfram’s story is exactly the sort that many people want to hear, because it matches the familiar beats of dramatic tales from science history that they already know: the lone genius (usually white and male), laboring in obscurity and rejected by the establishment, emerges from isolation, triumphantly grasping a piece of the Truth. But that is rarely—if ever—how scientific discovery actually unfolds. There are examples from the history of science that superficially fit this image: Think of Albert Einstein toiling away on relativity as an obscure Swiss patent clerk at the turn of the 20th century. Or, for a more recent example, consider mathematician Andrew Wiles working in his attic for years to prove Fermat’s last theorem before finally announcing his success in 1995. But portraying those discoveries as the work of a solo genius, romantic as it is, belies the real working process of science. Science is a group effort. Einstein was in close contact with researchers of his day, and Wiles’s work followed a path laid out by other mathematicians just a few years before he got started. Both of them were active, regular participants in the wider scientific community. And even so, they remain exceptions to the rule. Most major scientific breakthroughs are far more collaborative—quantum physics, for example, was developed slowly over a quarter-century by dozens of physicists around the world.

 “I think the popular notion that physicists are all in search of the eureka moment in which they will discover the theory of everything is an unfortunate one,” says Katie Mack, a cosmologist at North Carolina State University. “We do want to find better, more complete theories. But the way we go about that is to test and refine our models, look for inconsistencies and incrementally work our way toward better, more complete models.”

 Most scientists would readily tell you that their discipline is—and always has been—a collaborative, communal process. Nobody can revolutionize a scientific field without first getting the critical appraisal and eventual validation of their peers. Today this requirement is performed through peer review—a process Wolfram’s critics say he has circumvented with his announcement. “Certainly there’s no reason that Wolfram and his colleagues should be able to bypass formal peer review,” Mack says. “And they definitely have a much better chance of getting useful feedback from the physics community if they publish their results in a format we actually have the tools to deal with.”

 Mack is not alone in her concerns. “It’s hard to expect physicists to comb through hundreds of pages of a new theory out of the blue, with no buildup in the form of papers, seminars and conference presentations,” says Sean Carroll, a physicist at Caltech. “Personally, I feel it would be more effective to write short papers addressing specific problems with this kind of approach rather than proclaiming a breakthrough without much vetting.”

 So why did Wolfram announce his ideas this way? Why not go the traditional route? “I don't really believe in anonymous peer review,” he says. “I think it’s corrupt. It’s all a giant story of somewhat corrupt gaming, I would say. I think it’s sort of inevitable that happens with these very large systems. It’s a pity.”

 So what are Wolfram’s goals? He says he wants the attention and feedback of the physics community. But his unconventional approach—soliciting public comments on an exceedingly long paper—almost ensures it shall remain obscure. Wolfram says he wants physicists’ respect. The ones consulted for this story said gaining it would require him to recognize and engage with the prior work of others in the scientific community.

 And when provided with some of the responses from other physicists regarding his work, Wolfram is singularly unenthused. “I’m disappointed by the naivete of the questions that you’re communicating,” he grumbles. “I deserve better.”

 Rights & Permissions

 ABOUT THE AUTHOR(S)

 Adam Becker is a freelance science journalist and author of What Is Real?, about the sordid untold history of quantum physics. His writing has appeared in the New York Times, the BBC, NPR and elsewhere. He holds a Ph.D. in cosmology from the University of Michigan. Credit: Nick Higgins

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Stephen Wolfram blames himself for not changing the face of physics sooner.
“I do fault myself for not having done this 20 years ago,” the physicist turned software entrepreneur says. “To be fair, I also fault some people in the physics community for trying to prevent it happening 20 years ago. They were successful.” Back in 2002, after years of labor, Wolfram self-published A New Kind of Science, a 1,200-page magnum opus detailing the general idea that nature runs on ultrasimple computational rules. The book was an instant best seller and received glowing reviews: the New York Times called it “a first-class intellectual thrill.” But Wolfram’s arguments found few converts among scientists. Their work carried on, and he went back to running his software company Wolfram Research. And that is where things remained—until last month, when, accompanied by breathless press coverage (and a 448-page preprint paper), Wolfram announced a possible “path to the fundamental theory of physics” based on his unconventional ideas. Once again, physicists are unconvinced—in no small part, they say, because existing theories do a better job than his model.
At its heart, Wolfram’s new approach is a computational picture of the cosmos—one where the fundamental rules that the universe obeys resemble lines of computer code. This code acts on a graph, a network of points with connections between them, that grows and changes as the digital logic of the code clicks forward, one step at a time. According to Wolfram, this graph is the fundamental stuff of the universe. From the humble beginning of a small graph and a short set of rules, fabulously complex structures can rapidly appear. “Even when the underlying rules for a system are extremely simple, the behavior of the system as a whole can be essentially arbitrarily rich and complex,” he wrote in a blog post summarizing the idea. “And this got me thinking: Could the universe work this way?” Wolfram and his collaborator Jonathan Gorard, a physics Ph.D. candidate at the University of Cambridge and a consultant at Wolfram Research, found that this kind of model could reproduce some of the aspects of quantum theory and Einstein’s general theory of relativity, the two fundamental pillars of modern physics.
But Wolfram’s model’s ability to incorporate currently accepted physics is not necessarily that impressive. “It’s this sort of infinitely flexible philosophy where, regardless of what anyone said was true about physics, they could then assert, ‘Oh, yeah, you could graft something like that onto our model,’” says Scott Aaronson, a quantum computer scientist at the University of Texas at Austin.
When asked about such criticisms, Gorard agrees—to a point. “We’re just kind of fitting things,” he says. “But we're only doing that so we can actually go and do a systematized search” for specific rules that fit those of our universe.
Wolfram and Gorard have not yet found any computational rules meeting those requirements, however. And without those rules, they cannot make any definite, concrete new predictions that could be experimentally tested. Indeed, according to critics, Wolfram’s model has yet to even reproduce the most basic quantitative predictions of conventional physics. “The experimental predictions of [quantum physics and general relativity] have been confirmed to many decimal places—in some cases, to a precision of one part in [10 billion],” says Daniel Harlow, a physicist at the Massachusetts Institute of Technology. “So far I see no indication that this could be done using the simple kinds of [computational rules] advocated by Wolfram. The successes he claims are, at best, qualitative.” Further, even that qualitative success is limited: There are crucial features of modern physics missing from the model. And the parts of physics that it can qualitatively reproduce are mostly there because Wolfram and his colleagues put them in to begin with. This arrangement is akin to announcing, “‘If we suppose that a rabbit was coming out of the hat, then remarkably, this rabbit would be coming out of the hat,’” Aaronson says. “And then [going] on and on about how remarkable it is.”
Unsurprisingly, Wolfram disagrees. He claims that his model has replicated most of fundamental physics already. “From an extremely simple model, we’re able to reproduce special relativity, general relativity and the core results of quantum mechanics,” he says, “which, of course, are what have led to so many precise quantitative predictions of physics over the past century.”
Even Wolfram’s critics acknowledge he is right about at least one thing: it is genuinely interesting that simple computational rules can lead to such complex phenomena. But, they hasten to add, that is hardly an original discovery. The idea “goes back long before Wolfram,” Harlow says. He cites the work of computing pioneers Alan Turing in the 1930s and John von Neumann in the 1950s, as well as that of mathematician John Conway in the early 1970s. (Conway, a professor at Princeton University, died of COVID-19 last month.) To the contrary, Wolfram insists that he was the first to discover that virtually boundless complexity could arise from simple rules in the 1980s. “John von Neumann, he absolutely didn’t see this,” Wolfram says. “John Conway, same thing.”
From Prodigy to Prodigal Scientist
Born in London in 1959, Wolfram was a child prodigy who studied at Eton College and the University of Oxford before earning a Ph.D. in theoretical physics at the California Institute of Technology in 1979—at the age of 20. After his Ph.D., Caltech promptly hired Wolfram to work alongside his mentors, including physicist Richard Feynman. “I don’t know of any others in this field that have the wide range of understanding of Dr. Wolfram,” Feynman wrote in a letter recommending him for the first ever round of MacArthur “genius grants” in 1981. “He seems to have worked on everything and has some original or careful judgement on any topic.” Wolfram won the grant—at age 21, making him among the youngest ever to receive the award—and became a faculty member at Caltech and then a long-term member at the Institute for Advanced Study in Princeton, N.J. While at the latter, he became interested in simple computational systems and then moved to the University of Illinois in 1986 to start a research center to study the emergence of complex phenomena. In 1987 he founded Wolfram Research, and shortly after he left academia altogether. The software company’s flagship product, Mathematica, is a powerful and impressive piece of mathematics software that has sold millions of copies and is today nearly ubiquitous in physics and mathematics departments worldwide.
Then, in the 1990s, Wolfram decided to go back to scientific research—but without the support and input provided by a traditional research environment. By his own account, he sequestered himself for about a decade, putting together what would eventually become A New Kind of Science with the assistance of a small army of his employees.
Upon the release of the book, the media was ensorcelled by the romantic image of the heroic outsider returning from the wilderness to single-handedly change all of science. Wired dubbed Wolfram “the man who cracked the code to everything” on its cover. “Wolfram has earned some bragging rights,” the New York Times proclaimed. “No one has contributed more seminally to this new way of thinking about the world.” Yet then, as now, researchers largely ignored and derided his work. “There’s a tradition of scientists approaching senility to come up with grand, improbable theories,” the late physicist Freeman Dyson told Newsweek back in 2002. “Wolfram is unusual in that he’s doing this in his 40s.” 
Sign up for Scientific American’s free newsletters.
Wolfram’s story is exactly the sort that many people want to hear, because it matches the familiar beats of dramatic tales from science history that they already know: the lone genius (usually white and male), laboring in obscurity and rejected by the establishment, emerges from isolation, triumphantly grasping a piece of the Truth. But that is rarely—if ever—how scientific discovery actually unfolds. There are examples from the history of science that superficially fit this image: Think of Albert Einstein toiling away on relativity as an obscure Swiss patent clerk at the turn of the 20th century. Or, for a more recent example, consider mathematician Andrew Wiles working in his attic for years to prove Fermat’s last theorem before finally announcing his success in 1995. But portraying those discoveries as the work of a solo genius, romantic as it is, belies the real working process of science. Science is a group effort. Einstein was in close contact with researchers of his day, and Wiles’s work followed a path laid out by other mathematicians just a few years before he got started. Both of them were active, regular participants in the wider scientific community. And even so, they remain exceptions to the rule. Most major scientific breakthroughs are far more collaborative—quantum physics, for example, was developed slowly over a quarter-century by dozens of physicists around the world.
“I think the popular notion that physicists are all in search of the eureka moment in which they will discover the theory of everything is an unfortunate one,” says Katie Mack, a cosmologist at North Carolina State University. “We do want to find better, more complete theories. But the way we go about that is to test and refine our models, look for inconsistencies and incrementally work our way toward better, more complete models.”
Most scientists would readily tell you that their discipline is—and always has been—a collaborative, communal process. Nobody can revolutionize a scientific field without first getting the critical appraisal and eventual validation of their peers. Today this requirement is performed through peer review—a process Wolfram’s critics say he has circumvented with his announcement. “Certainly there’s no reason that Wolfram and his colleagues should be able to bypass formal peer review,” Mack says. “And they definitely have a much better chance of getting useful feedback from the physics community if they publish their results in a format we actually have the tools to deal with.”
Mack is not alone in her concerns. “It’s hard to expect physicists to comb through hundreds of pages of a new theory out of the blue, with no buildup in the form of papers, seminars and conference presentations,” says Sean Carroll, a physicist at Caltech. “Personally, I feel it would be more effective to write short papers addressing specific problems with this kind of approach rather than proclaiming a breakthrough without much vetting.”
So why did Wolfram announce his ideas this way? Why not go the traditional route? “I don't really believe in anonymous peer review,” he says. “I think it’s corrupt. It’s all a giant story of somewhat corrupt gaming, I would say. I think it’s sort of inevitable that happens with these very large systems. It’s a pity.”
So what are Wolfram’s goals? He says he wants the attention and feedback of the physics community. But his unconventional approach—soliciting public comments on an exceedingly long paper—almost ensures it shall remain obscure. Wolfram says he wants physicists’ respect. The ones consulted for this story said gaining it would require him to recognize and engage with the prior work of others in the scientific community.
And when provided with some of the responses from other physicists regarding his work, Wolfram is singularly unenthused. “I’m disappointed by the naivete of the questions that you’re communicating,” he grumbles. “I deserve better.”   ABOUT THE AUTHOR(S)        Adam Becker is a freelance science journalist and author of What Is Real?, about the sordid untold history of quantum physics. His writing has appeared in the New York Times, the BBC, NPR and elsewhere. He holds a Ph.D. in cosmology from the University of Michigan. Credit: Nick Higgins
Physicists Criticize Stephen Wolfram’s ‘Theory of Everything’
https://www.scientificamerican.com/article/physicists-criticize-stephen-wolframs-theory-of-everything/
https://www.amazon.com/New-Kind-Science-Stephen-Wolfram/dp/1579550088
A New Kind of Science 1st Edition
2002-05-14
Stephen Wolfram
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 <p><i>5.0 out of 5 stars</i>
 
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 <p>Reviewed in the United States 🇺🇸 on December 6, 2020</p><p><div><p>There are a vast range of physical, biological, and other systems that we are confronted with that exhibit an immense complexity. Throughout history, it has been seen as perhaps the work of a supernatural being. Also, often when we encounter something that seems complex, we take it for granted that the phenomenon must be the result of an underlying mechanism that is itself complex. The author shows us how simple programs can, in fact, produce great complexity. In regards to physics, the author states, “that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known feature of our universe will emerge.” We learn in this book, through virtually hundreds of pages of diagrams, how simple initial conditions can produce complex behavior. Wolfram argues that this phenomenon is responsible from most of the complexity we see in nature.<p>There is extensive discussion of cellular automata, where we have simple rules that result in complex behavior- so much in fact that I skipped through some of the material. It is believed that these behaviors are common in nature. An interesting finding is that “adding more complexity to the underlying rules does not yield behavior that it ultimately any more complex.” By chapter four, we are introduced to systems based on numbers. How does their behavior compare with other systems discussed? Even here it is possible to get behavior of considerable complexity by applying a series of operations based on simple arithmetic. Chapter 5 expands things to two dimension and beyond. The next chapter considers completely random initial conditions.</p><p>Finally, we get to the application of what was covered in the previous chapters – the study of actual phenomena in nature. There seems to be a similarity between the basic mechanisms in nature and those responsible for the phenomena seen in simple programs. It is important to note that even though the models are based on simple programs that may successfully reproduce the behavior of some system even though that system does not actually consist of discrete cells, it is just a model. Models provide “an abstract representation of effects that are important in determining the behavior of a system.” The author provides examples of snowflake formation and fluidic motion. Regarding evolution, it seems that natural selection may not be as important as once thought saying that features in complexity of biological organisms arise in a sense not because of natural selection but despite it. Also, it was noted that the high degree of complexity seen in nature “can arise in a sense quite effortlessly just as a consequence of following certain simple rules of growth.”</p><p>In the chapter on physics, this idea of simple initial conditions producing complex and seemingly random behavior is applied to the Second Law of Thermodynamics. The author discusses models for continuum fluid mechanics and even the universe. In fact, he believes that most physical laws discovered so far “are not truly fundamental but are emergent features of the large-scale behavior of some ultimate underlying rules.” The discussion veers into topics like gravity, spacetime, and quantum phenomena. It is the author’s “strong belief that in the end it will turn out that every detail of our universe does indeed follow rules that can be represented by a very simple program – and that everything we see will ultimately emerge just from running this program.” Wow, the thought went through my head: could we be living in a simulation? Who knows?</p><p>The next chapter deals with process of perception and analysis. I skimmed through parts of this section – a bit too much detail for me. But the question becomes could simple mechanisms underlie human thinking? The next chapter covers universality and how “in the end cellular automata can actually be made to emulate almost every single type of system” that has been discussed so far. In fact, a wide range of computations that can be performed by computers can be done by cellular automata.</p><p>In the final chapter, the author discusses something called the Principle of Computational Equivalence, where one can view all processes whether produced by humans or nature as computations. Another concept discussed is computational irreducibility, which implies that even if one has all the information to work out how some system will behave, it will still take an irreducible amount of computational work to do this. The concepts developed here apply not only to phenomena in nature but can provide new insights on fundamental issues in mathematics as well. Note that in discussing his work, the author supplies literally hundreds of charts, diagrams, and tables to back up his claims. The author concludes the main purpose of the book has been to build a new kind of basic science, but the most important consequence will have been to introduce a vast new range of systems helpful to technology.</p>
 
</p></div></p></div><div><p>Reviewed in the United States 🇺🇸 on October 5, 2021</p><p><div><p>I have just skimmed a few pages here and there in A New Kind of Science. I bought it because Stephen Wolfram is almost as brilliant as he would have you believe he is if you have read some of the promo material about him that has appeared on Wolfram Research's main page.<p>I used to use Mathematica years ago, when it was no longer amazing and trendy, but just a very accepted software tool college and grad students were expected to use in conjunction with many of their science, math and engineering classes. I never had to use it for a class or work, but it was often available for free or cheap when I was a grad student or F/T employee on campus so I got to like it because it was interesting and fun to use as a way to explore some "hobby" math problems.</p><p>This book is not one I expect to read cover to cover, but I would be surprised if I do not read large sections of it, with a few days of binge reading per section. After I had much parts of Ch1 and Ch2 when I had access to someone else's copy of the book a couple of years ago, I realized that Wolfram's notions regarding intuition as being a key to really understanding science [and math, by implication] are much more explicitly stated than how I have described what took me several years of my life to figure out.</p><p>Looking at other chapters later in the book, I expect I will dive into sections mentioning or covering subject matter that has fascinated me for some reason in the past, just because I have a hunch that what Wolfram writes about it might save me a lot of time learning some basics that I might not pick up in more years of very casual reading about that or similar topics.</p><p>Douglas R. Hostadter is mentioned in the index of A New Kind of Science [hereinafter: ANKoS] as he is referenced when recursive sequenced are discussed in this book. As a reader who found Godel, Escher, Bach: An Eternal Golden Braid (*) to be a life-changing book in the late 1970s when it was first published, I think books such as ANKoS are likely to be a way to keep expanding my knowledge of areas of science and math I enjoy without having to read several in-depth books covering those fields of study.</p><p>My S.W.A.G is that ANKos will be useful as a tour guide to topics in math and science I would like to learn more about, written by a guide who looks at different things and observes their mathematical and conceptual similarities or other relationships.</p><p>Sorry for the rather vague limited "first impression" done with a very broad brush, but people who will like this book for the reasons I am glad I already bought it (no, not cognitive dissonance! :-) might well want to risk $15 on a book that could enable them to continue studying certain subjects where the typical book price might be upwards of $40 and know enough to select just a few very appropriate titles.</p><p>In short if glancing through ANKoS for subjects that look interesting and maybe reading the first chapter or two to see if the style and manner of presentation work for you, makes you want to delve deeper into some of those subjects, ANKoS might be an excellent place to start.</p><p>Now I want to pick up a current edition of Mathematica, but unlike this book, that is not cheap if one does not have a university or corporate affiliation which makes it free or very affordable.</p><p>Regards,<br />TMG</p><p>* ( https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 )</p>
 
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 <div><p><i>1.0 out of 5 stars</i>
 
 
 Very light on technical detail and high on self-opinion at the very least
 
</p><p>Reviewed in the United Kingdom 🇬🇧 on July 20, 2019</p><p><div><p>20 years in the making &amp; revolutionary ? he is having a giraffe. First I thought he was telling the story of Lorenz in the first few pages and then going onto the lorenz attractor, chaos theory and Poincare sections etc .... but no ... it is all about himself, a self-obsession he exhibits on every page, and his supposedly startling, earth-shattering observations that he fails to appreciate have been observed by many others before and presented in more detail, are nothing of the sort and in the intervening time has not had any impact. Detail is sadly lacking in this tome with its extra wide margins and large font. He then compares the book ( his work ) to Newton's work as a new science, and raises an idea that he says he finally suppressed, which was the need to invent a whole new technical framework &amp; lexicon to explain these monumental concepts. Was he on the lash or on ego building supplements ?? It has been 17 odd years since this book came out and there's nothing in the book beyond Barnsley's 'Iterated function system' book presentation or Conway's game of life and cellular automata. The book is very light on technical detail and high on self-opinion at the very least.
 
</p></div></p></div><div><p><i>2.0 out of 5 stars</i>
 
 
 The automata are interesting, the remainder is bloated nonsense
 
</p><p>Reviewed in the United Kingdom 🇬🇧 on January 4, 2022</p><p><div><p>This book was written by Dr Wolfram is clever in ways that I will never be. Therefore, I was willing to be open to new idea, despite the scepticism of others. He starts with automata, where a new element is a combination of previous elements - The Game Of Life is a well-known example of this. There are some very interesting outcomes from this, and it would have been nice if he had continued with this theme. He does some work with Turing machines - I would have welcomed more.<p>Unfortunately, Dr Wolfram had other notions. He believes, without any evidence that I could detect, that these automata could be used to model scientific behaviour. An example given was snowflakes, and he showed a automata model of a snowflake at different stages in growth. He claimed that regular science cannot explain snowflakes, which is untrue. What regular science can explain is the way in which temperature and humidity interact to deliver the diverse forms of snowflake. All that his automata can do is make a hexagonal shape, on a hexagonal grid. It predicts nothing, and is therefore worthless.</p><p>Dr Wolfram does a presentation about relativity and quantum mechanics. This is worse than the snowflake example. It is page after page of utter nonsense.</p><p>Overall, the impression is of a bloated book, which could say more in fewer pages if it was properly edited. The automata are genuinely interesting - the remainder is just nonsense, and could have been omitted.</p>
 
</p></div></p></div><div><p><i>2.0 out of 5 stars</i>
 
 
 Long detailed exploration of fundimentals within thus new Math
 
</p><p>Reviewed in the United Kingdom 🇬🇧 on March 7, 2022</p><p><div><p>* Physical<p>This book is over 1201 pages. It's a door stop of a book. The paper is very good quality and it's printed very well.</p><p>* Target Audience: A-Level, H.N.D, Undergraduate, Graduate, Post Graduate, Masters?</p><p>It's open to a widespread A-level to Graduate.</p><p>* Whats does it cover?</p><p>1. the Foundation for a New Kind of Science, 2. The Crucial Experiment, 3. the World of Simple Programs, 4. System based on Numbers, 5. Two Dimensional and Beyond. 6. Starting from Randomness, 7. Mechanisms in Programs and Nature, 8. Implications for Everyday Systems, 9. Fundamental Physics, 10 Processes of Perception and Analysis,11 the Notion of Computation, 12. the Principle of Computational Equivalence, Notes index</p><p>* What are the best bits?</p><p>This book on its own is an examination of how simple programs can generate complex data. It is needed to remember that these simple methods are fields in, Mathematics, Physics, Biology, Social Sciences, Computer Science, Philosophy, Art, Technology, A.I, Artificial life, Catsrophe, Chaos theory, Complexity theory, Computational Complexity Theory, Cybernetics, Dynamic Systems, Theory, Evolution Theory, Experimental Mathematics, Fractals Geometry, General system Theory, Nano Technology, Nonlinear Dynamics, Scientific Computing, Self Organisation, Statistical Mechanics.</p><p>The book is very detailed explorations of subtle changes to simple programs and data is catalogued systematically to help organise the resulting fields of results and are reapplied to the next generations of programs. It's a lot of stuff. His style is chatty as if he's there with you explaining the results to you. The message is to organise your results and build on them. You could cut 25% of the chatty style and still wouldn't have harmed the structure.</p><p>* Summary</p><p>It's an education in that the simpler the program, the wider the potential useability of what's shown.</p>
 
</p></div></p></div><div><p><i>5.0 out of 5 stars</i>
 
 
 A cybernetic aid to understanding
 
</p><p>Reviewed in the United Kingdom 🇬🇧 on December 18, 2017</p><p><div><p>As an initial reaction, perhaps 'Wow!' is the applicable expression! Mr. Wolfram has a respected and productive history in the field of computer science; his primary accolade being the powerful 'Mathmatica' suite of symbolic software tools used ubiquitously in the science and engineering fields. But he hasn't stopped there: this book describes how these tools - each of which is a functional sub-routine - can be combined and manipulated in such ways as if they were (almost) primary source code for building complex algorithmic entities - aka 'Cellular Automata' - applicable to almost every field of human endeavour. Thus is born the 'Wolfram' language for knowledge-based programming, using which may reveal useful, hitherto undiscovered relationships and interactions between systems and processes. in my estimation, this will revolutionise the way in which research and development is undertaken - and not just in the physical sciences - gaining huge benefits in costs, time and effort. This is truly a conceptual and intellectual 'amplifier' which will surely figure as being of immense significance in our increasing drive to unravel the mysteries of nature. The writing style is pacy and bright, giving full credit to the lay reader who may not have sufficient knowledge or experience in computer science or physics. It is not a laborious read, but Mr. Wolfram does, however, tediously tend to blow his own trumpet at every opportunity - obviously holding great pride in own theories and achievements! Having said that, it is in no way a detraction of the book or the man; I applaud him for what he has achieved and for presenting it to the world with articulate and informal narrative.
 
</p></div></p></div><div><p><i>3.0 out of 5 stars</i>
 
 
 Interesting but you have to close your eyes to the repetitive "one of the most important discoveries I made" style language
 
</p><p>Reviewed in the United Kingdom 🇬🇧 on December 22, 2014</p><p><div><p>Verbose, over long, repetitive, insufferably arrogant and plagiaristic this book is (see other reviews), it does have one big advantage - It contains a wealth of information on cellular automata. I'm a quarter of the way through so far. I imagine it will ultimately reward a non specialist in the field like myself at the cost of many hours of my life.<p>No doubt the book completely annoys the pants off industry practitioners who, reading this book, one would think barely exist compared to the self-professed awesome brain of Mr Wolfram. At times, it almost falls into parody as if an intelligent version of Arnold Rimmer for Red Dwarf was writing it. As an 11-year old, I remember programming Conway's Game of Life and, like many other kids who did this, being fascinated by where the hell the information to describe all this comes from. But according to Mr. Wolfram, he was the first one to think like this. Forget about all the teenage Game of Life hobbyists out there, Zuse, Tegmark etc. Perhaps its true that Wolfram did independently come up with lots of discoveries and helped reinvigorate the subject, but even if he did, he should acknowledge others more in his writing and not boast about what a clever chappy he is the whole time. It's my first book in digital physics, so can't compare it to others out there, unfortunately.</p>
 
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 Starting from a collection of simple computer experiments illustrated by striking computer graphics Stephen Wolfram shows in this landmark book how their unexpected results force a whole new way of looking at the operation of our universe. Wolfram uses his approach to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe.
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  5.0 out of 5 stars good conditions
  Reviewed in the United States on August 19, 2010
 The book arrived in very good conditions and the price was excellent.Careful: It is bigger and heavier than you expect :)thanks
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  Randolph Eck
  VINE VOICE
  5.0 out of 5 stars Very interesting, if you can get through it all
  Reviewed in the United States 🇺🇸 on December 6, 2020
  Verified Purchase
  There are a vast range of physical, biological, and other systems that we are confronted with that exhibit an immense complexity. Throughout history, it has been seen as perhaps the work of a supernatural being. Also, often when we encounter something that seems complex, we take it for granted that the phenomenon must be the result of an underlying mechanism that is itself complex. The author shows us how simple programs can, in fact, produce great complexity. In regards to physics, the author states, “that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known feature of our universe will emerge.” We learn in this book, through virtually hundreds of pages of diagrams, how simple initial conditions can produce complex behavior. Wolfram argues that this phenomenon is responsible from most of the complexity we see in nature.

  There is extensive discussion of cellular automata, where we have simple rules that result in complex behavior- so much in fact that I skipped through some of the material. It is believed that these behaviors are common in nature. An interesting finding is that “adding more complexity to the underlying rules does not yield behavior that it ultimately any more complex.” By chapter four, we are introduced to systems based on numbers. How does their behavior compare with other systems discussed? Even here it is possible to get behavior of considerable complexity by applying a series of operations based on simple arithmetic. Chapter 5 expands things to two dimension and beyond. The next chapter considers completely random initial conditions.

  Finally, we get to the application of what was covered in the previous chapters – the study of actual phenomena in nature. There seems to be a similarity between the basic mechanisms in nature and those responsible for the phenomena seen in simple programs. It is important to note that even though the models are based on simple programs that may successfully reproduce the behavior of some system even though that system does not actually consist of discrete cells, it is just a model. Models provide “an abstract representation of effects that are important in determining the behavior of a system.” The author provides examples of snowflake formation and fluidic motion. Regarding evolution, it seems that natural selection may not be as important as once thought saying that features in complexity of biological organisms arise in a sense not because of natural selection but despite it. Also, it was noted that the high degree of complexity seen in nature “can arise in a sense quite effortlessly just as a consequence of following certain simple rules of growth.”

  In the chapter on physics, this idea of simple initial conditions producing complex and seemingly random behavior is applied to the Second Law of Thermodynamics. The author discusses models for continuum fluid mechanics and even the universe. In fact, he believes that most physical laws discovered so far “are not truly fundamental but are emergent features of the large-scale behavior of some ultimate underlying rules.” The discussion veers into topics like gravity, spacetime, and quantum phenomena. It is the author’s “strong belief that in the end it will turn out that every detail of our universe does indeed follow rules that can be represented by a very simple program – and that everything we see will ultimately emerge just from running this program.” Wow, the thought went through my head: could we be living in a simulation? Who knows?

  The next chapter deals with process of perception and analysis. I skimmed through parts of this section – a bit too much detail for me. But the question becomes could simple mechanisms underlie human thinking? The next chapter covers universality and how “in the end cellular automata can actually be made to emulate almost every single type of system” that has been discussed so far. In fact, a wide range of computations that can be performed by computers can be done by cellular automata.

  In the final chapter, the author discusses something called the Principle of Computational Equivalence, where one can view all processes whether produced by humans or nature as computations. Another concept discussed is computational irreducibility, which implies that even if one has all the information to work out how some system will behave, it will still take an irreducible amount of computational work to do this. The concepts developed here apply not only to phenomena in nature but can provide new insights on fundamental issues in mathematics as well. Note that in discussing his work, the author supplies literally hundreds of charts, diagrams, and tables to back up his claims. The author concludes the main purpose of the book has been to build a new kind of basic science, but the most important consequence will have been to introduce a vast new range of systems helpful to technology.
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  Mr. Wizard
  4.0 out of 5 stars A good guided tour to many interesting subjects involving mathematical patterns.
  Reviewed in the United States 🇺🇸 on October 5, 2021
  Verified Purchase
  I have just skimmed a few pages here and there in A New Kind of Science. I bought it because Stephen Wolfram is almost as brilliant as he would have you believe he is if you have read some of the promo material about him that has appeared on Wolfram Research's main page.

  I used to use Mathematica years ago, when it was no longer amazing and trendy, but just a very accepted software tool college and grad students were expected to use in conjunction with many of their science, math and engineering classes. I never had to use it for a class or work, but it was often available for free or cheap when I was a grad student or F/T employee on campus so I got to like it because it was interesting and fun to use as a way to explore some "hobby" math problems.

  This book is not one I expect to read cover to cover, but I would be surprised if I do not read large sections of it, with a few days of binge reading per section. After I had much parts of Ch1 and Ch2 when I had access to someone else's copy of the book a couple of years ago, I realized that Wolfram's notions regarding intuition as being a key to really understanding science [and math, by implication] are much more explicitly stated than how I have described what took me several years of my life to figure out.

  Looking at other chapters later in the book, I expect I will dive into sections mentioning or covering subject matter that has fascinated me for some reason in the past, just because I have a hunch that what Wolfram writes about it might save me a lot of time learning some basics that I might not pick up in more years of very casual reading about that or similar topics.

  Douglas R. Hostadter is mentioned in the index of A New Kind of Science [hereinafter: ANKoS] as he is referenced when recursive sequenced are discussed in this book. As a reader who found Godel, Escher, Bach: An Eternal Golden Braid (*) to be a life-changing book in the late 1970s when it was first published, I think books such as ANKoS are likely to be a way to keep expanding my knowledge of areas of science and math I enjoy without having to read several in-depth books covering those fields of study.

  My S.W.A.G is that ANKos will be useful as a tour guide to topics in math and science I would like to learn more about, written by a guide who looks at different things and observes their mathematical and conceptual similarities or other relationships.

  Sorry for the rather vague limited "first impression" done with a very broad brush, but people who will like this book for the reasons I am glad I already bought it (no, not cognitive dissonance! :-) might well want to risk $15 on a book that could enable them to continue studying certain subjects where the typical book price might be upwards of $40 and know enough to select just a few very appropriate titles.

  In short if glancing through ANKoS for subjects that look interesting and maybe reading the first chapter or two to see if the style and manner of presentation work for you, makes you want to delve deeper into some of those subjects, ANKoS might be an excellent place to start.

  Now I want to pick up a current edition of Mathematica, but unlike this book, that is not cheap if one does not have a university or corporate affiliation which makes it free or very affordable.

  Regards,
  TMG

  * ( https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 )
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  bluemarlin
  1.0 out of 5 stars Very light on technical detail and high on self-opinion at the very least
  Reviewed in the United Kingdom 🇬🇧 on July 20, 2019
  Verified Purchase
  20 years in the making & revolutionary ? he is having a giraffe. First I thought he was telling the story of Lorenz in the first few pages and then going onto the lorenz attractor, chaos theory and Poincare sections etc .... but no ... it is all about himself, a self-obsession he exhibits on every page, and his supposedly startling, earth-shattering observations that he fails to appreciate have been observed by many others before and presented in more detail, are nothing of the sort and in the intervening time has not had any impact. Detail is sadly lacking in this tome with its extra wide margins and large font. He then compares the book ( his work ) to Newton's work as a new science, and raises an idea that he says he finally suppressed, which was the need to invent a whole new technical framework & lexicon to explain these monumental concepts. Was he on the lash or on ego building supplements ?? It has been 17 odd years since this book came out and there's nothing in the book beyond Barnsley's 'Iterated function system' book presentation or Conway's game of life and cellular automata. The book is very light on technical detail and high on self-opinion at the very least.
  Read more
  30 people found this helpful
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  Mr. Francis A. King
  2.0 out of 5 stars The automata are interesting, the remainder is bloated nonsense
  Reviewed in the United Kingdom 🇬🇧 on January 4, 2022
  Verified Purchase
  This book was written by Dr Wolfram is clever in ways that I will never be. Therefore, I was willing to be open to new idea, despite the scepticism of others. He starts with automata, where a new element is a combination of previous elements - The Game Of Life is a well-known example of this. There are some very interesting outcomes from this, and it would have been nice if he had continued with this theme. He does some work with Turing machines - I would have welcomed more.

  Unfortunately, Dr Wolfram had other notions. He believes, without any evidence that I could detect, that these automata could be used to model scientific behaviour. An example given was snowflakes, and he showed a automata model of a snowflake at different stages in growth. He claimed that regular science cannot explain snowflakes, which is untrue. What regular science can explain is the way in which temperature and humidity interact to deliver the diverse forms of snowflake. All that his automata can do is make a hexagonal shape, on a hexagonal grid. It predicts nothing, and is therefore worthless.

  Dr Wolfram does a presentation about relativity and quantum mechanics. This is worse than the snowflake example. It is page after page of utter nonsense.

  Overall, the impression is of a bloated book, which could say more in fewer pages if it was properly edited. The automata are genuinely interesting - the remainder is just nonsense, and could have been omitted.
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  ab..c
  2.0 out of 5 stars Long detailed exploration of fundimentals within thus new Math
  Reviewed in the United Kingdom 🇬🇧 on March 7, 2022
  Verified Purchase
  * Physical

  This book is over 1201 pages. It's a door stop of a book. The paper is very good quality and it's printed very well.

  * Target Audience: A-Level, H.N.D, Undergraduate, Graduate, Post Graduate, Masters?

  It's open to a widespread A-level to Graduate.

  * Whats does it cover?

  1. the Foundation for a New Kind of Science, 2. The Crucial Experiment, 3. the World of Simple Programs, 4. System based on Numbers, 5. Two Dimensional and Beyond. 6. Starting from Randomness, 7. Mechanisms in Programs and Nature, 8. Implications for Everyday Systems, 9. Fundamental Physics, 10 Processes of Perception and Analysis,11 the Notion of Computation, 12. the Principle of Computational Equivalence, Notes index

  * What are the best bits?

  This book on its own is an examination of how simple programs can generate complex data. It is needed to remember that these simple methods are fields in, Mathematics, Physics, Biology, Social Sciences, Computer Science, Philosophy, Art, Technology, A.I, Artificial life, Catsrophe, Chaos theory, Complexity theory, Computational Complexity Theory, Cybernetics, Dynamic Systems, Theory, Evolution Theory, Experimental Mathematics, Fractals Geometry, General system Theory, Nano Technology, Nonlinear Dynamics, Scientific Computing, Self Organisation, Statistical Mechanics.

  The book is very detailed explorations of subtle changes to simple programs and data is catalogued systematically to help organise the resulting fields of results and are reapplied to the next generations of programs. It's a lot of stuff. His style is chatty as if he's there with you explaining the results to you. The message is to organise your results and build on them. You could cut 25% of the chatty style and still wouldn't have harmed the structure.

  * Summary

  It's an education in that the simpler the program, the wider the potential useability of what's shown.
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  Myrddyn
  5.0 out of 5 stars A cybernetic aid to understanding
  Reviewed in the United Kingdom 🇬🇧 on December 18, 2017
  Verified Purchase
  As an initial reaction, perhaps 'Wow!' is the applicable expression! Mr. Wolfram has a respected and productive history in the field of computer science; his primary accolade being the powerful 'Mathmatica' suite of symbolic software tools used ubiquitously in the science and engineering fields. But he hasn't stopped there: this book describes how these tools - each of which is a functional sub-routine - can be combined and manipulated in such ways as if they were (almost) primary source code for building complex algorithmic entities - aka 'Cellular Automata' - applicable to almost every field of human endeavour. Thus is born the 'Wolfram' language for knowledge-based programming, using which may reveal useful, hitherto undiscovered relationships and interactions between systems and processes. in my estimation, this will revolutionise the way in which research and development is undertaken - and not just in the physical sciences - gaining huge benefits in costs, time and effort. This is truly a conceptual and intellectual 'amplifier' which will surely figure as being of immense significance in our increasing drive to unravel the mysteries of nature. The writing style is pacy and bright, giving full credit to the lay reader who may not have sufficient knowledge or experience in computer science or physics. It is not a laborious read, but Mr. Wolfram does, however, tediously tend to blow his own trumpet at every opportunity - obviously holding great pride in own theories and achievements! Having said that, it is in no way a detraction of the book or the man; I applaud him for what he has achieved and for presenting it to the world with articulate and informal narrative.
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  MR A R BUTTS
  3.0 out of 5 stars Interesting but you have to close your eyes to the repetitive "one of the most important discoveries I made" style language
  Reviewed in the United Kingdom 🇬🇧 on December 22, 2014
  Verified Purchase
  Verbose, over long, repetitive, insufferably arrogant and plagiaristic this book is (see other reviews), it does have one big advantage - It contains a wealth of information on cellular automata. I'm a quarter of the way through so far. I imagine it will ultimately reward a non specialist in the field like myself at the cost of many hours of my life.

  No doubt the book completely annoys the pants off industry practitioners who, reading this book, one would think barely exist compared to the self-professed awesome brain of Mr Wolfram. At times, it almost falls into parody as if an intelligent version of Arnold Rimmer for Red Dwarf was writing it. As an 11-year old, I remember programming Conway's Game of Life and, like many other kids who did this, being fascinated by where the hell the information to describe all this comes from. But according to Mr. Wolfram, he was the first one to think like this. Forget about all the teenage Game of Life hobbyists out there, Zuse, Tegmark etc. Perhaps its true that Wolfram did independently come up with lots of discoveries and helped reinvigorate the subject, but even if he did, he should acknowledge others more in his writing and not boast about what a clever chappy he is the whole time. It's my first book in digital physics, so can't compare it to others out there, unfortunately.
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5.0 out of 5 stars
 
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 Reviewed in the United States on August 19, 2010
 
 
 
 
  
 
 The book arrived in very good conditions and the price was excellent.Careful: It is bigger and heavier than you expect :)thanks
 
 
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 Top reviews from the United States
     There was a problem filtering reviews right now. Please try again later.   
 Reviewed in the United States 🇺🇸 on December 6, 2020   There are a vast range of physical, biological, and other systems that we are confronted with that exhibit an immense complexity. Throughout history, it has been seen as perhaps the work of a supernatural being. Also, often when we encounter something that seems complex, we take it for granted that the phenomenon must be the result of an underlying mechanism that is itself complex. The author shows us how simple programs can, in fact, produce great complexity. In regards to physics, the author states, “that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known feature of our universe will emerge.” We learn in this book, through virtually hundreds of pages of diagrams, how simple initial conditions can produce complex behavior. Wolfram argues that this phenomenon is responsible from most of the complexity we see in nature.There is extensive discussion of cellular automata, where we have simple rules that result in complex behavior- so much in fact that I skipped through some of the material. It is believed that these behaviors are common in nature. An interesting finding is that “adding more complexity to the underlying rules does not yield behavior that it ultimately any more complex.” By chapter four, we are introduced to systems based on numbers. How does their behavior compare with other systems discussed? Even here it is possible to get behavior of considerable complexity by applying a series of operations based on simple arithmetic. Chapter 5 expands things to two dimension and beyond. The next chapter considers completely random initial conditions. Finally, we get to the application of what was covered in the previous chapters – the study of actual phenomena in nature. There seems to be a similarity between the basic mechanisms in nature and those responsible for the phenomena seen in simple programs. It is important to note that even though the models are based on simple programs that may successfully reproduce the behavior of some system even though that system does not actually consist of discrete cells, it is just a model. Models provide “an abstract representation of effects that are important in determining the behavior of a system.” The author provides examples of snowflake formation and fluidic motion. Regarding evolution, it seems that natural selection may not be as important as once thought saying that features in complexity of biological organisms arise in a sense not because of natural selection but despite it. Also, it was noted that the high degree of complexity seen in nature “can arise in a sense quite effortlessly just as a consequence of following certain simple rules of growth.” In the chapter on physics, this idea of simple initial conditions producing complex and seemingly random behavior is applied to the Second Law of Thermodynamics. The author discusses models for continuum fluid mechanics and even the universe. In fact, he believes that most physical laws discovered so far “are not truly fundamental but are emergent features of the large-scale behavior of some ultimate underlying rules.” The discussion veers into topics like gravity, spacetime, and quantum phenomena. It is the author’s “strong belief that in the end it will turn out that every detail of our universe does indeed follow rules that can be represented by a very simple program – and that everything we see will ultimately emerge just from running this program.” Wow, the thought went through my head: could we be living in a simulation? Who knows? The next chapter deals with process of perception and analysis. I skimmed through parts of this section – a bit too much detail for me. But the question becomes could simple mechanisms underlie human thinking? The next chapter covers universality and how “in the end cellular automata can actually be made to emulate almost every single type of system” that has been discussed so far. In fact, a wide range of computations that can be performed by computers can be done by cellular automata. In the final chapter, the author discusses something called the Principle of Computational Equivalence, where one can view all processes whether produced by humans or nature as computations. Another concept discussed is computational irreducibility, which implies that even if one has all the information to work out how some system will behave, it will still take an irreducible amount of computational work to do this. The concepts developed here apply not only to phenomena in nature but can provide new insights on fundamental issues in mathematics as well. Note that in discussing his work, the author supplies literally hundreds of charts, diagrams, and tables to back up his claims. The author concludes the main purpose of the book has been to build a new kind of basic science, but the most important consequence will have been to introduce a vast new range of systems helpful to technology.
 
     Reviewed in the United States 🇺🇸 on October 5, 2021   I have just skimmed a few pages here and there in A New Kind of Science. I bought it because Stephen Wolfram is almost as brilliant as he would have you believe he is if you have read some of the promo material about him that has appeared on Wolfram Research's main page.I used to use Mathematica years ago, when it was no longer amazing and trendy, but just a very accepted software tool college and grad students were expected to use in conjunction with many of their science, math and engineering classes. I never had to use it for a class or work, but it was often available for free or cheap when I was a grad student or F/T employee on campus so I got to like it because it was interesting and fun to use as a way to explore some "hobby" math problems. This book is not one I expect to read cover to cover, but I would be surprised if I do not read large sections of it, with a few days of binge reading per section. After I had much parts of Ch1 and Ch2 when I had access to someone else's copy of the book a couple of years ago, I realized that Wolfram's notions regarding intuition as being a key to really understanding science [and math, by implication] are much more explicitly stated than how I have described what took me several years of my life to figure out. Looking at other chapters later in the book, I expect I will dive into sections mentioning or covering subject matter that has fascinated me for some reason in the past, just because I have a hunch that what Wolfram writes about it might save me a lot of time learning some basics that I might not pick up in more years of very casual reading about that or similar topics. Douglas R. Hostadter is mentioned in the index of A New Kind of Science [hereinafter: ANKoS] as he is referenced when recursive sequenced are discussed in this book. As a reader who found Godel, Escher, Bach: An Eternal Golden Braid (*) to be a life-changing book in the late 1970s when it was first published, I think books such as ANKoS are likely to be a way to keep expanding my knowledge of areas of science and math I enjoy without having to read several in-depth books covering those fields of study. My S.W.A.G is that ANKos will be useful as a tour guide to topics in math and science I would like to learn more about, written by a guide who looks at different things and observes their mathematical and conceptual similarities or other relationships. Sorry for the rather vague limited "first impression" done with a very broad brush, but people who will like this book for the reasons I am glad I already bought it (no, not cognitive dissonance! :-) might well want to risk $15 on a book that could enable them to continue studying certain subjects where the typical book price might be upwards of $40 and know enough to select just a few very appropriate titles. In short if glancing through ANKoS for subjects that look interesting and maybe reading the first chapter or two to see if the style and manner of presentation work for you, makes you want to delve deeper into some of those subjects, ANKoS might be an excellent place to start. Now I want to pick up a current edition of Mathematica, but unlike this book, that is not cheap if one does not have a university or corporate affiliation which makes it free or very affordable. Regards,TMG * ( https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 )
 
      
 
 
 
 
  
 
 
 Top reviews from other countries
 
 
 
 
 
 
  
 
 
   1.0 out of 5 stars
 
 
 Very light on technical detail and high on self-opinion at the very least
 
 Reviewed in the United Kingdom 🇬🇧 on July 20, 2019   20 years in the making &amp; revolutionary ? he is having a giraffe. First I thought he was telling the story of Lorenz in the first few pages and then going onto the lorenz attractor, chaos theory and Poincare sections etc .... but no ... it is all about himself, a self-obsession he exhibits on every page, and his supposedly startling, earth-shattering observations that he fails to appreciate have been observed by many others before and presented in more detail, are nothing of the sort and in the intervening time has not had any impact. Detail is sadly lacking in this tome with its extra wide margins and large font. He then compares the book ( his work ) to Newton's work as a new science, and raises an idea that he says he finally suppressed, which was the need to invent a whole new technical framework &amp; lexicon to explain these monumental concepts. Was he on the lash or on ego building supplements ?? It has been 17 odd years since this book came out and there's nothing in the book beyond Barnsley's 'Iterated function system' book presentation or Conway's game of life and cellular automata. The book is very light on technical detail and high on self-opinion at the very least.
 
      2.0 out of 5 stars
 
 
 The automata are interesting, the remainder is bloated nonsense
 
 Reviewed in the United Kingdom 🇬🇧 on January 4, 2022   This book was written by Dr Wolfram is clever in ways that I will never be. Therefore, I was willing to be open to new idea, despite the scepticism of others. He starts with automata, where a new element is a combination of previous elements - The Game Of Life is a well-known example of this. There are some very interesting outcomes from this, and it would have been nice if he had continued with this theme. He does some work with Turing machines - I would have welcomed more.Unfortunately, Dr Wolfram had other notions. He believes, without any evidence that I could detect, that these automata could be used to model scientific behaviour. An example given was snowflakes, and he showed a automata model of a snowflake at different stages in growth. He claimed that regular science cannot explain snowflakes, which is untrue. What regular science can explain is the way in which temperature and humidity interact to deliver the diverse forms of snowflake. All that his automata can do is make a hexagonal shape, on a hexagonal grid. It predicts nothing, and is therefore worthless. Dr Wolfram does a presentation about relativity and quantum mechanics. This is worse than the snowflake example. It is page after page of utter nonsense. Overall, the impression is of a bloated book, which could say more in fewer pages if it was properly edited. The automata are genuinely interesting - the remainder is just nonsense, and could have been omitted.
 
      2.0 out of 5 stars
 
 
 Long detailed exploration of fundimentals within thus new Math
 
 Reviewed in the United Kingdom 🇬🇧 on March 7, 2022   * PhysicalThis book is over 1201 pages. It's a door stop of a book. The paper is very good quality and it's printed very well. * Target Audience: A-Level, H.N.D, Undergraduate, Graduate, Post Graduate, Masters? It's open to a widespread A-level to Graduate. * Whats does it cover? 1. the Foundation for a New Kind of Science, 2. The Crucial Experiment, 3. the World of Simple Programs, 4. System based on Numbers, 5. Two Dimensional and Beyond. 6. Starting from Randomness, 7. Mechanisms in Programs and Nature, 8. Implications for Everyday Systems, 9. Fundamental Physics, 10 Processes of Perception and Analysis,11 the Notion of Computation, 12. the Principle of Computational Equivalence, Notes index * What are the best bits? This book on its own is an examination of how simple programs can generate complex data. It is needed to remember that these simple methods are fields in, Mathematics, Physics, Biology, Social Sciences, Computer Science, Philosophy, Art, Technology, A.I, Artificial life, Catsrophe, Chaos theory, Complexity theory, Computational Complexity Theory, Cybernetics, Dynamic Systems, Theory, Evolution Theory, Experimental Mathematics, Fractals Geometry, General system Theory, Nano Technology, Nonlinear Dynamics, Scientific Computing, Self Organisation, Statistical Mechanics. The book is very detailed explorations of subtle changes to simple programs and data is catalogued systematically to help organise the resulting fields of results and are reapplied to the next generations of programs. It's a lot of stuff. His style is chatty as if he's there with you explaining the results to you. The message is to organise your results and build on them. You could cut 25% of the chatty style and still wouldn't have harmed the structure. * Summary It's an education in that the simpler the program, the wider the potential useability of what's shown.
 
      5.0 out of 5 stars
 
 
 A cybernetic aid to understanding
 
 Reviewed in the United Kingdom 🇬🇧 on December 18, 2017   As an initial reaction, perhaps 'Wow!' is the applicable expression! Mr. Wolfram has a respected and productive history in the field of computer science; his primary accolade being the powerful 'Mathmatica' suite of symbolic software tools used ubiquitously in the science and engineering fields. But he hasn't stopped there: this book describes how these tools - each of which is a functional sub-routine - can be combined and manipulated in such ways as if they were (almost) primary source code for building complex algorithmic entities - aka 'Cellular Automata' - applicable to almost every field of human endeavour. Thus is born the 'Wolfram' language for knowledge-based programming, using which may reveal useful, hitherto undiscovered relationships and interactions between systems and processes. in my estimation, this will revolutionise the way in which research and development is undertaken - and not just in the physical sciences - gaining huge benefits in costs, time and effort. This is truly a conceptual and intellectual 'amplifier' which will surely figure as being of immense significance in our increasing drive to unravel the mysteries of nature. The writing style is pacy and bright, giving full credit to the lay reader who may not have sufficient knowledge or experience in computer science or physics. It is not a laborious read, but Mr. Wolfram does, however, tediously tend to blow his own trumpet at every opportunity - obviously holding great pride in own theories and achievements! Having said that, it is in no way a detraction of the book or the man; I applaud him for what he has achieved and for presenting it to the world with articulate and informal narrative.
 
      3.0 out of 5 stars
 
 
 Interesting but you have to close your eyes to the repetitive "one of the most important discoveries I made" style language
 
 Reviewed in the United Kingdom 🇬🇧 on December 22, 2014   Verbose, over long, repetitive, insufferably arrogant and plagiaristic this book is (see other reviews), it does have one big advantage - It contains a wealth of information on cellular automata. I'm a quarter of the way through so far. I imagine it will ultimately reward a non specialist in the field like myself at the cost of many hours of my life.No doubt the book completely annoys the pants off industry practitioners who, reading this book, one would think barely exist compared to the self-professed awesome brain of Mr Wolfram. At times, it almost falls into parody as if an intelligent version of Arnold Rimmer for Red Dwarf was writing it. As an 11-year old, I remember programming Conway's Game of Life and, like many other kids who did this, being fascinated by where the hell the information to describe all this comes from. But according to Mr. Wolfram, he was the first one to think like this. Forget about all the teenage Game of Life hobbyists out there, Zuse, Tegmark etc. Perhaps its true that Wolfram did independently come up with lots of discoveries and helped reinvigorate the subject, but even if he did, he should acknowledge others more in his writing and not boast about what a clever chappy he is the whole time. It's my first book in digital physics, so can't compare it to others out there, unfortunately.
A New Kind of Science 1st Edition
https://www.amazon.com/New-Kind-Science-Stephen-Wolfram/dp/1579550088
https://writings.stephenwolfram.com/2017/05/a-new-kind-of-science-a-15-year-view/
A New Kind of Science: A 15-Year View—Stephen Wolfram Writings
2017-05-16
None
<div><div>
 <p><em>Starting now, in celebration of its 15th anniversary, <a href="http://wolframscience.com/nks">A New Kind of Science will be freely available</a> in its entirety, with high-resolution images, on the web or for download.</em></p>
<p><a href="http://www.wolframscience.com/nks/"></a></p>
<p>It’s now <a href="https://www.stephenwolfram.com/scrapbook/page8/">15 years since I published</a> my book <a href="http://wolframscience.com/nks">A New Kind of Science</a>—more than <a href="https://www.stephenwolfram.com/scrapbook/page7/">25 since I started writing it</a>, and more than <a href="https://www.stephenwolfram.com/scrapbook/page3/">35 since I started working towards it</a>. But with every passing year I feel I understand more about what the book is really about—and why it’s important. I wrote the book, as its title suggests, to contribute to the progress of science. But as the years have gone by, I’ve realized that the core of what’s in the book actually goes far beyond science—into many areas that will be increasingly important in defining our whole future.</p>
<div><p>This essay is also in:<br />
<a href="https://soundcloud.com/stephenwolfram/sets/revisiting-a-new-kind-of-science"><strong><em>SoundCloud</em> »</strong></a></p></div>
<p>So, viewed from a distance of 15 years, what is the book really about? At its core, it’s about something profoundly abstract: the theory of all possible theories, or the universe of all possible universes. But for me one of the achievements of the book is the realization that one can explore such fundamental things concretely—by doing actual experiments in the computational universe of possible programs. And in the end the book is full of what might at first seem like quite alien pictures made just by running very simple such programs.</p>
<p><a href="https://www.stephenwolfram.com/scrapbook/page2/">Back in 1980, when I made my living as a theoretical physicist</a>, if you’d asked me what I thought simple programs would do, I expect I would have said “not much”. I had been very interested in the kind of complexity one sees in nature, but I thought—like a typical reductionistic scientist—that the key to understanding it must lie in figuring out detailed features of the underlying component parts.</p>
<p>In retrospect <a href="http://wolframscience.com/nks/p17--the-personal-story-of-the-science-in-this-book/">I consider it incredibly lucky</a> that all those years ago I happened to have the right interests and the right skills to actually try what is in a sense the most basic experiment in the computational universe: to systematically take a sequence of the simplest possible programs, and run them.</p>
<p>I could tell as soon as I did this that there were interesting things going on, but it took a couple more years before I began to really appreciate the force of what I’d seen. For me it all started with one picture:</p>
<p></p>
<p>Or, in modern form:</p>
<p></p>
<p>I call it <a href="https://www.wolframscience.com/nks/p27--how-do-simple-programs-behave/">rule 30</a>. It’s my all-time favorite discovery, and today I carry it around everywhere on my <a href="https://twitter.com/stephen_wolfram/status/839868391674556417">business cards</a>. What is it? It’s one of the <a href="http://wolframscience.com/nks/p23--how-do-simple-programs-behave/">simplest programs one can imagine</a>. It operates on rows of black and white cells, starting from a single black cell, and then repeatedly applies the rules at the bottom. And the crucial point is that even though those rules are by any measure extremely simple, the pattern that emerges is not.</p>
<p>It’s a crucial—and utterly unexpected—feature of the computational universe: that even among the very simplest programs, it’s easy to get immensely complex behavior. It took me a solid decade to understand just how broad this phenomenon is. It doesn’t just happen in programs (“<a href="http://wolframscience.com/nks/p23--how-do-simple-programs-behave/">cellular automata</a>”) like rule 30. It <a href="http://wolframscience.com/nks/chap-3--the-world-of-simple-programs/">basically shows up whenever</a> you start enumerating possible rules or possible programs whose behavior isn’t obviously trivial.</p>
<p>Similar phenomena <a href="http://wolframscience.com/nks/p42--why-these-discoveries-were-not-made-before/">had actually been seen for centuries</a> in things like the <a href="http://wolframscience.com/nks/p136--mathematical-constants/">digits of pi</a> and the <a href="http://wolframscience.com/nks/p132--the-sequence-of-primes/">distribution of primes</a>—but they were basically just viewed as curiosities, and not as signs of something profoundly important. It’s been nearly 35 years since I first saw what happens in rule 30, and with every passing year I feel I come to understand more clearly and deeply what its significance is.</p>
<p>Four centuries ago it was the discovery of the moons of Jupiter and their regularities that sowed the seeds for modern exact science, and for the modern scientific approach to thinking. Could my little rule 30 now be the seed for another such intellectual revolution, and a new way of thinking about everything?</p>
<p>In some ways I might personally prefer not to take responsibility for shepherding such ideas (<a href="https://writings.stephenwolfram.com/2012/05/living-a-paradigm-shift-looking-back-on-reactions-to-a-new-kind-of-science/">“paradigm shifts” are hard and thankless work</a>). And certainly for years I have just quietly used such ideas to develop technology and my own thinking. But as computation and AI become increasingly central to our world, I think it’s important that the implications of what’s out there in the computational universe be more widely understood.</p>
<h2>Implications of the Computational Universe</h2>
<p>Here’s the way I see it today. From observing the <a href="http://www.wolframalpha.com/input/?i=moons+of+jupiter">moons of Jupiter </a>we came away with the idea that—if looked at right—the universe is an ordered and regular place, that we can ultimately understand. But now, in exploring the computational universe, we quickly come upon things like rule 30 where even the simplest rules seem to lead to irreducibly complex behavior.</p>
<p>One of the big ideas of A New Kind of Science is what I call the <a href="http://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a>. The first step is to think of every process—whether it’s happening with black and white squares, or in physics, or inside our brains—as a computation that somehow transforms input to output. What the Principle of Computational Equivalence says is that above an extremely low threshold, all processes correspond to computations of equivalent sophistication.</p>
<p>It might not be true. It might be that something like rule 30 corresponds to a fundamentally simpler computation than the fluid dynamics of a hurricane, or the processes in my brain as I write this. But what the Principle of Computational Equivalence says is that in fact all these things are computationally equivalent.</p>
<p>It’s a very important statement, with many deep implications. For one thing, it implies what I call <a href="http://wolframscience.com/nks/p737--computational-irreducibility/">computational irreducibility</a>. If something like rule 30 is doing a computation just as sophisticated as our brains or our mathematics, then there’s no way we can “outrun” it: to figure out what it will do, we have to do an irreducible amount of computation, effectively tracing each of its steps.</p>
<p>The mathematical tradition in exact science has emphasized the idea of predicting the behavior of systems by doing things like solving mathematical equations. But what computational irreducibility implies is that out in the computational universe that often won’t work, and instead the only way forward is just to explicitly run a computation to simulate the behavior of the system.</p>
<h2>A Shift in Looking at the World</h2>
<p>One of the things I did in A New Kind of Science was to show how simple programs can <a href="http://wolframscience.com/nks/chap-8--implications-for-everyday-systems/">serve as models</a> for the essential features of all sorts of <a href="http://wolframscience.com/nks/chap-8--implications-for-everyday-systems/">physical, biological and other systems</a>. Back when the book appeared, <a href="https://writings.stephenwolfram.com/2012/05/living-a-paradigm-shift-looking-back-on-reactions-to-a-new-kind-of-science/">some people were skeptical about this</a>. And indeed at that time there was a <a href="http://wolframscience.com/nks/notes-1-1--mathematics-in-science/">300-year unbroken tradition</a> that serious models in science should be based on mathematical equations.</p>
<p>But in the past 15 years <a href="https://writings.stephenwolfram.com/2012/05/its-been-10-years-whats-happened-with-a-new-kind-of-science/">something remarkable has happened</a>. For now, when new models are created—whether of animal patterns or web browsing behavior—they are overwhelmingly more often based on programs than on mathematical equations.</p>
<p>Year by year, it’s been a slow, almost silent, process. But by this point, it’s a dramatic shift. Three centuries ago pure philosophical reasoning was supplanted by mathematical equations. Now in these few short years, equations have been largely supplanted by programs. For now, it’s mostly been something practical and pragmatic: the models work better, and are more useful.</p>
<p>But when it comes to understanding the foundations of what’s going on, one’s led not to things like mathematical theorems and calculus, but instead to ideas like the <a href="http://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a>. Traditional mathematics-based ways of thinking have made concepts like force and momentum ubiquitous in the way we talk about the world. But now as we think in fundamentally computational terms we have to start talking in terms of concepts like <a href="http://www.wolframscience.com/nks/p753--undecidability-and-intractability/">undecidability</a> and <a href="http://www.wolframscience.com/nks/p737--computational-irreducibility/">computational irreducibility</a>.</p>
<p>Will some type of <a href="https://www.stephenwolfram.com/publications/biomedical-implications-a-new-kind-of-science/">tumor always stop growing in some particular model</a>? It might be undecidable. Is there a way to work out how a weather system will develop? It might be computationally irreducible.</p>
<p>These concepts are pretty important when it comes to understanding not only what can and cannot be modeled, but also what can and cannot be controlled in the world. Computational irreducibility in economics is going to limit what can be globally controlled. Computational irreducibility in biology is going to limit how generally effective therapies can be—and make highly personalized medicine a fundamental necessity.</p>
<p>And through ideas like the Principle of Computational Equivalence we can start to discuss just what it is that allows nature—seemingly so effortlessly—to generate so much that seems so complex to us. Or how even deterministic underlying rules can lead to computationally irreducible behavior that for all practical purposes can seem to show “<a href="http://www.wolframscience.com/nks/p750--the-phenomenon-of-free-will/">free will</a>”.</p>
<p></p>
<h2>Mining the Computational Universe</h2>
<p>A central lesson of A New Kind of Science is that there’s a lot of incredible richness out there in the computational universe. And one reason that’s important is that it means that there’s a lot of incredible stuff out there for us to “mine” and harness for our purposes.</p>
<p>Want to automatically make an interesting custom piece of art? Just start looking at simple programs and <a href="https://writings.stephenwolfram.com/2011/06/music-mathematica-and-the-computational-universe/">automatically pick out one you like</a>—as in our <a href="http://tones.wolfram.com/">WolframTones</a> music site from a decade ago. Want to find an optimal algorithm for something? Just search enough programs out there, and you’ll find one.</p>
<p>We’ve normally been used to creating things by building them up, step by step, with human effort—progressively creating architectural plans, or engineering drawings, or lines of code. But the discovery that there’s so much richness so easily accessible in the computational universe suggests a different approach: don’t try building anything; just define what you want, and then search for it in the computational universe.</p>
<p>Sometimes it’s really easy to find. Like let’s say you want to generate apparent randomness. Well, then just enumerate cellular automata (as I did in 1984), and very quickly you come upon rule 30—which turns out to be one of the very <a href="https://www.stephenwolfram.com/publications/academic/random-sequence-generation-cellular-automata.pdf">best known generators of apparent randomness</a> (look down the center column of cell values, for examples). In other situations you might have to search 100,000 cases (as I did in finding the <a href="http://www.wolframscience.com/nks/p808--implications-for-mathematics-and-its-foundations/">simplest axiom system for logic</a>, or the <a href="http://www.wolframscience.com/prizes/tm23/">simplest universal Turing machine</a>), or you might have to search millions or even trillions of cases. But in the past 25 years, we’ve had incredible success in just discovering algorithms out there in the computational universe—and we rely on many of them in implementing the <a href="https://www.wolfram.com/language/">Wolfram Language</a>.</p>
<p>At some level it’s quite sobering. One finds some tiny program out in the computational universe. One can tell it does what one wants. But when one looks at what it’s doing, one doesn’t have any real idea how it works. Maybe one can analyze some part—and be struck by how “clever” it is. But there just isn’t a way for us to understand the whole thing; it’s not something familiar from our usual patterns of thinking.</p>
<p>Of course, we’ve often had similar experiences before—when we use things from nature. We may notice that some particular substance is a useful drug or a great chemical catalyst, but we may have no idea why. But in doing engineering and in most of our modern efforts to build technology, the great emphasis has instead been on constructing things whose design and operation we can readily understand.</p>
<p>In the past we might have thought that was enough. But what our explorations of the computational universe show is that it’s not: selecting only things whose operation we can readily understand misses most of the immense power and richness that’s out there in the computational universe.</p>
<h2>A World of Discovered Technology</h2>
<p>What will the world look like when more of what we have is mined from the computational universe? Today the environment we build for ourselves is dominated by things like simple shapes and repetitive processes. But the more we use what’s out there in the computational universe, the less regular things will look. Sometimes they may look a bit “organic”, or like what we see in nature (since after all, nature follows similar kinds of rules). But sometimes they may look quite random, until perhaps suddenly and incomprehensibly they achieve something we recognize.</p>
<p>For several millennia we as a civilization have been on a path to understand more about what happens in our world—whether by using science to decode nature, or by creating our own environment through technology. But to use more of the richness of the computational universe we must at least to some extent forsake this path.</p>
<p>In the past, we somehow counted on the idea that between our brains and the tools we could create we would always have fundamentally greater computational power than the things around us—and as a result we would always be able to “understand” them. But what the Principle of Computational Equivalence says is that this isn’t true: out in the computational universe there are lots of things just as powerful as our brains or the tools we build. And as soon as we start using those things, we lose the “edge” we thought we had.</p>
<p>Today we still imagine we can identify discrete “bugs” in programs. But most of what’s powerful out there in the computational universe is rife with computational irreducibility—so the only real way to see what it does is just to run it and watch what happens.</p>
<p>We ourselves, as biological systems, are a great example of computation happening at a molecular scale—and we are no doubt rife with computational irreducibility (which is, at some fundamental level, why medicine is hard). I suppose it’s a tradeoff: we could limit our technology to consist only of things whose operation we understand. But then we would miss all that richness that’s out there in the computational universe. And we wouldn’t even be able to match the achievements of our own biology in the technology we create.</p>
<h2>Machine Learning and the Neural Net Renaissance</h2>
<p>There’s a common pattern I’ve noticed with intellectual fields. They go for decades and perhaps centuries with only incremental growth, and then suddenly, usually as a result of a methodological advance, there’s a burst of “hypergrowth” for perhaps 5 years, in which important new results arrive almost every week.</p>
<p>I was fortunate enough that my own very first field—particle physics—was in its period of hypergrowth right <a href="https://www.stephenwolfram.com/scrapbook/page2/">when I was involved</a> in the late 1970s. And for myself, the 1990s felt like a kind of personal period of hypergrowth for what became A New Kind of Science—and indeed that’s why I couldn’t pull myself away from it for more than a decade.</p>
<p>But today, the obvious field in hypergrowth is <a href="http://reference.wolfram.com/language/guide/MachineLearning.html">machine learning</a>, or, more specifically, <a href="http://reference.wolfram.com/language/guide/NeuralNetworks.html">neural nets</a>. It’s funny for me to see this. I actually <a href="https://writings.stephenwolfram.com/2015/05/wolfram-language-artificial-intelligence-the-image-identification-project/">worked on neural nets back in 1981</a>, before I started on cellular automata, and several years before I found rule 30. But I never managed to get neural nets to do anything very interesting—and actually I found them too messy and complicated for the fundamental questions I was concerned with.</p>
<p>And so I “<a href="http://www.wolframscience.com/nks/notes-2-3--my-work-on-cellular-automata/">simplified them</a>”—and wound up with cellular automata. (I was also inspired by things like the Ising model in statistical physics, etc.) <a href="http://www.wolframscience.com/nks/p17--the-personal-story-of-the-science-in-this-book/">At the outset</a>, I thought I might have simplified too far, and that my little cellular automata would never do anything interesting. But then I found things like rule 30. And I’ve been trying to understand its implications ever since.</p>
<p>In building <a href="https://www.wolfram.com/mathematica/">Mathematica</a> and the <a href="https://www.wolfram.com/language/">Wolfram Language</a>, I’d always kept track of neural nets, and occasionally we’d use them in some small way for some algorithm or another. But about 5 years ago I suddenly started hearing amazing things: that somehow the idea of training neural nets to do sophisticated things was actually working. At first I wasn’t sure. But then we started building neural net capabilities in the Wolfram Language, and finally <a href="https://writings.stephenwolfram.com/2015/05/wolfram-language-artificial-intelligence-the-image-identification-project/">two years ago we released</a> our <a href="https://www.imageidentify.com/">ImageIdentify.com</a> website—and now we’ve got our whole <a href="https://writings.stephenwolfram.com/2017/03/the-rd-pipeline-continues-launching-version-11-1/">symbolic neural net system</a>. And, yes, I’m impressed. There are lots of tasks that had traditionally been viewed as the unique domain of humans, but which now we can routinely do by computer.</p>
<p>But <a href="https://writings.stephenwolfram.com/2017/05/machine-learning-for-middle-schoolers/">what’s actually going on in a neural net</a>? It’s not really to do with the brain; that was just the inspiration (though in reality the brain probably works more or less the same way). A neural net is really a sequence of functions that operate on arrays of numbers, with each function typically taking quite a few inputs from around the array. It’s not so different from a cellular automaton. Except that in a cellular automaton, one’s usually dealing with, say, just 0s and 1s, not arbitrary numbers like 0.735. And instead of taking inputs from all over the place, in a cellular automaton each step takes inputs only from a very well-defined local region.</p>
<p>Now, to be fair, it’s pretty common to study “<a href="http://reference.wolfram.com/language/ref/ConvolutionLayer.html">convolutional neural nets</a>”, in which the patterns of inputs are very regular, just like in a cellular automaton. And it’s becoming clear that having precise (say 32-bit) numbers isn’t critical to the operation of neural nets; one can probably make do with just a few bits.</p>
<p>But a big feature of neural nets is that we know how to make them “learn”. In particular, they have enough features from traditional mathematics (like involving continuous numbers) that techniques like calculus can be applied to provide strategies to make them incrementally change their parameters to “fit their behavior” to whatever training examples they’re given.</p>
<p>It’s far from obvious how much computational effort, or how many training examples, will be needed. But the breakthrough of about five years ago was the discovery that for many important practical problems, what’s available with modern GPUs and modern web-collected training sets can be enough.</p>
<p>Pretty much nobody ends up explicitly setting or “engineering” the parameters in a neural net. Instead, what happens is that they’re found automatically. But unlike with simple programs like cellular automata, where one’s typically enumerating all possibilities, in current neural nets there’s an <a href="http://reference.wolfram.com/language/ref/NetTrain.html">incremental process</a>, essentially based on calculus, that manages to progressively improve the net—a little like the way biological evolution progressively improves the “fitness” of an organism.</p>
<p>It’s plenty remarkable what comes out from training a neural net in this way, and it’s plenty difficult to understand how the neural net does what it does. But in some sense the neural net isn’t venturing too far across the computational universe: it’s always basically keeping the same basic computational structure, and just changing its behavior by changing parameters.</p>
<p>But to me the success of today’s neural nets is a spectacular endorsement of the power of the computational universe, and another validation of the ideas of A New Kind of Science. Because it shows that out in the computational universe, away from the constraints of explicitly building systems whose detailed behavior one can foresee, there are immediately all sorts of rich and useful things to be found.</p>
<h2>NKS Meets Modern Machine Learning</h2>
<p>Is there a way to bring the full power of the computational universe—and the ideas of <a href="http://wolframscience.com/nks">A New Kind of Science</a>—to the kinds of things one does with neural nets? I suspect so. And in fact, as the details become clear, I wouldn’t be surprised if exploration of the computational universe saw its own period of hypergrowth: a “mining boom” of perhaps unprecedented proportions.</p>
<p>In current work on neural nets, there’s a definite tradeoff one sees. The more what’s going on inside the neural net is like a simple mathematical function with essentially arithmetic parameters, the easier it is to use ideas from calculus to train the network. But the more what’s going is like a discrete program, or like a computation whose whole structure can change, the more difficult it is to train the network.</p>
<p>It’s worth remembering, though, that the networks we’re routinely training now would have looked utterly impractical to train only a few years ago. It’s effectively just all those quadrillions of GPU operations that we can throw at the problem that makes training feasible. And I won’t be surprised if even quite pedestrian (say, local exhaustive search) techniques will fairly soon let one do significant training even in cases where no incremental numerical approach is possible. And perhaps even it will be possible to invent some major generalization of things like calculus that will operate in the full computational universe. (I have some suspicions, based on thinking about generalizing basic notions of geometry to cover things like cellular automaton rule spaces.)</p>
<p>What would this let one do? Likely it would let one find considerably simpler systems that could achieve particular computational goals. And maybe that would bring within reach some qualitatively new level of operations, perhaps beyond what we’re used to being possible with things like brains.</p>
<p>There’s a funny thing that’s going on with modeling these days. As neural nets become more successful, one begins to wonder: why bother to simulate what’s going on inside a system when one can just make a black-box model of its output using a neural net? Well, if we manage to get machine learning to reach deeper into the computational universe, we won’t have as much of this tradeoff any more—because we’ll be able to learn models of the mechanism as well as the output.</p>
<p>I’m pretty sure that bringing the full computational universe into the purview of machine learning will have spectacular consequences. But it’s worth realizing that computational universality—and the <a href="http://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a>—make it less a matter of principle. Because they imply that even neural nets of the kinds we have now are universal, and are capable of emulating anything any other system can do. (In fact, this universality result was essentially what launched the <a href="http://www.wolframscience.com/nks/notes-10-12--history-of-ideas-about-thinking/">whole modern idea of neural nets</a>, back in 1943.)</p>
<p>And as a practical matter, the fact that current neural net primitives are being built into hardware and so on will make them a desirable foundation for actual technology systems, though, even if they’re far from optimal. But my guess is that there are tasks where for the foreseeable future access to the full computational universe will be necessary to make them even vaguely practical.</p>
<h2>Finding AI</h2>
<p>What will it take to make artificial intelligence? As a kid, I was very interested in figuring out how to make a computer know things, and be able to answer questions from what it knew. And when I studied neural nets in 1981, it was partly in the context of trying to understand how to build such a system. As it happens, I had just developed <a href="https://www.stephenwolfram.com/publications/smp-symbolic-manipulation-program/">SMP</a>, which was a forerunner of Mathematica (and ultimately the Wolfram Language)—and which was very much based on symbolic pattern matching (“if you see this, transform it to that”). At the time, though, I imagined that artificial intelligence was somehow a “higher level of computation”, and I didn’t know how to achieve it.</p>
<p>I returned to the problem every so often, and kept putting it off. But then when I was working on A New Kind of Science it struck me: if I’m to take the Principle of Computational Equivalence seriously, then there can’t be any fundamentally “higher level of computation”—so AI must be achievable just with the standard ideas of computation that I already know.</p>
<p>And it was this realization <a href="https://writings.stephenwolfram.com/2016/04/my-life-in-technology-as-told-at-the-computer-history-museum/">that got me started</a> building <a href="http://www.wolframalpha.com/">Wolfram|Alpha</a>. And, yes, what I found is that lots of those very “AI-oriented things”, like natural language understanding, could be done just with “ordinary computation”, without any magic new AI invention. Now, to be fair, part of what was happening was that we were using ideas and methods from A New Kind of Science: we weren’t just engineering everything; we were often searching the computational universe for rules and algorithms to use.</p>
<p>So what about “general AI”? Well, I think at this point that with the tools and understanding we have, we’re in a good position to automate essentially anything we can define. But definition is a more difficult and central issue than we might imagine.</p>
<p>The way I see things at this point is that there’s a lot of computation even near at hand in the computational universe. And it’s powerful computation. As powerful as anything that happens in our brains. But we don’t recognize it as “intelligence” unless it’s aligned with our human goals and purposes.</p>
<p>Ever since I was writing A New Kind of Science, I’ve been fond of quoting the aphorism “<a href="http://www.wolframscience.com/nks/p844--historical-perspectives/">the weather has a mind of its own</a>”. It sounds so animistic and pre-scientific. But what the Principle of Computational Equivalence says is that actually, according to the most modern science, it’s true: the fluid dynamics of the weather is the same in its computational sophistication as the electrical processes that go on in our brains.</p>
<p>But is it “intelligent”? When I talk to people about <em>A New Kind of Science</em>, and about AI, I’ll often get asked when I think we’ll achieve “consciousness” in a machine. <a href="http://www.wolframscience.com/nks/notes-12-10--defining-life/">Life</a>, <a href="http://www.wolframscience.com/nks/notes-12-10--defining-intelligence/">intelligence</a>, consciousness: they are all concepts that we have a specific example of, here on Earth. But what are they in general? All life on Earth shares RNA and the structure of cell membranes. But surely that’s just because all life we know is part of one connected thread of history; it’s not that such details are fundamental to the very concept of life.</p>
<p>And so it is with intelligence. We have only one example we’re sure of: us humans. (We’re not even sure about animals.) But human intelligence as we experience it is deeply entangled with human civilization, human culture and ultimately also human physiology—even though none of these details are presumably relevant in the abstract definition of intelligence.</p>
<p>We might think about <a href="http://www.wolframscience.com/nks/p822--intelligence-in-the-universe/">extraterrestrial intelligence</a>. But what the Principle of Computational Equivalence implies is that actually there’s “alien intelligence” all around us. But somehow it’s just not quite aligned with human intelligence. We might look at rule 30, for example, and be able to see that it’s doing sophisticated computation, just like our brains. But somehow it just doesn’t seem to have any “point” to what it’s doing.</p>
<p>We imagine that in doing the things we humans do, we operate with certain goals or purposes. But rule 30, for example, just seems to be doing what it’s doing—just following some definite rule. In the end, though, one realizes we’re not so very different. After all, there are definite laws of nature that govern our brains. So anything we do is at some level just playing out those laws.</p>
<p>Any process can actually be described either in terms of mechanism (“the stone is moving according to <a href="http://www.wolframalpha.com/input/?i=newton%27s+laws">Newton’s laws</a>”), or in terms of goals (“the stone is moving so as to minimize potential energy”). The description in terms of mechanism is usually what’s most useful in connecting with science. But the description in terms of goals is usually what’s most useful in connecting with human intelligence.</p>
<p>And this is crucial in thinking about AI. We know we can have computational systems whose operations are as sophisticated as anything. But can we get them to do things that are aligned with human goals and purposes?</p>
<p>In a sense this is what I now view as the key problem of AI: it’s not about achieving underlying computational sophistication, but instead it’s about communicating what we want from this computation.</p>
<h2>The Importance of Language</h2>
<p>I’ve spent much of my life as a computer language designer—most importantly creating what is now the <a href="https://www.wolfram.com/language/">Wolfram Language</a>. I’d always seen my role as a language designer being to imagine the possible computations people might want to do, then—like a reductionist scientist—trying to “drill down” to find good primitives from which all these computations could be built up. But somehow from A New Kind of Science, and from thinking about AI, I’ve come to think about it a little differently.</p>
<p>Now what I more see myself as doing is making a <a href="https://writings.stephenwolfram.com/2015/11/how-should-we-talk-to-ais/">bridge between our patterns of human thinking, and what the computational universe is capable of</a>. There are all sorts of amazing things that can in principle be done by computation. But what the language does is to provide a way for us humans to express what we want done, or want to achieve—and then to get this actually executed, as automatically as possible.</p>
<p>Language design has to start from what we know and are familiar with. In the Wolfram Language, we name the built-in primitives with English words, leveraging the meanings that those words have acquired. But the Wolfram Language is not like natural language. It’s something more structured, and more powerful. It’s based on the words and concepts that we’re familiar with through the shared corpus of human knowledge. But it gives us a way to build up arbitrarily sophisticated programs that in effect express arbitrarily complex goals.</p>
<p>Yes, the computational universe is capable of remarkable things. But they’re not necessarily things that we humans can describe or relate to. But in building the Wolfram Language my goal is to do the best I can in capturing everything we humans want—and being able to express it in executable computational terms.</p>
<p>When we look at the computational universe, it’s hard not to be struck by the limitations of what we know how to describe or think about. Modern neural nets provide an interesting example. For the <a href="http://reference.wolfram.com/language/ref/ImageIdentify.html">ImageIdentify</a> function of the Wolfram Language we’ve trained a neural net to identify thousands of kinds of things in the world. And to cater to our human purposes, what the network ultimately does is to describe what it sees in terms of concepts that we can name with words—tables, chairs, elephants, etc.</p>
<p>But internally what the network is doing is to identify a series of features of any object in the world. Is it green? Is it round? And so on. And what happens as the neural network is trained is that it identifies features it finds useful for distinguishing different kinds of things in the world. But the point is that almost none of these features are ones to which we happen to have assigned words in human language.</p>
<p>Out in the computational universe it’s possible to find what may be incredibly useful ways to describe things. But they’re alien to us humans. They’re not something we know how to express, based on the corpus of knowledge our civilization has developed.</p>
<p>Now of course new concepts are being added to the corpus of human knowledge all the time. Back a century ago, if someone <a href="http://www.wolframscience.com/nks/notes-2-3--ornamental-art/">saw a nested pattern</a> they wouldn’t have any way to describe it. But now we’d just say “it’s a fractal”. But the problem is that in the computational universe there’s an infinite collection of “potentially useful concepts”—with which we can never hope to ultimately keep up.</p>
<h2>The Analogy in Mathematics</h2>
<p>When I wrote A New Kind of Science I viewed it in no small part as an effort to break away from the use of mathematics—at least as a foundation for science. But one of the things I realized is that the ideas in the book also have a lot of <a href="http://www.wolframscience.com/nks/p772--implications-for-mathematics-and-its-foundations/">implications for pure mathematics itself</a>.</p>
<p>What is mathematics? Well, it’s a study of certain abstract kinds of systems, based on things like numbers and geometry. In a sense it’s exploring a small corner of the computational universe of all possible abstract systems. But still, plenty has been done in mathematics: indeed, the 3 million or so published theorems of mathematics represent perhaps <a href="https://writings.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/">the largest single coherent intellectual structure</a> that our species has built.</p>
<p>Ever since <a href="http://www.wolframalpha.com/input/?i=euclid">Euclid</a>, people have at least notionally imagined that mathematics starts from certain axioms (say, a+b=b+a, a+0=a, etc.), then builds up derivations of theorems. Why is math hard? The answer is fundamentally rooted in the phenomenon of computational irreducibility—which here is manifest in the fact that there’s no general way to shortcut the series of steps needed to derive a theorem. In other words, it can be arbitrarily hard to get a result in mathematics. But worse than that—as <a href="http://www.wolframalpha.com/input/?i=godel%27s+incompleteness+theorem">Gödel’s Theorem</a> showed—there can be mathematical statements where there just aren’t any finite ways to prove or disprove them from the axioms. And in such cases, the statements just have to be considered “undecidable”.</p>
<p>And in a sense what’s remarkable about math is that one can usefully do it at all. Because it could be that most mathematical results one cares about would be undecidable. So why doesn’t that happen?</p>
<p>Well, if one considers arbitrary abstract systems it happens a lot. Take a typical cellular automaton—or a Turing machine—and ask whether it’s true that the system, say, always settles down to periodic behavior regardless of its initial state. Even something as simple as that will often be undecidable.</p>
<p>So why doesn’t this happen in mathematics? Maybe there’s something special about the particular axioms used in mathematics. And certainly if one thinks they’re the ones that uniquely describe science and the world there might be a reason for that. But one of the whole points of the book is that actually there’s a whole computational universe of possible rules that can be useful for doing science and describing the world.</p>
<p>And in fact I don’t think there’s <a href="http://www.wolframscience.com/nks/p772--implications-for-mathematics-and-its-foundations/">anything abstractly special</a> about the particular axioms that have traditionally been used in mathematics: I think they’re just accidents of history.</p>
<p>What about the theorems that people investigate in mathematics? Again, I think there’s a strong historical character to them. For all but the most trivial areas of mathematics, there’s a whole sea of undecidability out there. But somehow mathematics picks the islands where theorems can actually be proved—often particularly priding itself on places close to the sea of undecidability where the proof can only be done with great effort.</p>
<p>I’ve been interested in the <a href="https://writings.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/">whole network of published theorems in mathematics</a> (it’s a <a href="https://writings.stephenwolfram.com/2010/09/making-the-worlds-data-computable/">thing to curate</a>, like wars in history, or properties of chemicals). And one of the things I’m curious about is whether there’s an inexorable sequence to the mathematics that’s done, or whether, in a sense, random parts are being picked.</p>
<p>And here, I think, there’s a considerable analogy to the kind of thing we were discussing before with language. What is a proof? Basically it’s a way of explaining to someone why something is true. I’ve made all sorts of <a href="http://www.wolframscience.com/nks/p810--implications-for-mathematics-and-its-foundations/">automated proofs</a> in which there are hundreds of steps, each perfectly verifiable by computer. But—like the innards of a neural net—what’s going on looks alien and not understandable by a human.</p>
<p>For a human to understand, there have to be familiar “conceptual waypoints”. It’s pretty much like with words in languages. If some particular part of a proof has a name (“Smith’s Theorem”), and has a known meaning, then it’s useful to us. But if it’s just a lump of undifferentiated computation, it won’t be meaningful to us.</p>
<p>In pretty much any axiom system, there’s an infinite set of possible theorems. But which ones are “interesting”? That’s really a human question. And basically it’s going to end up being ones with “stories”. In the book <a href="http://www.wolframscience.com/nks/p817--implications-for-mathematics-and-its-foundations/">I show that for the simple case of basic logic</a>, the theorems that have historically been considered interesting enough to be given names happen to be precisely the ones that are in some sense minimal.</p>
<p>But my guess is that for richer axiom systems pretty much anything that’s going to be considered “interesting” is going to have to be reached from things that are already considered interesting. It’s like building up words or concepts: you don’t get to introduce new ones unless you can directly relate them to existing ones.</p>
<p>In recent years I’ve wondered quite a bit about how inexorable or not progress is in a field like mathematics. Is there just one historical path that can be taken, say from arithmetic to algebra to the higher reaches of modern mathematics? Or are there an infinite diversity of possible paths, with completely different histories for mathematics?</p>
<p>The answer is going to depend on—in a sense—the “structure of metamathematical space”: just what is the network of true theorems that avoid the sea of undecidability? Maybe it’ll be different for different fields of mathematics, and some will be more “inexorable” (so it feels like the math is being “discovered”) than others (where it seems more like the math is arbitrary, and “invented”).</p>
<p>But to me one of the most interesting things is how close—when viewed in these kinds of terms—questions about the nature and character of mathematics end up being to questions about the nature and character of intelligence and AI. And it’s this kind of commonality that makes me realize just how powerful and general the ideas in A New Kind of Science actually are.</p>
<h2>When Is There a Science?</h2>
<p>There are some areas of science—like physics and astronomy—where the traditional mathematical approach has done quite well. But there are others—like biology, social science and linguistics—where it’s had a lot less to say. And one of the things I’ve long believed is that what’s needed to make progress in these areas is to generalize the kinds of models one’s using, to consider a broader range of what’s out there in the computational universe.</p>
<p>And indeed in the past 15 or so years there’s been increasing success in doing this. And there are lots of biological and social systems, for example, where models have now been constructed using simple programs.</p>
<p>But unlike with mathematical models which can potentially be “solved”, these computational models often show computational irreducibility, and are typically used by doing explicit simulations. This can be perfectly successful for making particular predictions, or for applying the models in technology. But a bit like for the automated proofs of mathematical theorems one might still ask, “is this really science?”.</p>
<p>Yes, one can simulate what a system does, but does one “understand” it? Well, the problem is that computational irreducibility implies that in some fundamental sense one can’t always “understand” things. There might be no useful “story” that can be told; there may be no “conceptual waypoints”—only lots of detailed computation.</p>
<p>Imagine that one’s trying to make a science of how the brain understands language—one of the big goals of linguistics. Well, perhaps we’ll get an adequate model of the precise rules which determine the firing of neurons or some other low-level representation of the brain. And then we look at the patterns generated in understanding some whole collection of sentences.</p>
<p>Well, what if those patterns look like the behavior of rule 30? Or, closer at hand, the innards of some recurrent neural network? Can we “tell a story” about what’s happening? To do so would basically require that we create some kind of higher-level symbolic representation: something where we effectively have words for core elements of what’s going on.</p>
<p>But computational irreducibility implies that there may ultimately be no way to create such a thing. Yes, it will always be possible to find patches of computational reducibility, where some things can be said. But there won’t be a complete story that can be told. And one might say there won’t be a useful reductionistic piece of science to be done. But that’s just one of the things that happens when one’s dealing with (as the title says) a new kind of science.</p>
<h2>Controlling the AIs</h2>
<p>People have gotten very worried about AI in recent years. They wonder what’s going to happen when AIs “get much smarter” than us humans. Well, the <a href="http://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> has one piece of good news: at some fundamental level, AIs will never be “smarter”—they’ll just be able to do computations that are ultimately equivalent to what our brains do, or, for that matter, what all sorts of simple programs do.</p>
<p>As a practical matter, of course, AIs will be able to process larger amounts of data more quickly than actual brains. And no doubt we’ll choose to have them run many aspects of the world for us—from medical devices, to central banks to transportation systems, and much more.</p>
<p>So then it’s important to figure <a href="https://writings.stephenwolfram.com/2015/11/how-should-we-talk-to-ais/">how we’ll tell them what to do</a>. As soon as we’re making serious use of what’s out there in the computational universe, we’re not going to be able to give a line-by-line description of what the AIs are going to do. Rather, we’re going to have to define goals for the AIs, then let them figure out how best to achieve those goals.</p>
<p>In a sense we’ve already been doing something like this for years in the <a href="http://www.wolfram.com/language/">Wolfram Language</a>. There’s some high-level function that describes something you want to do (“<a href="http://reference.wolfram.com/language/ref/Graph.html">lay out a graph</a>”, “<a href="http://reference.wolfram.com/language/ref/Classify.html">classify data</a>”, etc.). Then it’s up to the language to automatically figure out the best way to do it.</p>
<p>And in the end the real challenge is to find a way to describe goals. Yes, you want to search for cellular automata that will make a “nice carpet pattern”, or a “good edge detector”. But what exactly do those things mean? What you need is a language that a human can use to say as precisely as possible what they mean.</p>
<p>It’s really the same problem as I’ve been talking about a lot here. One has to have a way for humans to be able to talk about things they care about. There’s infinite detail out there in the computational universe. But through our civilization and our shared cultural history we’ve come to identify certain concepts that are important to us. And when we describe our goals, it’s in terms of these concepts.</p>
<p>Three hundred years ago people like <a href="https://writings.stephenwolfram.com/2013/05/dropping-in-on-gottfried-leibniz/">Leibniz</a> were interested in finding a precise symbolic way to represent the content of human thoughts and human discourse. He was far too early. But now I <a href="https://writings.stephenwolfram.com/2016/10/computational-law-symbolic-discourse-and-the-ai-constitution/">think we’re finally in a position</a> to actually make this work. In fact, we’ve already gotten a long way with the <a href="https://www.wolfram.com/language/">Wolfram Language</a> in being able to describe real things in the world. And I’m hoping it’ll be possible to construct a fairly complete “<a href="https://writings.stephenwolfram.com/2016/10/computational-law-symbolic-discourse-and-the-ai-constitution/">symbolic discourse language</a>” that lets us talk about the things we care about.</p>
<p>Right now we write legal contracts in “legalese” as a way to make them slightly more precise than ordinary natural language. But with a symbolic discourse language we’ll be able to write true “computational contracts” that describe in high-level terms what we want to have happen—and then machines will automatically be able to verify or execute the contract.</p>
<p>But what about the AIs? Well, we need to tell them what we generally want them to do. We need to have a contract with them. Or maybe we need to have a <a href="https://writings.stephenwolfram.com/2016/10/computational-law-symbolic-discourse-and-the-ai-constitution/">constitution for them</a>. And it’ll be written in some kind of symbolic discourse language, that both allows us humans to express what we want, and is executable by the AIs.</p>
<p>There’s lots to say about what should be in an AI Constitution, and how the construction of such things might map onto the political and cultural landscape of the world. But one of the obvious questions is: can the constitution be simple, like <a href="http://www.wolframalpha.com/input/?i=three+laws+of+robotics">Asimov’s Laws of Robotics</a>?</p>
<p>And here what we know from A New Kind of Science tells us the answer: it can’t be. In a sense the constitution is an attempt to sculpt what can happen in the world and what can’t. But computational irreducibility says that there will be an unbounded collection of cases to consider. </p>
<p>For me it’s interesting to see how theoretical ideas like computational irreducibility end up impinging on these very practical—and central—societal issues. Yes, it all started with questions about things like the theory of all possible theories. But in the end it turns into issues that everyone in society is going to end up being concerned about.</p>
<h2>There’s an Endless Frontier</h2>
<p>Will we reach the end of science? Will we—or our AIs—eventually invent everything there is to be invented?</p>
<p>For mathematics, it’s easy to see that there’s an infinite number of possible theorems one can construct. For science, there’s an infinite number of possible detailed questions to ask. And there’s also an infinite array of possible inventions one can construct.</p>
<p>But the real question is: will there always be interesting new things out there?</p>
<p>Well, computational irreducibility says there will always be new things that need an irreducible amount of computational work to reach from what’s already there. So in a sense there’ll always be “surprises”, that aren’t immediately evident from what’s come before.</p>
<p>But will it just be like an endless array of different weirdly shaped rocks? Or will there be fundamental new features that appear, that we humans consider interesting?</p>
<p>It’s back to the very same issue we’ve encountered several times before: for us humans to find things “interesting” we have to have a conceptual framework that we can use to think about them. Yes, we can identify a “<a href="http://www.wolframscience.com/nks/p281--structures-in-class-4-systems/">persistent structure</a>” in a cellular automaton. Then maybe we can start talking about “collisions between structures”. But when we just see a whole mess of stuff going on, it’s not going to be “interesting” to us unless we have some higher-level symbolic way to talk about it.</p>
<p>In a sense, then, the rate of “interesting discovery” isn’t going to be limited by our ability to go out into the computational universe and find things. Instead, it’s going to be limited by our ability as humans to build a conceptual framework for what we’re finding.</p>
<p>It’s a bit like what happened in the whole development of what became A New Kind of Science. People had seen <a href="http://www.wolframscience.com/nks/p42--why-these-discoveries-were-not-made-before/">related phenomena for centuries if not millennia</a> (distribution of primes, digits of pi, etc.). But without a conceptual framework they just didn’t seem “interesting”, and nothing was built around them. And indeed as I understand more about what’s out there in the computational universe—and even about things I saw long ago there—I gradually build up a conceptual framework that lets me go further.</p>
<p>By the way, it’s worth realizing that inventions work a little differently from discoveries. One can see something new happen in the computational universe, and that might be a discovery. But an invention is about figuring out how something can be achieved in the computational universe.</p>
<p>And—like in patent law—it isn’t really an invention if you just say “look, this does that”. You have to somehow understand a purpose that it’s achieving.</p>
<p>In the past, the focus of the process of invention has tended to be on actually getting something to work (“find the lightbulb filament that works”, etc.). But in the computational universe, the focus shifts to the question of what you want the invention to do. Because once you’ve described the goal, finding a way to achieve it is something that can be automated.</p>
<p>That’s not to say that it will always be easy. In fact, computational irreducibility implies that it can be arbitrarily difficult. Let’s say you know the precise rules by which some chemicals can interact. Can you find a chemical synthesis pathway that will let you get to some particular chemical structure? There may be a way, but computational irreducibility implies that there may be no way to find out how long the pathway may be. And if you haven’t found a pathway you may never be sure if it’s because there isn’t one, or just because you didn’t reach it yet.</p>
<h2>The Fundamental Theory of Physics</h2>
<p>If one thinks about reaching the edge of science, one cannot help but wonder about the <a href="http://www.wolframscience.com/nks/chap-9--fundamental-physics/">fundamental theory of physics</a>. Given everything we’ve seen in the computational universe, is it conceivable that our physical universe could just correspond to one of those programs out there in the computational universe?</p>
<p>Of course, we won’t really know until or unless we find it. But in the years since A New Kind of Science appeared, I’ve become ever more optimistic about the possibilities.</p>
<p>Needless to say, it would be a big change for physics. Today there are basically two major frameworks for thinking about fundamental physics: <a href="http://www.wolframalpha.com/input/?i=general+relativity">general relativity</a> and <a href="http://www.wolframalpha.com/input/?i=quantum+field+theory">quantum field theory</a>. General relativity is a bit more than 100 years old; quantum field theory maybe 90. And both have achieved spectacular things. But neither has succeeded in delivering us a complete fundamental theory of physics. And if nothing else, I think after all this time, it’s worth trying something new.</p>
<p>But there’s another thing: from actually exploring the computational universe, we have a huge amount of new intuition about what’s possible, even in very simple models. We might have thought that the kind of richness we know exists in physics would require some very elaborate underlying model. But what’s become clear is that that kind of richness can perfectly well emerge even from a very simple underlying model.</p>
<p><a href="https://writings.stephenwolfram.com/2015/12/what-is-spacetime-really/">What might the underlying model be like</a>? I’m not going to discuss this in great detail here, but suffice it to say that I think the most important thing about the model is that it should have as little as possible built in. We shouldn’t have the hubris to think we know how the universe is constructed; we should just take a general type of model that’s as unstructured as possible, and do what we typically do in the computational universe: just search for a program that does what we want.</p>
<p>My favorite formulation for a model that’s as unstructured as possible is a <a href="https://writings.stephenwolfram.com/2015/12/what-is-spacetime-really/">network</a>: just a collection of nodes with connections between them. It’s perfectly possible to formulate such a model as an algebraic-like structure, and probably many other kinds of things. But we can think of it as a network. And in the way I’ve imagined setting it up, it’s a network that’s somehow “underneath” space and time: every aspect of space and time as we know it must emerge from the actual behavior of the network.</p>
<p>Over the past decade or so there’s been increasing interest in things like loop quantum gravity and spin networks. They’re related to what I’ve been doing in the same way that they also involve networks. And maybe there’s some deeper relationship. But in their usual formulation, they’re much more mathematically elaborate.</p>
<p>From the point of view of the traditional methods of physics, this might seem like a good idea. But with the intuition we have from studying the computational universe—and using it for science and technology—it seems completely unnecessary. Yes, we don’t yet know the fundamental theory of physics. But it seems sensible to start with the simplest hypothesis. And that’s definitely something like a simple network of the kind I’ve studied.</p>
<p>At the outset, it’ll look pretty alien to people (including myself) trained in traditional theoretical physics. But some of what emerges isn’t so alien. <a href="http://www.wolframscience.com/nks/p530--the-phenomenon-of-gravity/">A big result</a> I found nearly 20 years ago (that still hasn’t been widely understood) is that when you look at a large enough network of the kind I studied you can show that its averaged behavior follows Einstein’s equations for gravity. In other words, without putting any fancy physics into the underlying model, it ends up automatically emerging. I think it’s pretty exciting.</p>
<p>People ask a lot about <a href="http://www.wolframscience.com/nks/p537--quantum-phenomena/">quantum mechanics</a>. Yes, my underlying model doesn’t build in quantum mechanics (just as it doesn’t build in general relativity). Now, it’s a little difficult to pin down exactly what the essence of “being quantum mechanical” actually is. But there are some very suggestive signs that my simple networks actually end up showing what amounts to quantum behavior—just like in the physics we know.</p>
<p>OK, so how should one set about actually finding the fundamental theory of physics if it’s out there in the computational universe of possible programs? Well, the obvious thing is to just start searching for it, starting with the simplest programs.</p>
<p><a href="https://writings.stephenwolfram.com/2007/09/my-hobby-hunting-for-our-universe/">I’ve been doing this</a>—more sporadically than I would like—for the past 15 years or so. And my main discovery so far is that it’s actually quite easy to find programs that aren’t obviously not our universe. There are plenty of programs where space or time are obviously completely different from the way they are in our universe, or there’s some other pathology. But it turns out it’s not so difficult to find candidate universes that aren’t obviously not our universe.</p>
<p>But we’re immediately bitten by computational irreducibility. We can simulate the candidate universe for billions of steps. But we don’t know what it’s going to do—and whether it’s going to grow up to be like our universe, or completely different.</p>
<p>It’s pretty unlikely that in looking at that tiny fragment of the very beginning of a universe we’re going to ever be able to see anything familiar, like a photon. And it’s not at all obvious that we’ll be able to construct any kind of descriptive theory, or effective physics. But in a sense the problem is bizarrely similar to the one we have even in systems like neural networks: there’s computation going on there, but can we identify “conceptual waypoints” from which we can build up a theory that we might understand?</p>
<p>It’s not at all clear our universe has to be understandable at that level, and it’s quite possible that for a very long time we’ll be left in the strange situation of thinking we might have “found our universe” out in the computational universe, but not being sure.</p>
<p>Of course, we might be lucky, and it might be possible to deduce an effective physics, and see that some little program that we found ends up reproducing our whole universe. It would be a remarkable moment for science. But it would immediately raise a host of new questions—like why this universe, and not another?</p>
<h2>Box of a Trillion Souls</h2>
<p>Right now us humans exist as biological systems. But in the future it’s certainly going to be technologically possible to reproduce all the processes in our brains in some purely digital—computational—form. So insofar as those processes represent “us”, we’re going to be able to be “virtualized” on pretty much any computational substrate. And in this case we might imagine that the whole future of a civilization could wind up in effect as a “<a href="https://www.edge.org/conversation/stephen_wolfram-the-ark-of-ai">box of a trillion souls</a>”.</p>
<p>Inside that box there would be all kinds of computations going on, representing the thoughts and experiences of all those disembodied souls. Those computations would reflect the rich history of our civilization, and all the things that have happened to us. But at some level they wouldn’t be anything special.</p>
<p>It’s perhaps a bit disappointing, but the <a href="http://www.wolframscience.com/nks/chap-12--the-principle-of-computational-equivalence/">Principle of Computational Equivalence</a> tells us that ultimately these computations will be no more sophisticated than the ones that go on in all sorts of other systems—even ones with simple rules, and no elaborate history of civilization. Yes, the details will reflect all that history. But in a sense without knowing what to look for—or what to care about—one won’t be able to tell that there’s anything special about it.</p>
<p>OK, but what about for the “souls” themselves? Will one be able to understand their behavior by seeing that they achieve certain purposes? Well, in our current biological existence, we have all sorts of constraints and features that give us goals and purposes. But in a virtualized “uploaded” form, most of these just go away.</p>
<p>I’ve thought quite a bit about how “human” purposes might evolve in such a situation, recognizing, of course, that in virtualized form there’s little difference between human and AI. The disappointing vision is that perhaps the future of our civilization consists in disembodied souls in effect “playing videogames” for the rest of eternity.</p>
<p>But what I’ve slowly realized is that it’s actually quite unrealistic to project our view of goals and purposes from our experience today into that future situation. Imagine talking to someone from a thousand years ago and trying to explain that people in the future would be walking on treadmills every day, or continually sending photographs to their friends. The point is that such activities don’t make sense until the cultural framework around them has developed.</p>
<p>It’s the same story yet again as with trying to characterize what’s interesting or what’s explainable. It relies on the development of a whole network of conceptual waypoints.</p>
<p>Can we imagine what the mathematics of 100 years from now will be like? It depends on concepts we don’t yet know. So similarly if we try to imagine human motivation in the future, it’s going to rely on concepts we don’t know. Our best description from today’s viewpoint might be that those disembodied souls are just “playing videogames”. But to them there might be a whole subtle motivation structure that they could only explain by rewinding all sorts of steps in history and cultural development.</p>
<p>By the way, if we know the fundamental theory of physics then in a sense we can make the virtualization complete, at least in principle: we can just run a simulation of the universe for those disembodied souls. Of course, if that’s what’s happening, then there’s no particular reason it has to be a simulation of our particular universe. It could as well be any universe from out in the computational universe.</p>
<p>Now, as I’ve mentioned, even in any given universe one will never in a sense run out of things to do, or discover. But I suppose I myself at least find it amusing to imagine that at some point those disembodied souls might get bored with just being in a simulated version of our physical universe—and might decide it’s more fun (whatever that means to them) to go out and explore the broader computational universe. Which would mean that in a sense the future of humanity would be an infinite voyage of discovery in the context of none other than A New Kind of Science!</p>
<h2>The Economics of the Computational Universe</h2>
<p>Long before we have to think about disembodied human souls, we’ll have to confront the issue of what humans should be doing in a world where more and more can be done automatically by AIs. Now in a sense this issue is nothing new: it’s just an extension of the long-running story of technology and automation. But somehow this time it feels different.</p>
<p>And I think the reason is in a sense just that there’s so much out there in the computational universe, that’s so easy to get to. Yes, we can build a machine that automates some particular task. We can even have a general-purpose computer that can be programmed to do a full range of different tasks. But even though these kinds of automation extend what we can do, it still feels like there’s effort that we have to put into them.</p>
<p>But the picture now is different—because in effect what we’re saying is that if we can just define the goal we want to achieve, then everything else will be automatic. All sorts of computation, and, yes, “thinking”, may have to be done, but the idea is that it’s just going to happen, without human effort.</p>
<p>At first, something seems wrong. How could we get all that benefit, without putting in more effort? It’s a bit like asking how nature could manage to make all the complexity it does—even though when we build artifacts, even with great effort, they end up far less complex. The answer, I think, is it’s mining the computational universe. And it’s exactly the same thing for us: by mining the computational universe, we can achieve essentially an unbounded level of automation.</p>
<p>If we look at the important resources in today’s world, many of them still depend on actual materials. And often these materials are literally mined from the Earth. Of course, there are accidents of geography and geology that determine by whom and where that mining can be done. And in the end there’s a limit (if often very large) to the amount of material that’ll ever be available.</p>
<p>But when it comes to the computational universe, there’s in a sense an inexhaustible supply of material—and it’s accessible to anyone. Yes, there are technical issues about how to “do the mining”, and there’s a whole stack of technology associated with doing it well. But the ultimate resource of the computational universe is a global and infinite one. There’s no scarcity and no reason to be “expensive”. One just has to understand that it’s there, and take advantage of it.</p>
<h2>The Path to Computational Thinking</h2>
<p>Probably the greatest intellectual shift of the past century has been the one towards the computational way of thinking about things. I’ve often said that if one picks almost any field “X”, from archaeology to zoology, then by now there either is, or soon will be, a field called “computational X”—and it’s going to be the future of the field.</p>
<p>I myself have been deeply involved in trying to enable such computational fields, in particular through the development of the Wolfram Language. But I’ve also been interested in what is essentially the meta problem: how should one <a href="https://writings.stephenwolfram.com/2016/09/how-to-teach-computational-thinking/">teach abstract computational thinking</a>, for example to kids? The Wolfram Language is certainly important as a practical tool. But what about the conceptual, theoretical foundations?</p>
<p>Well, that’s where A New Kind of Science comes in. Because at its core it’s discussing the pure abstract phenomenon of computation, independent of its applications to particular fields or tasks. It’s a bit like with elementary mathematics: there are things to teach and understand just to introduce the ideas of mathematical thinking, independent of their specific applications. And so it is too with the core of A New Kind of Science. There are things to learn about the computational universe that give intuition and introduce patterns of computational thinking—quite independent of detailed applications.</p>
<p>One can think of it as a kind of “pre computer science” , or “pre computational X”. Before one gets into discussing the specifics of particular computational processes, one can just study the simple but pure things one finds in the computational universe.</p>
<p>And, yes, even before kids learn to do arithmetic, it’s perfectly possible for them to fill out something like a cellular automaton coloring book—or to execute for themselves or on a computer a whole range of different simple programs. What does it teach? Well, it certainly teaches the idea that there can be definite rules or algorithms for things—and that if one follows them one can create useful and interesting results. And, yes, it helps that systems like cellular automata make obvious visual patterns, that for example one can even find in nature (say on mollusc shells).</p>
<p>As the world becomes more computational—and more things are done by AIs and by mining the computational universe—there’s going to an extremely high value not only in understanding computational thinking, but also in having the kind of intuition that develops from exploring the computational universe and that is, in a sense, the foundation for <em>A New Kind of Science</em>.</p>
<h2>What’s Left to Figure Out?</h2>
<p>My goal over the decade that I spent writing <em>A New Kind of Science</em> was, as much as possible, to answer all the first round of “obvious questions” about the computational universe. And looking back 15 years later I think that worked out pretty well. Indeed, today, when I wonder about something to do with the computational universe, I find it’s incredibly likely that somewhere in the main text or notes of the book I already said something about it.</p>
<p>But one of the biggest things that’s changed over the past 15 years is that I’ve gradually begun to understand more of the implications of what the book describes. There are lots of specific ideas and discoveries in the book. But in the longer term I think what’s most significant is how they serve as foundations, both practical and conceptual, for a whole range of new things that one can now understand and explore.</p>
<p>But even in terms of the basic science of the computational universe, there are certainly specific results one would still like to get. For example, it would be great to get more evidence for or against the Principle of Computational Equivalence, and its domain of applicability.</p>
<p>Like most general principles in science, the whole <a href="http://www.wolframscience.com/nks/p719--the-content-of-the-principle/">epistemological status of the Principle of Computational Equivalence</a> is somewhat complicated. Is it like a mathematical theorem that can be proved? Is it like a law of nature that might (or might not) be true about the universe? Or is it like a definition, say of the very concept of computation? Well, much like, say, the Second Law of Thermodynamics or Evolution by Natural Selection, it’s a combination of these.</p>
<p>But one thing that’s significant is that it’s possible to get concrete evidence for (or against) the Principle of Computational Equivalence. The principle says that even systems with very simple rules should be capable of arbitrarily sophisticated computation—so that in particular they should be able to act as universal computers.</p>
<p>And indeed one of the results of the book is that this is <a href="http://www.wolframscience.com/nks/p675--the-rule-110-cellular-automaton/">true for one of the simplest possible cellular automata (rule 110)</a>. Five years after the book was published I decided to put up a prize for evidence about another case: the <a href="http://www.wolframscience.com/prizes/tm23/">simplest conceivably universal Turing machine</a>. And I was very pleased that in just a few months the prize was won, the Turing machine was proved universal, and there was another piece of evidence for the Principle of Computational Equivalence.</p>
<p>There’s a lot to do in developing the applications of <em>A New Kind of Science</em>. There are models to be made of all sorts of systems. There’s technology to be found. Art to be created. There’s also a lot to do in understanding the implications.</p>
<p>But it’s important not to forget the pure investigation of the computational universe. In the analogy of mathematics, there are applications to be pursued. But there’s also a “pure mathematics” that’s worth pursuing in its own right. And so it is with the computational universe: there’s a huge amount to explore just at an abstract level. And indeed (as the title of the book implies) there’s enough to define a whole new kind of science: a pure science of the computational universe. And it’s the opening of that new kind of science that I think is the core achievement of <em>A New Kind of Science</em>—and the one of which I am most proud.</p>
<hr />
<p>For the 10th anniversary of <em>A New Kind of Science</em>, I wrote three posts:</p>
<ul>
<li><a href="https://writings.stephenwolfram.com/2012/05/its-been-10-years-whats-happened-with-a-new-kind-of-science/">It’s Been 10 Years: What’s Happened with <em>A New Kind of Science</em>?</a></li>
<li><a href="https://writings.stephenwolfram.com/2012/05/living-a-paradigm-shift-looking-back-on-reactions-to-a-new-kind-of-science/">Living a Paradigm Shift: Looking Back on Reactions to <em>A New Kind of Science</em></a></li>
<li><a href="https://writings.stephenwolfram.com/2012/05/looking-to-the-future-of-a-new-kind-of-science/">Looking to the Future of <em>A New Kind of Science</em></a></li>
</ul>
<p><em>The complete high-resolution <em>A New Kind of Science</em> is <a href="http://www.wolframscience.com/nks/">now available on the web</a>. There are also a limited number of print copies of the <a href="http://store.wolfram.com/arrive.cgi?URI=https://store.wolfram.com/view/book/ISBN1579550088.str">book still available</a> (all individually coded!).</em></p>
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 Stephen Wolfram

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 Contents
 * Top
 * Implications of the Computational Universe
 * A Shift in Looking at the World
 * Mining the Computational Universe
 * A World of Discovered Technology
 * Machine Learning and the Neural Net Renaissance
 * NKS Meets Modern Machine Learning
 * Finding AI
 * The Importance of Language
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 * Controlling the AIs
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 * What’s Left to Figure Out?
 A New Kind of Science : A 15-Year View

 A New Kind of Science: A 15-Year View

 May 16, 2017

 Starting now, in celebration of its 15th anniversary, A New Kind of Science will be freely available in its entirety, with high-resolution images, on the web or for download.

 It’s now 15 years since I published my book A New Kind of Science —more than 25 since I started writing it , and more than 35 since I started working towards it . But with every passing year I feel I understand more about what the book is really about—and why it’s important. I wrote the book, as its title suggests, to contribute to the progress of science. But as the years have gone by, I’ve realized that the core of what’s in the book actually goes far beyond science—into many areas that will be increasingly important in defining our whole future.

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 So, viewed from a distance of 15 years, what is the book really about? At its core, it’s about something profoundly abstract: the theory of all possible theories, or the universe of all possible universes. But for me one of the achievements of the book is the realization that one can explore such fundamental things concretely—by doing actual experiments in the computational universe of possible programs. And in the end the book is full of what might at first seem like quite alien pictures made just by running very simple such programs.

 Back in 1980, when I made my living as a theoretical physicist , if you’d asked me what I thought simple programs would do, I expect I would have said “not much”. I had been very interested in the kind of complexity one sees in nature, but I thought—like a typical reductionistic scientist—that the key to understanding it must lie in figuring out detailed features of the underlying component parts.

 In retrospect I consider it incredibly lucky that all those years ago I happened to have the right interests and the right skills to actually try what is in a sense the most basic experiment in the computational universe: to systematically take a sequence of the simplest possible programs, and run them.

 I could tell as soon as I did this that there were interesting things going on, but it took a couple more years before I began to really appreciate the force of what I’d seen. For me it all started with one picture:

 Or, in modern form:

 I call it rule 30 . It’s my all-time favorite discovery, and today I carry it around everywhere on my business cards . What is it? It’s one of the simplest programs one can imagine . It operates on rows of black and white cells, starting from a single black cell, and then repeatedly applies the rules at the bottom. And the crucial point is that even though those rules are by any measure extremely simple, the pattern that emerges is not.

 It’s a crucial—and utterly unexpected—feature of the computational universe: that even among the very simplest programs, it’s easy to get immensely complex behavior. It took me a solid decade to understand just how broad this phenomenon is. It doesn’t just happen in programs (“ cellular automata ”) like rule 30. It basically shows up whenever you start enumerating possible rules or possible programs whose behavior isn’t obviously trivial.

 Similar phenomena had actually been seen for centuries in things like the digits of pi and the distribution of primes —but they were basically just viewed as curiosities, and not as signs of something profoundly important. It’s been nearly 35 years since I first saw what happens in rule 30, and with every passing year I feel I come to understand more clearly and deeply what its significance is.

 Four centuries ago it was the discovery of the moons of Jupiter and their regularities that sowed the seeds for modern exact science, and for the modern scientific approach to thinking. Could my little rule 30 now be the seed for another such intellectual revolution, and a new way of thinking about everything?

 In some ways I might personally prefer not to take responsibility for shepherding such ideas ( “paradigm shifts” are hard and thankless work ). And certainly for years I have just quietly used such ideas to develop technology and my own thinking. But as computation and AI become increasingly central to our world, I think it’s important that the implications of what’s out there in the computational universe be more widely understood.

 Implications of the Computational Universe

 Here’s the way I see it today. From observing the moons of Jupiter we came away with the idea that—if looked at right—the universe is an ordered and regular place, that we can ultimately understand. But now, in exploring the computational universe, we quickly come upon things like rule 30 where even the simplest rules seem to lead to irreducibly complex behavior.

 One of the big ideas of A New Kind of Science is what I call the Principle of Computational Equivalence . The first step is to think of every process—whether it’s happening with black and white squares, or in physics, or inside our brains—as a computation that somehow transforms input to output. What the Principle of Computational Equivalence says is that above an extremely low threshold, all processes correspond to computations of equivalent sophistication.

 It might not be true. It might be that something like rule 30 corresponds to a fundamentally simpler computation than the fluid dynamics of a hurricane, or the processes in my brain as I write this. But what the Principle of Computational Equivalence says is that in fact all these things are computationally equivalent.

 It’s a very important statement, with many deep implications. For one thing, it implies what I call computational irreducibility . If something like rule 30 is doing a computation just as sophisticated as our brains or our mathematics, then there’s no way we can “outrun” it: to figure out what it will do, we have to do an irreducible amount of computation, effectively tracing each of its steps.

 The mathematical tradition in exact science has emphasized the idea of predicting the behavior of systems by doing things like solving mathematical equations. But what computational irreducibility implies is that out in the computational universe that often won’t work, and instead the only way forward is just to explicitly run a computation to simulate the behavior of the system.

 A Shift in Looking at the World

 One of the things I did in A New Kind of Science was to show how simple programs can serve as models for the essential features of all sorts of physical, biological and other systems . Back when the book appeared, some people were skeptical about this . And indeed at that time there was a 300-year unbroken tradition that serious models in science should be based on mathematical equations.

 But in the past 15 years something remarkable has happened . For now, when new models are created—whether of animal patterns or web browsing behavior—they are overwhelmingly more often based on programs than on mathematical equations.

 Year by year, it’s been a slow, almost silent, process. But by this point, it’s a dramatic shift. Three centuries ago pure philosophical reasoning was supplanted by mathematical equations. Now in these few short years, equations have been largely supplanted by programs. For now, it’s mostly been something practical and pragmatic: the models work better, and are more useful.

 But when it comes to understanding the foundations of what’s going on, one’s led not to things like mathematical theorems and calculus, but instead to ideas like the Principle of Computational Equivalence . Traditional mathematics-based ways of thinking have made concepts like force and momentum ubiquitous in the way we talk about the world. But now as we think in fundamentally computational terms we have to start talking in terms of concepts like undecidability and computational irreducibility .

 Will some type of tumor always stop growing in some particular model ? It might be undecidable. Is there a way to work out how a weather system will develop? It might be computationally irreducible.

 These concepts are pretty important when it comes to understanding not only what can and cannot be modeled, but also what can and cannot be controlled in the world. Computational irreducibility in economics is going to limit what can be globally controlled. Computational irreducibility in biology is going to limit how generally effective therapies can be—and make highly personalized medicine a fundamental necessity.

 And through ideas like the Principle of Computational Equivalence we can start to discuss just what it is that allows nature—seemingly so effortlessly—to generate so much that seems so complex to us. Or how even deterministic underlying rules can lead to computationally irreducible behavior that for all practical purposes can seem to show “ free will ”.

 Mining the Computational Universe

 A central lesson of A New Kind of Science is that there’s a lot of incredible richness out there in the computational universe. And one reason that’s important is that it means that there’s a lot of incredible stuff out there for us to “mine” and harness for our purposes.

 Want to automatically make an interesting custom piece of art? Just start looking at simple programs and automatically pick out one you like —as in our WolframTones music site from a decade ago. Want to find an optimal algorithm for something? Just search enough programs out there, and you’ll find one.

 We’ve normally been used to creating things by building them up, step by step, with human effort—progressively creating architectural plans, or engineering drawings, or lines of code. But the discovery that there’s so much richness so easily accessible in the computational universe suggests a different approach: don’t try building anything; just define what you want, and then search for it in the computational universe.

 Sometimes it’s really easy to find. Like let’s say you want to generate apparent randomness. Well, then just enumerate cellular automata (as I did in 1984), and very quickly you come upon rule 30—which turns out to be one of the very best known generators of apparent randomness (look down the center column of cell values, for examples). In other situations you might have to search 100,000 cases (as I did in finding the simplest axiom system for logic , or the simplest universal Turing machine ), or you might have to search millions or even trillions of cases. But in the past 25 years, we’ve had incredible success in just discovering algorithms out there in the computational universe—and we rely on many of them in implementing the Wolfram Language .

 At some level it’s quite sobering. One finds some tiny program out in the computational universe. One can tell it does what one wants. But when one looks at what it’s doing, one doesn’t have any real idea how it works. Maybe one can analyze some part—and be struck by how “clever” it is. But there just isn’t a way for us to understand the whole thing; it’s not something familiar from our usual patterns of thinking.

 Of course, we’ve often had similar experiences before—when we use things from nature. We may notice that some particular substance is a useful drug or a great chemical catalyst, but we may have no idea why. But in doing engineering and in most of our modern efforts to build technology, the great emphasis has instead been on constructing things whose design and operation we can readily understand.

 In the past we might have thought that was enough. But what our explorations of the computational universe show is that it’s not: selecting only things whose operation we can readily understand misses most of the immense power and richness that’s out there in the computational universe.

 A World of Discovered Technology

 What will the world look like when more of what we have is mined from the computational universe? Today the environment we build for ourselves is dominated by things like simple shapes and repetitive processes. But the more we use what’s out there in the computational universe, the less regular things will look. Sometimes they may look a bit “organic”, or like what we see in nature (since after all, nature follows similar kinds of rules). But sometimes they may look quite random, until perhaps suddenly and incomprehensibly they achieve something we recognize.

 For several millennia we as a civilization have been on a path to understand more about what happens in our world—whether by using science to decode nature, or by creating our own environment through technology. But to use more of the richness of the computational universe we must at least to some extent forsake this path.

 In the past, we somehow counted on the idea that between our brains and the tools we could create we would always have fundamentally greater computational power than the things around us—and as a result we would always be able to “understand” them. But what the Principle of Computational Equivalence says is that this isn’t true: out in the computational universe there are lots of things just as powerful as our brains or the tools we build. And as soon as we start using those things, we lose the “edge” we thought we had.

 Today we still imagine we can identify discrete “bugs” in programs. But most of what’s powerful out there in the computational universe is rife with computational irreducibility—so the only real way to see what it does is just to run it and watch what happens.

 We ourselves, as biological systems, are a great example of computation happening at a molecular scale—and we are no doubt rife with computational irreducibility (which is, at some fundamental level, why medicine is hard). I suppose it’s a tradeoff: we could limit our technology to consist only of things whose operation we understand. But then we would miss all that richness that’s out there in the computational universe. And we wouldn’t even be able to match the achievements of our own biology in the technology we create.

 Machine Learning and the Neural Net Renaissance

 There’s a common pattern I’ve noticed with intellectual fields. They go for decades and perhaps centuries with only incremental growth, and then suddenly, usually as a result of a methodological advance, there’s a burst of “hypergrowth” for perhaps 5 years, in which important new results arrive almost every week.

 I was fortunate enough that my own very first field—particle physics—was in its period of hypergrowth right when I was involved in the late 1970s. And for myself, the 1990s felt like a kind of personal period of hypergrowth for what became A New Kind of Science —and indeed that’s why I couldn’t pull myself away from it for more than a decade.

 But today, the obvious field in hypergrowth is machine learning , or, more specifically, neural nets . It’s funny for me to see this. I actually worked on neural nets back in 1981 , before I started on cellular automata, and several years before I found rule 30. But I never managed to get neural nets to do anything very interesting—and actually I found them too messy and complicated for the fundamental questions I was concerned with.

 And so I “ simplified them ”—and wound up with cellular automata. (I was also inspired by things like the Ising model in statistical physics, etc.) At the outset , I thought I might have simplified too far, and that my little cellular automata would never do anything interesting. But then I found things like rule 30. And I’ve been trying to understand its implications ever since.

 In building Mathematica and the Wolfram Language , I’d always kept track of neural nets, and occasionally we’d use them in some small way for some algorithm or another. But about 5 years ago I suddenly started hearing amazing things: that somehow the idea of training neural nets to do sophisticated things was actually working. At first I wasn’t sure. But then we started building neural net capabilities in the Wolfram Language, and finally two years ago we released our ImageIdentify.com website—and now we’ve got our whole symbolic neural net system . And, yes, I’m impressed. There are lots of tasks that had traditionally been viewed as the unique domain of humans, but which now we can routinely do by computer.

 But what’s actually going on in a neural net ? It’s not really to do with the brain; that was just the inspiration (though in reality the brain probably works more or less the same way). A neural net is really a sequence of functions that operate on arrays of numbers, with each function typically taking quite a few inputs from around the array. It’s not so different from a cellular automaton. Except that in a cellular automaton, one’s usually dealing with, say, just 0s and 1s, not arbitrary numbers like 0.735. And instead of taking inputs from all over the place, in a cellular automaton each step takes inputs only from a very well-defined local region.

 Now, to be fair, it’s pretty common to study “ convolutional neural nets ”, in which the patterns of inputs are very regular, just like in a cellular automaton. And it’s becoming clear that having precise (say 32-bit) numbers isn’t critical to the operation of neural nets; one can probably make do with just a few bits.

 But a big feature of neural nets is that we know how to make them “learn”. In particular, they have enough features from traditional mathematics (like involving continuous numbers) that techniques like calculus can be applied to provide strategies to make them incrementally change their parameters to “fit their behavior” to whatever training examples they’re given.

 It’s far from obvious how much computational effort, or how many training examples, will be needed. But the breakthrough of about five years ago was the discovery that for many important practical problems, what’s available with modern GPUs and modern web-collected training sets can be enough.

 Pretty much nobody ends up explicitly setting or “engineering” the parameters in a neural net. Instead, what happens is that they’re found automatically. But unlike with simple programs like cellular automata, where one’s typically enumerating all possibilities, in current neural nets there’s an incremental process , essentially based on calculus, that manages to progressively improve the net—a little like the way biological evolution progressively improves the “fitness” of an organism.

 It’s plenty remarkable what comes out from training a neural net in this way, and it’s plenty difficult to understand how the neural net does what it does. But in some sense the neural net isn’t venturing too far across the computational universe: it’s always basically keeping the same basic computational structure, and just changing its behavior by changing parameters.

 But to me the success of today’s neural nets is a spectacular endorsement of the power of the computational universe, and another validation of the ideas of A New Kind of Science . Because it shows that out in the computational universe, away from the constraints of explicitly building systems whose detailed behavior one can foresee, there are immediately all sorts of rich and useful things to be found.

 NKS Meets Modern Machine Learning

 Is there a way to bring the full power of the computational universe—and the ideas of A New Kind of Science —to the kinds of things one does with neural nets? I suspect so. And in fact, as the details become clear, I wouldn’t be surprised if exploration of the computational universe saw its own period of hypergrowth: a “mining boom” of perhaps unprecedented proportions.

 In current work on neural nets, there’s a definite tradeoff one sees. The more what’s going on inside the neural net is like a simple mathematical function with essentially arithmetic parameters, the easier it is to use ideas from calculus to train the network. But the more what’s going is like a discrete program, or like a computation whose whole structure can change, the more difficult it is to train the network.

 It’s worth remembering, though, that the networks we’re routinely training now would have looked utterly impractical to train only a few years ago. It’s effectively just all those quadrillions of GPU operations that we can throw at the problem that makes training feasible. And I won’t be surprised if even quite pedestrian (say, local exhaustive search) techniques will fairly soon let one do significant training even in cases where no incremental numerical approach is possible. And perhaps even it will be possible to invent some major generalization of things like calculus that will operate in the full computational universe. (I have some suspicions, based on thinking about generalizing basic notions of geometry to cover things like cellular automaton rule spaces.)

 What would this let one do? Likely it would let one find considerably simpler systems that could achieve particular computational goals. And maybe that would bring within reach some qualitatively new level of operations, perhaps beyond what we’re used to being possible with things like brains.

 There’s a funny thing that’s going on with modeling these days. As neural nets become more successful, one begins to wonder: why bother to simulate what’s going on inside a system when one can just make a black-box model of its output using a neural net? Well, if we manage to get machine learning to reach deeper into the computational universe, we won’t have as much of this tradeoff any more—because we’ll be able to learn models of the mechanism as well as the output.

 I’m pretty sure that bringing the full computational universe into the purview of machine learning will have spectacular consequences. But it’s worth realizing that computational universality—and the Principle of Computational Equivalence —make it less a matter of principle. Because they imply that even neural nets of the kinds we have now are universal, and are capable of emulating anything any other system can do. (In fact, this universality result was essentially what launched the whole modern idea of neural nets , back in 1943.)

 And as a practical matter, the fact that current neural net primitives are being built into hardware and so on will make them a desirable foundation for actual technology systems, though, even if they’re far from optimal. But my guess is that there are tasks where for the foreseeable future access to the full computational universe will be necessary to make them even vaguely practical.

 Finding AI

 What will it take to make artificial intelligence? As a kid, I was very interested in figuring out how to make a computer know things, and be able to answer questions from what it knew. And when I studied neural nets in 1981, it was partly in the context of trying to understand how to build such a system. As it happens, I had just developed SMP , which was a forerunner of Mathematica (and ultimately the Wolfram Language)—and which was very much based on symbolic pattern matching (“if you see this, transform it to that”). At the time, though, I imagined that artificial intelligence was somehow a “higher level of computation”, and I didn’t know how to achieve it.

 I returned to the problem every so often, and kept putting it off. But then when I was working on A New Kind of Science it struck me: if I’m to take the Principle of Computational Equivalence seriously, then there can’t be any fundamentally “higher level of computation”—so AI must be achievable just with the standard ideas of computation that I already know.

 And it was this realization that got me started building Wolfram|Alpha . And, yes, what I found is that lots of those very “AI-oriented things”, like natural language understanding, could be done just with “ordinary computation”, without any magic new AI invention. Now, to be fair, part of what was happening was that we were using ideas and methods from A New Kind of Science : we weren’t just engineering everything; we were often searching the computational universe for rules and algorithms to use.

 So what about “general AI”? Well, I think at this point that with the tools and understanding we have, we’re in a good position to automate essentially anything we can define. But definition is a more difficult and central issue than we might imagine.

 The way I see things at this point is that there’s a lot of computation even near at hand in the computational universe. And it’s powerful computation. As powerful as anything that happens in our brains. But we don’t recognize it as “intelligence” unless it’s aligned with our human goals and purposes.

 Ever since I was writing A New Kind of Science , I’ve been fond of quoting the aphorism “ the weather has a mind of its own ”. It sounds so animistic and pre-scientific. But what the Principle of Computational Equivalence says is that actually, according to the most modern science, it’s true: the fluid dynamics of the weather is the same in its computational sophistication as the electrical processes that go on in our brains.

 But is it “intelligent”? When I talk to people about A New Kind of Science, and about AI, I’ll often get asked when I think we’ll achieve “consciousness” in a machine. Life , intelligence , consciousness: they are all concepts that we have a specific example of, here on Earth. But what are they in general? All life on Earth shares RNA and the structure of cell membranes. But surely that’s just because all life we know is part of one connected thread of history; it’s not that such details are fundamental to the very concept of life.

 And so it is with intelligence. We have only one example we’re sure of: us humans. (We’re not even sure about animals.) But human intelligence as we experience it is deeply entangled with human civilization, human culture and ultimately also human physiology—even though none of these details are presumably relevant in the abstract definition of intelligence.

 We might think about extraterrestrial intelligence . But what the Principle of Computational Equivalence implies is that actually there’s “alien intelligence” all around us. But somehow it’s just not quite aligned with human intelligence. We might look at rule 30, for example, and be able to see that it’s doing sophisticated computation, just like our brains. But somehow it just doesn’t seem to have any “point” to what it’s doing.

 We imagine that in doing the things we humans do, we operate with certain goals or purposes. But rule 30, for example, just seems to be doing what it’s doing—just following some definite rule. In the end, though, one realizes we’re not so very different. After all, there are definite laws of nature that govern our brains. So anything we do is at some level just playing out those laws.

 Any process can actually be described either in terms of mechanism (“the stone is moving according to Newton’s laws ”), or in terms of goals (“the stone is moving so as to minimize potential energy”). The description in terms of mechanism is usually what’s most useful in connecting with science. But the description in terms of goals is usually what’s most useful in connecting with human intelligence.

 And this is crucial in thinking about AI. We know we can have computational systems whose operations are as sophisticated as anything. But can we get them to do things that are aligned with human goals and purposes?

 In a sense this is what I now view as the key problem of AI: it’s not about achieving underlying computational sophistication, but instead it’s about communicating what we want from this computation.

 The Importance of Language

 I’ve spent much of my life as a computer language designer—most importantly creating what is now the Wolfram Language . I’d always seen my role as a language designer being to imagine the possible computations people might want to do, then—like a reductionist scientist—trying to “drill down” to find good primitives from which all these computations could be built up. But somehow from A New Kind of Science , and from thinking about AI, I’ve come to think about it a little differently.

 Now what I more see myself as doing is making a bridge between our patterns of human thinking, and what the computational universe is capable of . There are all sorts of amazing things that can in principle be done by computation. But what the language does is to provide a way for us humans to express what we want done, or want to achieve—and then to get this actually executed, as automatically as possible.

 Language design has to start from what we know and are familiar with. In the Wolfram Language, we name the built-in primitives with English words, leveraging the meanings that those words have acquired. But the Wolfram Language is not like natural language. It’s something more structured, and more powerful. It’s based on the words and concepts that we’re familiar with through the shared corpus of human knowledge. But it gives us a way to build up arbitrarily sophisticated programs that in effect express arbitrarily complex goals.

 Yes, the computational universe is capable of remarkable things. But they’re not necessarily things that we humans can describe or relate to. But in building the Wolfram Language my goal is to do the best I can in capturing everything we humans want—and being able to express it in executable computational terms.

 When we look at the computational universe, it’s hard not to be struck by the limitations of what we know how to describe or think about. Modern neural nets provide an interesting example. For the ImageIdentify function of the Wolfram Language we’ve trained a neural net to identify thousands of kinds of things in the world. And to cater to our human purposes, what the network ultimately does is to describe what it sees in terms of concepts that we can name with words—tables, chairs, elephants, etc.

 But internally what the network is doing is to identify a series of features of any object in the world. Is it green? Is it round? And so on. And what happens as the neural network is trained is that it identifies features it finds useful for distinguishing different kinds of things in the world. But the point is that almost none of these features are ones to which we happen to have assigned words in human language.

 Out in the computational universe it’s possible to find what may be incredibly useful ways to describe things. But they’re alien to us humans. They’re not something we know how to express, based on the corpus of knowledge our civilization has developed.

 Now of course new concepts are being added to the corpus of human knowledge all the time. Back a century ago, if someone saw a nested pattern they wouldn’t have any way to describe it. But now we’d just say “it’s a fractal”. But the problem is that in the computational universe there’s an infinite collection of “potentially useful concepts”—with which we can never hope to ultimately keep up.

 The Analogy in Mathematics

 When I wrote A New Kind of Science I viewed it in no small part as an effort to break away from the use of mathematics—at least as a foundation for science. But one of the things I realized is that the ideas in the book also have a lot of implications for pure mathematics itself .

 What is mathematics? Well, it’s a study of certain abstract kinds of systems, based on things like numbers and geometry. In a sense it’s exploring a small corner of the computational universe of all possible abstract systems. But still, plenty has been done in mathematics: indeed, the 3 million or so published theorems of mathematics represent perhaps the largest single coherent intellectual structure that our species has built.

 Ever since Euclid , people have at least notionally imagined that mathematics starts from certain axioms (say, a + b = b + a , a +0= a , etc.), then builds up derivations of theorems. Why is math hard? The answer is fundamentally rooted in the phenomenon of computational irreducibility—which here is manifest in the fact that there’s no general way to shortcut the series of steps needed to derive a theorem. In other words, it can be arbitrarily hard to get a result in mathematics. But worse than that—as Gödel’s Theorem showed—there can be mathematical statements where there just aren’t any finite ways to prove or disprove them from the axioms. And in such cases, the statements just have to be considered “undecidable”.

 And in a sense what’s remarkable about math is that one can usefully do it at all. Because it could be that most mathematical results one cares about would be undecidable. So why doesn’t that happen?

 Well, if one considers arbitrary abstract systems it happens a lot. Take a typical cellular automaton—or a Turing machine—and ask whether it’s true that the system, say, always settles down to periodic behavior regardless of its initial state. Even something as simple as that will often be undecidable.

 So why doesn’t this happen in mathematics? Maybe there’s something special about the particular axioms used in mathematics. And certainly if one thinks they’re the ones that uniquely describe science and the world there might be a reason for that. But one of the whole points of the book is that actually there’s a whole computational universe of possible rules that can be useful for doing science and describing the world.

 And in fact I don’t think there’s anything abstractly special about the particular axioms that have traditionally been used in mathematics: I think they’re just accidents of history.

 What about the theorems that people investigate in mathematics? Again, I think there’s a strong historical character to them. For all but the most trivial areas of mathematics, there’s a whole sea of undecidability out there. But somehow mathematics picks the islands where theorems can actually be proved—often particularly priding itself on places close to the sea of undecidability where the proof can only be done with great effort.

 I’ve been interested in the whole network of published theorems in mathematics (it’s a thing to curate , like wars in history, or properties of chemicals). And one of the things I’m curious about is whether there’s an inexorable sequence to the mathematics that’s done, or whether, in a sense, random parts are being picked.

 And here, I think, there’s a considerable analogy to the kind of thing we were discussing before with language. What is a proof? Basically it’s a way of explaining to someone why something is true. I’ve made all sorts of automated proofs in which there are hundreds of steps, each perfectly verifiable by computer. But—like the innards of a neural net—what’s going on looks alien and not understandable by a human.

 For a human to understand, there have to be familiar “conceptual waypoints”. It’s pretty much like with words in languages. If some particular part of a proof has a name (“Smith’s Theorem”), and has a known meaning, then it’s useful to us. But if it’s just a lump of undifferentiated computation, it won’t be meaningful to us.

 In pretty much any axiom system, there’s an infinite set of possible theorems. But which ones are “interesting”? That’s really a human question. And basically it’s going to end up being ones with “stories”. In the book I show that for the simple case of basic logic , the theorems that have historically been considered interesting enough to be given names happen to be precisely the ones that are in some sense minimal.

 But my guess is that for richer axiom systems pretty much anything that’s going to be considered “interesting” is going to have to be reached from things that are already considered interesting. It’s like building up words or concepts: you don’t get to introduce new ones unless you can directly relate them to existing ones.

 In recent years I’ve wondered quite a bit about how inexorable or not progress is in a field like mathematics. Is there just one historical path that can be taken, say from arithmetic to algebra to the higher reaches of modern mathematics? Or are there an infinite diversity of possible paths, with completely different histories for mathematics?

 The answer is going to depend on—in a sense—the “structure of metamathematical space”: just what is the network of true theorems that avoid the sea of undecidability? Maybe it’ll be different for different fields of mathematics, and some will be more “inexorable” (so it feels like the math is being “discovered”) than others (where it seems more like the math is arbitrary, and “invented”).

 But to me one of the most interesting things is how close—when viewed in these kinds of terms—questions about the nature and character of mathematics end up being to questions about the nature and character of intelligence and AI. And it’s this kind of commonality that makes me realize just how powerful and general the ideas in A New Kind of Science actually are.

 When Is There a Science?

 There are some areas of science—like physics and astronomy—where the traditional mathematical approach has done quite well. But there are others—like biology, social science and linguistics—where it’s had a lot less to say. And one of the things I’ve long believed is that what’s needed to make progress in these areas is to generalize the kinds of models one’s using, to consider a broader range of what’s out there in the computational universe.

 And indeed in the past 15 or so years there’s been increasing success in doing this. And there are lots of biological and social systems, for example, where models have now been constructed using simple programs.

 But unlike with mathematical models which can potentially be “solved”, these computational models often show computational irreducibility, and are typically used by doing explicit simulations. This can be perfectly successful for making particular predictions, or for applying the models in technology. But a bit like for the automated proofs of mathematical theorems one might still ask, “is this really science?”.

 Yes, one can simulate what a system does, but does one “understand” it? Well, the problem is that computational irreducibility implies that in some fundamental sense one can’t always “understand” things. There might be no useful “story” that can be told; there may be no “conceptual waypoints”—only lots of detailed computation.

 Imagine that one’s trying to make a science of how the brain understands language—one of the big goals of linguistics. Well, perhaps we’ll get an adequate model of the precise rules which determine the firing of neurons or some other low-level representation of the brain. And then we look at the patterns generated in understanding some whole collection of sentences.

 Well, what if those patterns look like the behavior of rule 30? Or, closer at hand, the innards of some recurrent neural network? Can we “tell a story” about what’s happening? To do so would basically require that we create some kind of higher-level symbolic representation: something where we effectively have words for core elements of what’s going on.

 But computational irreducibility implies that there may ultimately be no way to create such a thing. Yes, it will always be possible to find patches of computational reducibility, where some things can be said. But there won’t be a complete story that can be told. And one might say there won’t be a useful reductionistic piece of science to be done. But that’s just one of the things that happens when one’s dealing with (as the title says) a new kind of science.

 Controlling the AIs

 People have gotten very worried about AI in recent years. They wonder what’s going to happen when AIs “get much smarter” than us humans. Well, the Principle of Computational Equivalence has one piece of good news: at some fundamental level, AIs will never be “smarter”—they’ll just be able to do computations that are ultimately equivalent to what our brains do, or, for that matter, what all sorts of simple programs do.

 As a practical matter, of course, AIs will be able to process larger amounts of data more quickly than actual brains. And no doubt we’ll choose to have them run many aspects of the world for us—from medical devices, to central banks to transportation systems, and much more.

 So then it’s important to figure how we’ll tell them what to do . As soon as we’re making serious use of what’s out there in the computational universe, we’re not going to be able to give a line-by-line description of what the AIs are going to do. Rather, we’re going to have to define goals for the AIs, then let them figure out how best to achieve those goals.

 In a sense we’ve already been doing something like this for years in the Wolfram Language . There’s some high-level function that describes something you want to do (“ lay out a graph ”, “ classify data ”, etc.). Then it’s up to the language to automatically figure out the best way to do it.

 And in the end the real challenge is to find a way to describe goals. Yes, you want to search for cellular automata that will make a “nice carpet pattern”, or a “good edge detector”. But what exactly do those things mean? What you need is a language that a human can use to say as precisely as possible what they mean.

 It’s really the same problem as I’ve been talking about a lot here. One has to have a way for humans to be able to talk about things they care about. There’s infinite detail out there in the computational universe. But through our civilization and our shared cultural history we’ve come to identify certain concepts that are important to us. And when we describe our goals, it’s in terms of these concepts.

 Three hundred years ago people like Leibniz were interested in finding a precise symbolic way to represent the content of human thoughts and human discourse. He was far too early. But now I think we’re finally in a position to actually make this work. In fact, we’ve already gotten a long way with the Wolfram Language in being able to describe real things in the world. And I’m hoping it’ll be possible to construct a fairly complete “ symbolic discourse language ” that lets us talk about the things we care about.

 Right now we write legal contracts in “legalese” as a way to make them slightly more precise than ordinary natural language. But with a symbolic discourse language we’ll be able to write true “computational contracts” that describe in high-level terms what we want to have happen—and then machines will automatically be able to verify or execute the contract.

 But what about the AIs? Well, we need to tell them what we generally want them to do. We need to have a contract with them. Or maybe we need to have a constitution for them . And it’ll be written in some kind of symbolic discourse language, that both allows us humans to express what we want, and is executable by the AIs.

 There’s lots to say about what should be in an AI Constitution, and how the construction of such things might map onto the political and cultural landscape of the world. But one of the obvious questions is: can the constitution be simple, like Asimov’s Laws of Robotics ?

 And here what we know from A New Kind of Science tells us the answer: it can’t be. In a sense the constitution is an attempt to sculpt what can happen in the world and what can’t. But computational irreducibility says that there will be an unbounded collection of cases to consider.

 For me it’s interesting to see how theoretical ideas like computational irreducibility end up impinging on these very practical—and central—societal issues. Yes, it all started with questions about things like the theory of all possible theories. But in the end it turns into issues that everyone in society is going to end up being concerned about.

 There’s an Endless Frontier

 Will we reach the end of science? Will we—or our AIs—eventually invent everything there is to be invented?

 For mathematics, it’s easy to see that there’s an infinite number of possible theorems one can construct. For science, there’s an infinite number of possible detailed questions to ask. And there’s also an infinite array of possible inventions one can construct.

 But the real question is: will there always be interesting new things out there?

 Well, computational irreducibility says there will always be new things that need an irreducible amount of computational work to reach from what’s already there. So in a sense there’ll always be “surprises”, that aren’t immediately evident from what’s come before.

 But will it just be like an endless array of different weirdly shaped rocks? Or will there be fundamental new features that appear, that we humans consider interesting?

 It’s back to the very same issue we’ve encountered several times before: for us humans to find things “interesting” we have to have a conceptual framework that we can use to think about them. Yes, we can identify a “ persistent structure ” in a cellular automaton. Then maybe we can start talking about “collisions between structures”. But when we just see a whole mess of stuff going on, it’s not going to be “interesting” to us unless we have some higher-level symbolic way to talk about it.

 In a sense, then, the rate of “interesting discovery” isn’t going to be limited by our ability to go out into the computational universe and find things. Instead, it’s going to be limited by our ability as humans to build a conceptual framework for what we’re finding.

 It’s a bit like what happened in the whole development of what became A New Kind of Science . People had seen related phenomena for centuries if not millennia (distribution of primes, digits of pi, etc.). But without a conceptual framework they just didn’t seem “interesting”, and nothing was built around them. And indeed as I understand more about what’s out there in the computational universe—and even about things I saw long ago there—I gradually build up a conceptual framework that lets me go further.

 By the way, it’s worth realizing that inventions work a little differently from discoveries. One can see something new happen in the computational universe, and that might be a discovery. But an invention is about figuring out how something can be achieved in the computational universe.

 And—like in patent law—it isn’t really an invention if you just say “look, this does that”. You have to somehow understand a purpose that it’s achieving.

 In the past, the focus of the process of invention has tended to be on actually getting something to work (“find the lightbulb filament that works”, etc.). But in the computational universe, the focus shifts to the question of what you want the invention to do. Because once you’ve described the goal, finding a way to achieve it is something that can be automated.

 That’s not to say that it will always be easy. In fact, computational irreducibility implies that it can be arbitrarily difficult. Let’s say you know the precise rules by which some chemicals can interact. Can you find a chemical synthesis pathway that will let you get to some particular chemical structure? There may be a way, but computational irreducibility implies that there may be no way to find out how long the pathway may be. And if you haven’t found a pathway you may never be sure if it’s because there isn’t one, or just because you didn’t reach it yet.

 The Fundamental Theory of Physics

 If one thinks about reaching the edge of science, one cannot help but wonder about the fundamental theory of physics . Given everything we’ve seen in the computational universe, is it conceivable that our physical universe could just correspond to one of those programs out there in the computational universe?

 Of course, we won’t really know until or unless we find it. But in the years since A New Kind of Science appeared, I’ve become ever more optimistic about the possibilities.

 Needless to say, it would be a big change for physics. Today there are basically two major frameworks for thinking about fundamental physics: general relativity and quantum field theory . General relativity is a bit more than 100 years old; quantum field theory maybe 90. And both have achieved spectacular things. But neither has succeeded in delivering us a complete fundamental theory of physics. And if nothing else, I think after all this time, it’s worth trying something new.

 But there’s another thing: from actually exploring the computational universe, we have a huge amount of new intuition about what’s possible, even in very simple models. We might have thought that the kind of richness we know exists in physics would require some very elaborate underlying model. But what’s become clear is that that kind of richness can perfectly well emerge even from a very simple underlying model.

 What might the underlying model be like ? I’m not going to discuss this in great detail here, but suffice it to say that I think the most important thing about the model is that it should have as little as possible built in. We shouldn’t have the hubris to think we know how the universe is constructed; we should just take a general type of model that’s as unstructured as possible, and do what we typically do in the computational universe: just search for a program that does what we want.

 My favorite formulation for a model that’s as unstructured as possible is a network : just a collection of nodes with connections between them. It’s perfectly possible to formulate such a model as an algebraic-like structure, and probably many other kinds of things. But we can think of it as a network. And in the way I’ve imagined setting it up, it’s a network that’s somehow “underneath” space and time: every aspect of space and time as we know it must emerge from the actual behavior of the network.

 Over the past decade or so there’s been increasing interest in things like loop quantum gravity and spin networks. They’re related to what I’ve been doing in the same way that they also involve networks. And maybe there’s some deeper relationship. But in their usual formulation, they’re much more mathematically elaborate.

 From the point of view of the traditional methods of physics, this might seem like a good idea. But with the intuition we have from studying the computational universe—and using it for science and technology—it seems completely unnecessary. Yes, we don’t yet know the fundamental theory of physics. But it seems sensible to start with the simplest hypothesis. And that’s definitely something like a simple network of the kind I’ve studied.

 At the outset, it’ll look pretty alien to people (including myself) trained in traditional theoretical physics. But some of what emerges isn’t so alien. A big result I found nearly 20 years ago (that still hasn’t been widely understood) is that when you look at a large enough network of the kind I studied you can show that its averaged behavior follows Einstein’s equations for gravity. In other words, without putting any fancy physics into the underlying model, it ends up automatically emerging. I think it’s pretty exciting.

 People ask a lot about quantum mechanics . Yes, my underlying model doesn’t build in quantum mechanics (just as it doesn’t build in general relativity). Now, it’s a little difficult to pin down exactly what the essence of “being quantum mechanical” actually is. But there are some very suggestive signs that my simple networks actually end up showing what amounts to quantum behavior—just like in the physics we know.

 OK, so how should one set about actually finding the fundamental theory of physics if it’s out there in the computational universe of possible programs? Well, the obvious thing is to just start searching for it, starting with the simplest programs.

 I’ve been doing this —more sporadically than I would like—for the past 15 years or so. And my main discovery so far is that it’s actually quite easy to find programs that aren’t obviously not our universe. There are plenty of programs where space or time are obviously completely different from the way they are in our universe, or there’s some other pathology. But it turns out it’s not so difficult to find candidate universes that aren’t obviously not our universe.

 But we’re immediately bitten by computational irreducibility. We can simulate the candidate universe for billions of steps. But we don’t know what it’s going to do—and whether it’s going to grow up to be like our universe, or completely different.

 It’s pretty unlikely that in looking at that tiny fragment of the very beginning of a universe we’re going to ever be able to see anything familiar, like a photon. And it’s not at all obvious that we’ll be able to construct any kind of descriptive theory, or effective physics. But in a sense the problem is bizarrely similar to the one we have even in systems like neural networks: there’s computation going on there, but can we identify “conceptual waypoints” from which we can build up a theory that we might understand?

 It’s not at all clear our universe has to be understandable at that level, and it’s quite possible that for a very long time we’ll be left in the strange situation of thinking we might have “found our universe” out in the computational universe, but not being sure.

 Of course, we might be lucky, and it might be possible to deduce an effective physics, and see that some little program that we found ends up reproducing our whole universe. It would be a remarkable moment for science. But it would immediately raise a host of new questions—like why this universe, and not another?

 Box of a Trillion Souls

 Right now us humans exist as biological systems. But in the future it’s certainly going to be technologically possible to reproduce all the processes in our brains in some purely digital—computational—form. So insofar as those processes represent “us”, we’re going to be able to be “virtualized” on pretty much any computational substrate. And in this case we might imagine that the whole future of a civilization could wind up in effect as a “ box of a trillion souls ”.

 Inside that box there would be all kinds of computations going on, representing the thoughts and experiences of all those disembodied souls. Those computations would reflect the rich history of our civilization, and all the things that have happened to us. But at some level they wouldn’t be anything special.

 It’s perhaps a bit disappointing, but the Principle of Computational Equivalence tells us that ultimately these computations will be no more sophisticated than the ones that go on in all sorts of other systems—even ones with simple rules, and no elaborate history of civilization. Yes, the details will reflect all that history. But in a sense without knowing what to look for—or what to care about—one won’t be able to tell that there’s anything special about it.

 OK, but what about for the “souls” themselves? Will one be able to understand their behavior by seeing that they achieve certain purposes? Well, in our current biological existence, we have all sorts of constraints and features that give us goals and purposes. But in a virtualized “uploaded” form, most of these just go away.

 I’ve thought quite a bit about how “human” purposes might evolve in such a situation, recognizing, of course, that in virtualized form there’s little difference between human and AI. The disappointing vision is that perhaps the future of our civilization consists in disembodied souls in effect “playing videogames” for the rest of eternity.

 But what I’ve slowly realized is that it’s actually quite unrealistic to project our view of goals and purposes from our experience today into that future situation. Imagine talking to someone from a thousand years ago and trying to explain that people in the future would be walking on treadmills every day, or continually sending photographs to their friends. The point is that such activities don’t make sense until the cultural framework around them has developed.

 It’s the same story yet again as with trying to characterize what’s interesting or what’s explainable. It relies on the development of a whole network of conceptual waypoints.

 Can we imagine what the mathematics of 100 years from now will be like? It depends on concepts we don’t yet know. So similarly if we try to imagine human motivation in the future, it’s going to rely on concepts we don’t know. Our best description from today’s viewpoint might be that those disembodied souls are just “playing videogames”. But to them there might be a whole subtle motivation structure that they could only explain by rewinding all sorts of steps in history and cultural development.

 By the way, if we know the fundamental theory of physics then in a sense we can make the virtualization complete, at least in principle: we can just run a simulation of the universe for those disembodied souls. Of course, if that’s what’s happening, then there’s no particular reason it has to be a simulation of our particular universe. It could as well be any universe from out in the computational universe.

 Now, as I’ve mentioned, even in any given universe one will never in a sense run out of things to do, or discover. But I suppose I myself at least find it amusing to imagine that at some point those disembodied souls might get bored with just being in a simulated version of our physical universe—and might decide it’s more fun (whatever that means to them) to go out and explore the broader computational universe. Which would mean that in a sense the future of humanity would be an infinite voyage of discovery in the context of none other than A New Kind of Science !

 The Economics of the Computational Universe

 Long before we have to think about disembodied human souls, we’ll have to confront the issue of what humans should be doing in a world where more and more can be done automatically by AIs. Now in a sense this issue is nothing new: it’s just an extension of the long-running story of technology and automation. But somehow this time it feels different.

 And I think the reason is in a sense just that there’s so much out there in the computational universe, that’s so easy to get to. Yes, we can build a machine that automates some particular task. We can even have a general-purpose computer that can be programmed to do a full range of different tasks. But even though these kinds of automation extend what we can do, it still feels like there’s effort that we have to put into them.

 But the picture now is different—because in effect what we’re saying is that if we can just define the goal we want to achieve, then everything else will be automatic. All sorts of computation, and, yes, “thinking”, may have to be done, but the idea is that it’s just going to happen, without human effort.

 At first, something seems wrong. How could we get all that benefit, without putting in more effort? It’s a bit like asking how nature could manage to make all the complexity it does—even though when we build artifacts, even with great effort, they end up far less complex. The answer, I think, is it’s mining the computational universe. And it’s exactly the same thing for us: by mining the computational universe, we can achieve essentially an unbounded level of automation.

 If we look at the important resources in today’s world, many of them still depend on actual materials. And often these materials are literally mined from the Earth. Of course, there are accidents of geography and geology that determine by whom and where that mining can be done. And in the end there’s a limit (if often very large) to the amount of material that’ll ever be available.

 But when it comes to the computational universe, there’s in a sense an inexhaustible supply of material—and it’s accessible to anyone. Yes, there are technical issues about how to “do the mining”, and there’s a whole stack of technology associated with doing it well. But the ultimate resource of the computational universe is a global and infinite one. There’s no scarcity and no reason to be “expensive”. One just has to understand that it’s there, and take advantage of it.

 The Path to Computational Thinking

 Probably the greatest intellectual shift of the past century has been the one towards the computational way of thinking about things. I’ve often said that if one picks almost any field “X”, from archaeology to zoology, then by now there either is, or soon will be, a field called “computational X”—and it’s going to be the future of the field.

 I myself have been deeply involved in trying to enable such computational fields, in particular through the development of the Wolfram Language. But I’ve also been interested in what is essentially the meta problem: how should one teach abstract computational thinking , for example to kids? The Wolfram Language is certainly important as a practical tool. But what about the conceptual, theoretical foundations?

 Well, that’s where A New Kind of Science comes in. Because at its core it’s discussing the pure abstract phenomenon of computation, independent of its applications to particular fields or tasks. It’s a bit like with elementary mathematics: there are things to teach and understand just to introduce the ideas of mathematical thinking, independent of their specific applications. And so it is too with the core of A New Kind of Science . There are things to learn about the computational universe that give intuition and introduce patterns of computational thinking—quite independent of detailed applications.

 One can think of it as a kind of “pre computer science” , or “pre computational X”. Before one gets into discussing the specifics of particular computational processes, one can just study the simple but pure things one finds in the computational universe.

 And, yes, even before kids learn to do arithmetic, it’s perfectly possible for them to fill out something like a cellular automaton coloring book—or to execute for themselves or on a computer a whole range of different simple programs. What does it teach? Well, it certainly teaches the idea that there can be definite rules or algorithms for things—and that if one follows them one can create useful and interesting results. And, yes, it helps that systems like cellular automata make obvious visual patterns, that for example one can even find in nature (say on mollusc shells).

 As the world becomes more computational—and more things are done by AIs and by mining the computational universe—there’s going to an extremely high value not only in understanding computational thinking, but also in having the kind of intuition that develops from exploring the computational universe and that is, in a sense, the foundation for A New Kind of Science.

 What’s Left to Figure Out?

 My goal over the decade that I spent writing A New Kind of Science was, as much as possible, to answer all the first round of “obvious questions” about the computational universe. And looking back 15 years later I think that worked out pretty well. Indeed, today, when I wonder about something to do with the computational universe, I find it’s incredibly likely that somewhere in the main text or notes of the book I already said something about it.

 But one of the biggest things that’s changed over the past 15 years is that I’ve gradually begun to understand more of the implications of what the book describes. There are lots of specific ideas and discoveries in the book. But in the longer term I think what’s most significant is how they serve as foundations, both practical and conceptual, for a whole range of new things that one can now understand and explore.

 But even in terms of the basic science of the computational universe, there are certainly specific results one would still like to get. For example, it would be great to get more evidence for or against the Principle of Computational Equivalence, and its domain of applicability.

 Like most general principles in science, the whole epistemological status of the Principle of Computational Equivalence is somewhat complicated. Is it like a mathematical theorem that can be proved? Is it like a law of nature that might (or might not) be true about the universe? Or is it like a definition, say of the very concept of computation? Well, much like, say, the Second Law of Thermodynamics or Evolution by Natural Selection, it’s a combination of these.

 But one thing that’s significant is that it’s possible to get concrete evidence for (or against) the Principle of Computational Equivalence. The principle says that even systems with very simple rules should be capable of arbitrarily sophisticated computation—so that in particular they should be able to act as universal computers.

 And indeed one of the results of the book is that this is true for one of the simplest possible cellular automata (rule 110) . Five years after the book was published I decided to put up a prize for evidence about another case: the simplest conceivably universal Turing machine . And I was very pleased that in just a few months the prize was won, the Turing machine was proved universal, and there was another piece of evidence for the Principle of Computational Equivalence.

 There’s a lot to do in developing the applications of A New Kind of Science. There are models to be made of all sorts of systems. There’s technology to be found. Art to be created. There’s also a lot to do in understanding the implications.

 But it’s important not to forget the pure investigation of the computational universe. In the analogy of mathematics, there are applications to be pursued. But there’s also a “pure mathematics” that’s worth pursuing in its own right. And so it is with the computational universe: there’s a huge amount to explore just at an abstract level. And indeed (as the title of the book implies) there’s enough to define a whole new kind of science: a pure science of the computational universe. And it’s the opening of that new kind of science that I think is the core achievement of A New Kind of Science—and the one of which I am most proud.

 For the 10th anniversary of A New Kind of Science, I wrote three posts:

 * It’s Been 10 Years: What’s Happened with A New Kind of Science ?
 * Living a Paradigm Shift: Looking Back on Reactions to A New Kind of Science
 * Looking to the Future of A New Kind of Science

 The complete high-resolution A New Kind of Science is now available on the web . There are also a limited number of print copies of the book still available (all individually coded!).

 Posted in: Artificial Intelligence, Big Picture, New Kind of Science, Physics, Ruliology

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 6 comments

 1. Occasionally when I search the Pigmentation Patterns of Mollusc shells.
 I found your website(www.wolframscience.com) and THIS book. It is so great. I will introduce it to my friends.
 My major is biology, I hope I could translate it into Chinese though I am lack of some mathematical and computer knowledge.
 Wish you a happy new year 2019.

 Yang Xiaoming
 December 26, 2018 at 9:51 pm

 2. Thanks for a great review and update. Your belief that mathematical tools often fail to fully capture the discrete local interactions going on in many system is spot on. For the past 25 years I have been hopeful that neural networks and now deep networks will be able to capture these subtle discrete relationships of features in medical imaging studies as well in medical decision support systems. I do not think deep networks alone will do this but instead a combination of deep networks and actionable clinical practice guideline hypergraphs where the deep networks fine tune the decision points in the hypergraph. The clinical practice guideline hypergraphs are built from our current knowledge, constraints, and goals and act as a starting point that can be fine tuned and optimized over time. The deep learning assisted decisions we make improve the odds of better outcomes over time that feeds back both to the deep network as well as clinical practice guideline hypergraphs in a continuous non-stationary feedback loop since the field of medicine is constantly changing over time with new diagnostic tools, new treatments, as well as new diseases.

 Richard Dell MD
 March 28, 2019 at 2:08 pm

 3. Second Law of Thermodynamics … Evolution by Natural Selection … Principle of Computational Equivalence … epistemological status of these general principles in science is somewhat complicated … can you write an article one day explaining that … thanks …

 Dan Ellwein
 May 24, 2019 at 4:34 am

 4. Thank you for posting this! It was a fantastic read.

 Bryn Loftness
 October 27, 2019 at 2:52 pm

 5. At the beginning of this post, it is said that the book will be freely available for download. But the link gives only the “online” version. Is there an official PDF? Thanks a lot!

 Mike
 May 25, 2020 at 8:02 am

 + Hi Mike,

 For ease of download, each chapter (and section) of A New Kind of Science is available individually as a PDF. You can navigate to a chapter from the Table of Contents, then select “Download PDF” at the bottom of the page. Happy reading!

 Administrator
 May 29, 2020 at 9:22 am

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Starting now, in celebration of its 15th anniversary, A New Kind of Science will be freely available in its entirety, with high-resolution images, on the web or for download. 
   
It’s now 15 years since I published my book A New Kind of Science—more than 25 since I started writing it, and more than 35 since I started working towards it. But with every passing year I feel I understand more about what the book is really about—and why it’s important. I wrote the book, as its title suggests, to contribute to the progress of science. But as the years have gone by, I’ve realized that the core of what’s in the book actually goes far beyond science—into many areas that will be increasingly important in defining our whole future.
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So, viewed from a distance of 15 years, what is the book really about? At its core, it’s about something profoundly abstract: the theory of all possible theories, or the universe of all possible universes. But for me one of the achievements of the book is the realization that one can explore such fundamental things concretely—by doing actual experiments in the computational universe of possible programs. And in the end the book is full of what might at first seem like quite alien pictures made just by running very simple such programs.
 Back in 1980, when I made my living as a theoretical physicist, if you’d asked me what I thought simple programs would do, I expect I would have said “not much”. I had been very interested in the kind of complexity one sees in nature, but I thought—like a typical reductionistic scientist—that the key to understanding it must lie in figuring out detailed features of the underlying component parts.
In retrospect I consider it incredibly lucky that all those years ago I happened to have the right interests and the right skills to actually try what is in a sense the most basic experiment in the computational universe: to systematically take a sequence of the simplest possible programs, and run them.
I could tell as soon as I did this that there were interesting things going on, but it took a couple more years before I began to really appreciate the force of what I’d seen. For me it all started with one picture:
 
Or, in modern form:
 
I call it rule 30. It’s my all-time favorite discovery, and today I carry it around everywhere on my business cards. What is it? It’s one of the simplest programs one can imagine. It operates on rows of black and white cells, starting from a single black cell, and then repeatedly applies the rules at the bottom. And the crucial point is that even though those rules are by any measure extremely simple, the pattern that emerges is not.
It’s a crucial—and utterly unexpected—feature of the computational universe: that even among the very simplest programs, it’s easy to get immensely complex behavior. It took me a solid decade to understand just how broad this phenomenon is. It doesn’t just happen in programs (“cellular automata”) like rule 30. It basically shows up whenever you start enumerating possible rules or possible programs whose behavior isn’t obviously trivial.
Similar phenomena had actually been seen for centuries in things like the digits of pi and the distribution of primes—but they were basically just viewed as curiosities, and not as signs of something profoundly important. It’s been nearly 35 years since I first saw what happens in rule 30, and with every passing year I feel I come to understand more clearly and deeply what its significance is.
Four centuries ago it was the discovery of the moons of Jupiter and their regularities that sowed the seeds for modern exact science, and for the modern scientific approach to thinking. Could my little rule 30 now be the seed for another such intellectual revolution, and a new way of thinking about everything?
In some ways I might personally prefer not to take responsibility for shepherding such ideas (“paradigm shifts” are hard and thankless work). And certainly for years I have just quietly used such ideas to develop technology and my own thinking. But as computation and AI become increasingly central to our world, I think it’s important that the implications of what’s out there in the computational universe be more widely understood.
Implications of the Computational Universe
Here’s the way I see it today. From observing the moons of Jupiter we came away with the idea that—if looked at right—the universe is an ordered and regular place, that we can ultimately understand. But now, in exploring the computational universe, we quickly come upon things like rule 30 where even the simplest rules seem to lead to irreducibly complex behavior.
One of the big ideas of A New Kind of Science is what I call the Principle of Computational Equivalence. The first step is to think of every process—whether it’s happening with black and white squares, or in physics, or inside our brains—as a computation that somehow transforms input to output. What the Principle of Computational Equivalence says is that above an extremely low threshold, all processes correspond to computations of equivalent sophistication.
It might not be true. It might be that something like rule 30 corresponds to a fundamentally simpler computation than the fluid dynamics of a hurricane, or the processes in my brain as I write this. But what the Principle of Computational Equivalence says is that in fact all these things are computationally equivalent.
It’s a very important statement, with many deep implications. For one thing, it implies what I call computational irreducibility. If something like rule 30 is doing a computation just as sophisticated as our brains or our mathematics, then there’s no way we can “outrun” it: to figure out what it will do, we have to do an irreducible amount of computation, effectively tracing each of its steps.
The mathematical tradition in exact science has emphasized the idea of predicting the behavior of systems by doing things like solving mathematical equations. But what computational irreducibility implies is that out in the computational universe that often won’t work, and instead the only way forward is just to explicitly run a computation to simulate the behavior of the system.
A Shift in Looking at the World
One of the things I did in A New Kind of Science was to show how simple programs can serve as models for the essential features of all sorts of physical, biological and other systems. Back when the book appeared, some people were skeptical about this. And indeed at that time there was a 300-year unbroken tradition that serious models in science should be based on mathematical equations.
But in the past 15 years something remarkable has happened. For now, when new models are created—whether of animal patterns or web browsing behavior—they are overwhelmingly more often based on programs than on mathematical equations.
Year by year, it’s been a slow, almost silent, process. But by this point, it’s a dramatic shift. Three centuries ago pure philosophical reasoning was supplanted by mathematical equations. Now in these few short years, equations have been largely supplanted by programs. For now, it’s mostly been something practical and pragmatic: the models work better, and are more useful.
But when it comes to understanding the foundations of what’s going on, one’s led not to things like mathematical theorems and calculus, but instead to ideas like the Principle of Computational Equivalence. Traditional mathematics-based ways of thinking have made concepts like force and momentum ubiquitous in the way we talk about the world. But now as we think in fundamentally computational terms we have to start talking in terms of concepts like undecidability and computational irreducibility.
Will some type of tumor always stop growing in some particular model? It might be undecidable. Is there a way to work out how a weather system will develop? It might be computationally irreducible.
These concepts are pretty important when it comes to understanding not only what can and cannot be modeled, but also what can and cannot be controlled in the world. Computational irreducibility in economics is going to limit what can be globally controlled. Computational irreducibility in biology is going to limit how generally effective therapies can be—and make highly personalized medicine a fundamental necessity.
And through ideas like the Principle of Computational Equivalence we can start to discuss just what it is that allows nature—seemingly so effortlessly—to generate so much that seems so complex to us. Or how even deterministic underlying rules can lead to computationally irreducible behavior that for all practical purposes can seem to show “free will”.
 
Mining the Computational Universe
A central lesson of A New Kind of Science is that there’s a lot of incredible richness out there in the computational universe. And one reason that’s important is that it means that there’s a lot of incredible stuff out there for us to “mine” and harness for our purposes.
Want to automatically make an interesting custom piece of art? Just start looking at simple programs and automatically pick out one you like—as in our WolframTones music site from a decade ago. Want to find an optimal algorithm for something? Just search enough programs out there, and you’ll find one.
We’ve normally been used to creating things by building them up, step by step, with human effort—progressively creating architectural plans, or engineering drawings, or lines of code. But the discovery that there’s so much richness so easily accessible in the computational universe suggests a different approach: don’t try building anything; just define what you want, and then search for it in the computational universe.
Sometimes it’s really easy to find. Like let’s say you want to generate apparent randomness. Well, then just enumerate cellular automata (as I did in 1984), and very quickly you come upon rule 30—which turns out to be one of the very best known generators of apparent randomness (look down the center column of cell values, for examples). In other situations you might have to search 100,000 cases (as I did in finding the simplest axiom system for logic, or the simplest universal Turing machine), or you might have to search millions or even trillions of cases. But in the past 25 years, we’ve had incredible success in just discovering algorithms out there in the computational universe—and we rely on many of them in implementing the Wolfram Language.
At some level it’s quite sobering. One finds some tiny program out in the computational universe. One can tell it does what one wants. But when one looks at what it’s doing, one doesn’t have any real idea how it works. Maybe one can analyze some part—and be struck by how “clever” it is. But there just isn’t a way for us to understand the whole thing; it’s not something familiar from our usual patterns of thinking.
Of course, we’ve often had similar experiences before—when we use things from nature. We may notice that some particular substance is a useful drug or a great chemical catalyst, but we may have no idea why. But in doing engineering and in most of our modern efforts to build technology, the great emphasis has instead been on constructing things whose design and operation we can readily understand.
In the past we might have thought that was enough. But what our explorations of the computational universe show is that it’s not: selecting only things whose operation we can readily understand misses most of the immense power and richness that’s out there in the computational universe.
A World of Discovered Technology
What will the world look like when more of what we have is mined from the computational universe? Today the environment we build for ourselves is dominated by things like simple shapes and repetitive processes. But the more we use what’s out there in the computational universe, the less regular things will look. Sometimes they may look a bit “organic”, or like what we see in nature (since after all, nature follows similar kinds of rules). But sometimes they may look quite random, until perhaps suddenly and incomprehensibly they achieve something we recognize.
For several millennia we as a civilization have been on a path to understand more about what happens in our world—whether by using science to decode nature, or by creating our own environment through technology. But to use more of the richness of the computational universe we must at least to some extent forsake this path.
In the past, we somehow counted on the idea that between our brains and the tools we could create we would always have fundamentally greater computational power than the things around us—and as a result we would always be able to “understand” them. But what the Principle of Computational Equivalence says is that this isn’t true: out in the computational universe there are lots of things just as powerful as our brains or the tools we build. And as soon as we start using those things, we lose the “edge” we thought we had.
Today we still imagine we can identify discrete “bugs” in programs. But most of what’s powerful out there in the computational universe is rife with computational irreducibility—so the only real way to see what it does is just to run it and watch what happens.
We ourselves, as biological systems, are a great example of computation happening at a molecular scale—and we are no doubt rife with computational irreducibility (which is, at some fundamental level, why medicine is hard). I suppose it’s a tradeoff: we could limit our technology to consist only of things whose operation we understand. But then we would miss all that richness that’s out there in the computational universe. And we wouldn’t even be able to match the achievements of our own biology in the technology we create.
Machine Learning and the Neural Net Renaissance
There’s a common pattern I’ve noticed with intellectual fields. They go for decades and perhaps centuries with only incremental growth, and then suddenly, usually as a result of a methodological advance, there’s a burst of “hypergrowth” for perhaps 5 years, in which important new results arrive almost every week.
I was fortunate enough that my own very first field—particle physics—was in its period of hypergrowth right when I was involved in the late 1970s. And for myself, the 1990s felt like a kind of personal period of hypergrowth for what became A New Kind of Science—and indeed that’s why I couldn’t pull myself away from it for more than a decade.
But today, the obvious field in hypergrowth is machine learning, or, more specifically, neural nets. It’s funny for me to see this. I actually worked on neural nets back in 1981, before I started on cellular automata, and several years before I found rule 30. But I never managed to get neural nets to do anything very interesting—and actually I found them too messy and complicated for the fundamental questions I was concerned with.
And so I “simplified them”—and wound up with cellular automata. (I was also inspired by things like the Ising model in statistical physics, etc.) At the outset, I thought I might have simplified too far, and that my little cellular automata would never do anything interesting. But then I found things like rule 30. And I’ve been trying to understand its implications ever since.
In building Mathematica and the Wolfram Language, I’d always kept track of neural nets, and occasionally we’d use them in some small way for some algorithm or another. But about 5 years ago I suddenly started hearing amazing things: that somehow the idea of training neural nets to do sophisticated things was actually working. At first I wasn’t sure. But then we started building neural net capabilities in the Wolfram Language, and finally two years ago we released our ImageIdentify.com website—and now we’ve got our whole symbolic neural net system. And, yes, I’m impressed. There are lots of tasks that had traditionally been viewed as the unique domain of humans, but which now we can routinely do by computer.
But what’s actually going on in a neural net? It’s not really to do with the brain; that was just the inspiration (though in reality the brain probably works more or less the same way). A neural net is really a sequence of functions that operate on arrays of numbers, with each function typically taking quite a few inputs from around the array. It’s not so different from a cellular automaton. Except that in a cellular automaton, one’s usually dealing with, say, just 0s and 1s, not arbitrary numbers like 0.735. And instead of taking inputs from all over the place, in a cellular automaton each step takes inputs only from a very well-defined local region.
Now, to be fair, it’s pretty common to study “convolutional neural nets”, in which the patterns of inputs are very regular, just like in a cellular automaton. And it’s becoming clear that having precise (say 32-bit) numbers isn’t critical to the operation of neural nets; one can probably make do with just a few bits.
But a big feature of neural nets is that we know how to make them “learn”. In particular, they have enough features from traditional mathematics (like involving continuous numbers) that techniques like calculus can be applied to provide strategies to make them incrementally change their parameters to “fit their behavior” to whatever training examples they’re given.
It’s far from obvious how much computational effort, or how many training examples, will be needed. But the breakthrough of about five years ago was the discovery that for many important practical problems, what’s available with modern GPUs and modern web-collected training sets can be enough.
Pretty much nobody ends up explicitly setting or “engineering” the parameters in a neural net. Instead, what happens is that they’re found automatically. But unlike with simple programs like cellular automata, where one’s typically enumerating all possibilities, in current neural nets there’s an incremental process, essentially based on calculus, that manages to progressively improve the net—a little like the way biological evolution progressively improves the “fitness” of an organism.
It’s plenty remarkable what comes out from training a neural net in this way, and it’s plenty difficult to understand how the neural net does what it does. But in some sense the neural net isn’t venturing too far across the computational universe: it’s always basically keeping the same basic computational structure, and just changing its behavior by changing parameters.
But to me the success of today’s neural nets is a spectacular endorsement of the power of the computational universe, and another validation of the ideas of A New Kind of Science. Because it shows that out in the computational universe, away from the constraints of explicitly building systems whose detailed behavior one can foresee, there are immediately all sorts of rich and useful things to be found.
NKS Meets Modern Machine Learning
Is there a way to bring the full power of the computational universe—and the ideas of A New Kind of Science—to the kinds of things one does with neural nets? I suspect so. And in fact, as the details become clear, I wouldn’t be surprised if exploration of the computational universe saw its own period of hypergrowth: a “mining boom” of perhaps unprecedented proportions.
In current work on neural nets, there’s a definite tradeoff one sees. The more what’s going on inside the neural net is like a simple mathematical function with essentially arithmetic parameters, the easier it is to use ideas from calculus to train the network. But the more what’s going is like a discrete program, or like a computation whose whole structure can change, the more difficult it is to train the network.
It’s worth remembering, though, that the networks we’re routinely training now would have looked utterly impractical to train only a few years ago. It’s effectively just all those quadrillions of GPU operations that we can throw at the problem that makes training feasible. And I won’t be surprised if even quite pedestrian (say, local exhaustive search) techniques will fairly soon let one do significant training even in cases where no incremental numerical approach is possible. And perhaps even it will be possible to invent some major generalization of things like calculus that will operate in the full computational universe. (I have some suspicions, based on thinking about generalizing basic notions of geometry to cover things like cellular automaton rule spaces.)
What would this let one do? Likely it would let one find considerably simpler systems that could achieve particular computational goals. And maybe that would bring within reach some qualitatively new level of operations, perhaps beyond what we’re used to being possible with things like brains.
There’s a funny thing that’s going on with modeling these days. As neural nets become more successful, one begins to wonder: why bother to simulate what’s going on inside a system when one can just make a black-box model of its output using a neural net? Well, if we manage to get machine learning to reach deeper into the computational universe, we won’t have as much of this tradeoff any more—because we’ll be able to learn models of the mechanism as well as the output.
I’m pretty sure that bringing the full computational universe into the purview of machine learning will have spectacular consequences. But it’s worth realizing that computational universality—and the Principle of Computational Equivalence—make it less a matter of principle. Because they imply that even neural nets of the kinds we have now are universal, and are capable of emulating anything any other system can do. (In fact, this universality result was essentially what launched the whole modern idea of neural nets, back in 1943.)
And as a practical matter, the fact that current neural net primitives are being built into hardware and so on will make them a desirable foundation for actual technology systems, though, even if they’re far from optimal. But my guess is that there are tasks where for the foreseeable future access to the full computational universe will be necessary to make them even vaguely practical.
Finding AI
What will it take to make artificial intelligence? As a kid, I was very interested in figuring out how to make a computer know things, and be able to answer questions from what it knew. And when I studied neural nets in 1981, it was partly in the context of trying to understand how to build such a system. As it happens, I had just developed SMP, which was a forerunner of Mathematica (and ultimately the Wolfram Language)—and which was very much based on symbolic pattern matching (“if you see this, transform it to that”). At the time, though, I imagined that artificial intelligence was somehow a “higher level of computation”, and I didn’t know how to achieve it.
I returned to the problem every so often, and kept putting it off. But then when I was working on A New Kind of Science it struck me: if I’m to take the Principle of Computational Equivalence seriously, then there can’t be any fundamentally “higher level of computation”—so AI must be achievable just with the standard ideas of computation that I already know.
And it was this realization that got me started building Wolfram|Alpha. And, yes, what I found is that lots of those very “AI-oriented things”, like natural language understanding, could be done just with “ordinary computation”, without any magic new AI invention. Now, to be fair, part of what was happening was that we were using ideas and methods from A New Kind of Science: we weren’t just engineering everything; we were often searching the computational universe for rules and algorithms to use.
So what about “general AI”? Well, I think at this point that with the tools and understanding we have, we’re in a good position to automate essentially anything we can define. But definition is a more difficult and central issue than we might imagine.
The way I see things at this point is that there’s a lot of computation even near at hand in the computational universe. And it’s powerful computation. As powerful as anything that happens in our brains. But we don’t recognize it as “intelligence” unless it’s aligned with our human goals and purposes.
Ever since I was writing A New Kind of Science, I’ve been fond of quoting the aphorism “the weather has a mind of its own”. It sounds so animistic and pre-scientific. But what the Principle of Computational Equivalence says is that actually, according to the most modern science, it’s true: the fluid dynamics of the weather is the same in its computational sophistication as the electrical processes that go on in our brains.
But is it “intelligent”? When I talk to people about A New Kind of Science, and about AI, I’ll often get asked when I think we’ll achieve “consciousness” in a machine. Life, intelligence, consciousness: they are all concepts that we have a specific example of, here on Earth. But what are they in general? All life on Earth shares RNA and the structure of cell membranes. But surely that’s just because all life we know is part of one connected thread of history; it’s not that such details are fundamental to the very concept of life.
And so it is with intelligence. We have only one example we’re sure of: us humans. (We’re not even sure about animals.) But human intelligence as we experience it is deeply entangled with human civilization, human culture and ultimately also human physiology—even though none of these details are presumably relevant in the abstract definition of intelligence.
We might think about extraterrestrial intelligence. But what the Principle of Computational Equivalence implies is that actually there’s “alien intelligence” all around us. But somehow it’s just not quite aligned with human intelligence. We might look at rule 30, for example, and be able to see that it’s doing sophisticated computation, just like our brains. But somehow it just doesn’t seem to have any “point” to what it’s doing.
We imagine that in doing the things we humans do, we operate with certain goals or purposes. But rule 30, for example, just seems to be doing what it’s doing—just following some definite rule. In the end, though, one realizes we’re not so very different. After all, there are definite laws of nature that govern our brains. So anything we do is at some level just playing out those laws.
Any process can actually be described either in terms of mechanism (“the stone is moving according to Newton’s laws”), or in terms of goals (“the stone is moving so as to minimize potential energy”). The description in terms of mechanism is usually what’s most useful in connecting with science. But the description in terms of goals is usually what’s most useful in connecting with human intelligence.
And this is crucial in thinking about AI. We know we can have computational systems whose operations are as sophisticated as anything. But can we get them to do things that are aligned with human goals and purposes?
In a sense this is what I now view as the key problem of AI: it’s not about achieving underlying computational sophistication, but instead it’s about communicating what we want from this computation.
The Importance of Language
I’ve spent much of my life as a computer language designer—most importantly creating what is now the Wolfram Language. I’d always seen my role as a language designer being to imagine the possible computations people might want to do, then—like a reductionist scientist—trying to “drill down” to find good primitives from which all these computations could be built up. But somehow from A New Kind of Science, and from thinking about AI, I’ve come to think about it a little differently.
Now what I more see myself as doing is making a bridge between our patterns of human thinking, and what the computational universe is capable of. There are all sorts of amazing things that can in principle be done by computation. But what the language does is to provide a way for us humans to express what we want done, or want to achieve—and then to get this actually executed, as automatically as possible.
Language design has to start from what we know and are familiar with. In the Wolfram Language, we name the built-in primitives with English words, leveraging the meanings that those words have acquired. But the Wolfram Language is not like natural language. It’s something more structured, and more powerful. It’s based on the words and concepts that we’re familiar with through the shared corpus of human knowledge. But it gives us a way to build up arbitrarily sophisticated programs that in effect express arbitrarily complex goals.
Yes, the computational universe is capable of remarkable things. But they’re not necessarily things that we humans can describe or relate to. But in building the Wolfram Language my goal is to do the best I can in capturing everything we humans want—and being able to express it in executable computational terms.
When we look at the computational universe, it’s hard not to be struck by the limitations of what we know how to describe or think about. Modern neural nets provide an interesting example. For the ImageIdentify function of the Wolfram Language we’ve trained a neural net to identify thousands of kinds of things in the world. And to cater to our human purposes, what the network ultimately does is to describe what it sees in terms of concepts that we can name with words—tables, chairs, elephants, etc.
But internally what the network is doing is to identify a series of features of any object in the world. Is it green? Is it round? And so on. And what happens as the neural network is trained is that it identifies features it finds useful for distinguishing different kinds of things in the world. But the point is that almost none of these features are ones to which we happen to have assigned words in human language.
Out in the computational universe it’s possible to find what may be incredibly useful ways to describe things. But they’re alien to us humans. They’re not something we know how to express, based on the corpus of knowledge our civilization has developed.
Now of course new concepts are being added to the corpus of human knowledge all the time. Back a century ago, if someone saw a nested pattern they wouldn’t have any way to describe it. But now we’d just say “it’s a fractal”. But the problem is that in the computational universe there’s an infinite collection of “potentially useful concepts”—with which we can never hope to ultimately keep up.
The Analogy in Mathematics
When I wrote A New Kind of Science I viewed it in no small part as an effort to break away from the use of mathematics—at least as a foundation for science. But one of the things I realized is that the ideas in the book also have a lot of implications for pure mathematics itself.
What is mathematics? Well, it’s a study of certain abstract kinds of systems, based on things like numbers and geometry. In a sense it’s exploring a small corner of the computational universe of all possible abstract systems. But still, plenty has been done in mathematics: indeed, the 3 million or so published theorems of mathematics represent perhaps the largest single coherent intellectual structure that our species has built.
Ever since Euclid, people have at least notionally imagined that mathematics starts from certain axioms (say, a+b=b+a, a+0=a, etc.), then builds up derivations of theorems. Why is math hard? The answer is fundamentally rooted in the phenomenon of computational irreducibility—which here is manifest in the fact that there’s no general way to shortcut the series of steps needed to derive a theorem. In other words, it can be arbitrarily hard to get a result in mathematics. But worse than that—as Gödel’s Theorem showed—there can be mathematical statements where there just aren’t any finite ways to prove or disprove them from the axioms. And in such cases, the statements just have to be considered “undecidable”.
And in a sense what’s remarkable about math is that one can usefully do it at all. Because it could be that most mathematical results one cares about would be undecidable. So why doesn’t that happen?
Well, if one considers arbitrary abstract systems it happens a lot. Take a typical cellular automaton—or a Turing machine—and ask whether it’s true that the system, say, always settles down to periodic behavior regardless of its initial state. Even something as simple as that will often be undecidable.
So why doesn’t this happen in mathematics? Maybe there’s something special about the particular axioms used in mathematics. And certainly if one thinks they’re the ones that uniquely describe science and the world there might be a reason for that. But one of the whole points of the book is that actually there’s a whole computational universe of possible rules that can be useful for doing science and describing the world.
And in fact I don’t think there’s anything abstractly special about the particular axioms that have traditionally been used in mathematics: I think they’re just accidents of history.
What about the theorems that people investigate in mathematics? Again, I think there’s a strong historical character to them. For all but the most trivial areas of mathematics, there’s a whole sea of undecidability out there. But somehow mathematics picks the islands where theorems can actually be proved—often particularly priding itself on places close to the sea of undecidability where the proof can only be done with great effort.
I’ve been interested in the whole network of published theorems in mathematics (it’s a thing to curate, like wars in history, or properties of chemicals). And one of the things I’m curious about is whether there’s an inexorable sequence to the mathematics that’s done, or whether, in a sense, random parts are being picked.
And here, I think, there’s a considerable analogy to the kind of thing we were discussing before with language. What is a proof? Basically it’s a way of explaining to someone why something is true. I’ve made all sorts of automated proofs in which there are hundreds of steps, each perfectly verifiable by computer. But—like the innards of a neural net—what’s going on looks alien and not understandable by a human.
For a human to understand, there have to be familiar “conceptual waypoints”. It’s pretty much like with words in languages. If some particular part of a proof has a name (“Smith’s Theorem”), and has a known meaning, then it’s useful to us. But if it’s just a lump of undifferentiated computation, it won’t be meaningful to us.
In pretty much any axiom system, there’s an infinite set of possible theorems. But which ones are “interesting”? That’s really a human question. And basically it’s going to end up being ones with “stories”. In the book I show that for the simple case of basic logic, the theorems that have historically been considered interesting enough to be given names happen to be precisely the ones that are in some sense minimal.
But my guess is that for richer axiom systems pretty much anything that’s going to be considered “interesting” is going to have to be reached from things that are already considered interesting. It’s like building up words or concepts: you don’t get to introduce new ones unless you can directly relate them to existing ones.
In recent years I’ve wondered quite a bit about how inexorable or not progress is in a field like mathematics. Is there just one historical path that can be taken, say from arithmetic to algebra to the higher reaches of modern mathematics? Or are there an infinite diversity of possible paths, with completely different histories for mathematics?
The answer is going to depend on—in a sense—the “structure of metamathematical space”: just what is the network of true theorems that avoid the sea of undecidability? Maybe it’ll be different for different fields of mathematics, and some will be more “inexorable” (so it feels like the math is being “discovered”) than others (where it seems more like the math is arbitrary, and “invented”).
But to me one of the most interesting things is how close—when viewed in these kinds of terms—questions about the nature and character of mathematics end up being to questions about the nature and character of intelligence and AI. And it’s this kind of commonality that makes me realize just how powerful and general the ideas in A New Kind of Science actually are.
When Is There a Science?
There are some areas of science—like physics and astronomy—where the traditional mathematical approach has done quite well. But there are others—like biology, social science and linguistics—where it’s had a lot less to say. And one of the things I’ve long believed is that what’s needed to make progress in these areas is to generalize the kinds of models one’s using, to consider a broader range of what’s out there in the computational universe.
And indeed in the past 15 or so years there’s been increasing success in doing this. And there are lots of biological and social systems, for example, where models have now been constructed using simple programs.
But unlike with mathematical models which can potentially be “solved”, these computational models often show computational irreducibility, and are typically used by doing explicit simulations. This can be perfectly successful for making particular predictions, or for applying the models in technology. But a bit like for the automated proofs of mathematical theorems one might still ask, “is this really science?”.
Yes, one can simulate what a system does, but does one “understand” it? Well, the problem is that computational irreducibility implies that in some fundamental sense one can’t always “understand” things. There might be no useful “story” that can be told; there may be no “conceptual waypoints”—only lots of detailed computation.
Imagine that one’s trying to make a science of how the brain understands language—one of the big goals of linguistics. Well, perhaps we’ll get an adequate model of the precise rules which determine the firing of neurons or some other low-level representation of the brain. And then we look at the patterns generated in understanding some whole collection of sentences.
Well, what if those patterns look like the behavior of rule 30? Or, closer at hand, the innards of some recurrent neural network? Can we “tell a story” about what’s happening? To do so would basically require that we create some kind of higher-level symbolic representation: something where we effectively have words for core elements of what’s going on.
But computational irreducibility implies that there may ultimately be no way to create such a thing. Yes, it will always be possible to find patches of computational reducibility, where some things can be said. But there won’t be a complete story that can be told. And one might say there won’t be a useful reductionistic piece of science to be done. But that’s just one of the things that happens when one’s dealing with (as the title says) a new kind of science.
Controlling the AIs
People have gotten very worried about AI in recent years. They wonder what’s going to happen when AIs “get much smarter” than us humans. Well, the Principle of Computational Equivalence has one piece of good news: at some fundamental level, AIs will never be “smarter”—they’ll just be able to do computations that are ultimately equivalent to what our brains do, or, for that matter, what all sorts of simple programs do.
As a practical matter, of course, AIs will be able to process larger amounts of data more quickly than actual brains. And no doubt we’ll choose to have them run many aspects of the world for us—from medical devices, to central banks to transportation systems, and much more.
So then it’s important to figure how we’ll tell them what to do. As soon as we’re making serious use of what’s out there in the computational universe, we’re not going to be able to give a line-by-line description of what the AIs are going to do. Rather, we’re going to have to define goals for the AIs, then let them figure out how best to achieve those goals.
In a sense we’ve already been doing something like this for years in the Wolfram Language. There’s some high-level function that describes something you want to do (“lay out a graph”, “classify data”, etc.). Then it’s up to the language to automatically figure out the best way to do it.
And in the end the real challenge is to find a way to describe goals. Yes, you want to search for cellular automata that will make a “nice carpet pattern”, or a “good edge detector”. But what exactly do those things mean? What you need is a language that a human can use to say as precisely as possible what they mean.
It’s really the same problem as I’ve been talking about a lot here. One has to have a way for humans to be able to talk about things they care about. There’s infinite detail out there in the computational universe. But through our civilization and our shared cultural history we’ve come to identify certain concepts that are important to us. And when we describe our goals, it’s in terms of these concepts.
Three hundred years ago people like Leibniz were interested in finding a precise symbolic way to represent the content of human thoughts and human discourse. He was far too early. But now I think we’re finally in a position to actually make this work. In fact, we’ve already gotten a long way with the Wolfram Language in being able to describe real things in the world. And I’m hoping it’ll be possible to construct a fairly complete “symbolic discourse language” that lets us talk about the things we care about.
Right now we write legal contracts in “legalese” as a way to make them slightly more precise than ordinary natural language. But with a symbolic discourse language we’ll be able to write true “computational contracts” that describe in high-level terms what we want to have happen—and then machines will automatically be able to verify or execute the contract.
But what about the AIs? Well, we need to tell them what we generally want them to do. We need to have a contract with them. Or maybe we need to have a constitution for them. And it’ll be written in some kind of symbolic discourse language, that both allows us humans to express what we want, and is executable by the AIs.
There’s lots to say about what should be in an AI Constitution, and how the construction of such things might map onto the political and cultural landscape of the world. But one of the obvious questions is: can the constitution be simple, like Asimov’s Laws of Robotics?
And here what we know from A New Kind of Science tells us the answer: it can’t be. In a sense the constitution is an attempt to sculpt what can happen in the world and what can’t. But computational irreducibility says that there will be an unbounded collection of cases to consider. 
For me it’s interesting to see how theoretical ideas like computational irreducibility end up impinging on these very practical—and central—societal issues. Yes, it all started with questions about things like the theory of all possible theories. But in the end it turns into issues that everyone in society is going to end up being concerned about.
There’s an Endless Frontier
Will we reach the end of science? Will we—or our AIs—eventually invent everything there is to be invented?
For mathematics, it’s easy to see that there’s an infinite number of possible theorems one can construct. For science, there’s an infinite number of possible detailed questions to ask. And there’s also an infinite array of possible inventions one can construct.
But the real question is: will there always be interesting new things out there?
Well, computational irreducibility says there will always be new things that need an irreducible amount of computational work to reach from what’s already there. So in a sense there’ll always be “surprises”, that aren’t immediately evident from what’s come before.
But will it just be like an endless array of different weirdly shaped rocks? Or will there be fundamental new features that appear, that we humans consider interesting?
It’s back to the very same issue we’ve encountered several times before: for us humans to find things “interesting” we have to have a conceptual framework that we can use to think about them. Yes, we can identify a “persistent structure” in a cellular automaton. Then maybe we can start talking about “collisions between structures”. But when we just see a whole mess of stuff going on, it’s not going to be “interesting” to us unless we have some higher-level symbolic way to talk about it.
In a sense, then, the rate of “interesting discovery” isn’t going to be limited by our ability to go out into the computational universe and find things. Instead, it’s going to be limited by our ability as humans to build a conceptual framework for what we’re finding.
It’s a bit like what happened in the whole development of what became A New Kind of Science. People had seen related phenomena for centuries if not millennia (distribution of primes, digits of pi, etc.). But without a conceptual framework they just didn’t seem “interesting”, and nothing was built around them. And indeed as I understand more about what’s out there in the computational universe—and even about things I saw long ago there—I gradually build up a conceptual framework that lets me go further.
By the way, it’s worth realizing that inventions work a little differently from discoveries. One can see something new happen in the computational universe, and that might be a discovery. But an invention is about figuring out how something can be achieved in the computational universe.
And—like in patent law—it isn’t really an invention if you just say “look, this does that”. You have to somehow understand a purpose that it’s achieving.
In the past, the focus of the process of invention has tended to be on actually getting something to work (“find the lightbulb filament that works”, etc.). But in the computational universe, the focus shifts to the question of what you want the invention to do. Because once you’ve described the goal, finding a way to achieve it is something that can be automated.
That’s not to say that it will always be easy. In fact, computational irreducibility implies that it can be arbitrarily difficult. Let’s say you know the precise rules by which some chemicals can interact. Can you find a chemical synthesis pathway that will let you get to some particular chemical structure? There may be a way, but computational irreducibility implies that there may be no way to find out how long the pathway may be. And if you haven’t found a pathway you may never be sure if it’s because there isn’t one, or just because you didn’t reach it yet.
The Fundamental Theory of Physics
If one thinks about reaching the edge of science, one cannot help but wonder about the fundamental theory of physics. Given everything we’ve seen in the computational universe, is it conceivable that our physical universe could just correspond to one of those programs out there in the computational universe?
Of course, we won’t really know until or unless we find it. But in the years since A New Kind of Science appeared, I’ve become ever more optimistic about the possibilities.
Needless to say, it would be a big change for physics. Today there are basically two major frameworks for thinking about fundamental physics: general relativity and quantum field theory. General relativity is a bit more than 100 years old; quantum field theory maybe 90. And both have achieved spectacular things. But neither has succeeded in delivering us a complete fundamental theory of physics. And if nothing else, I think after all this time, it’s worth trying something new.
But there’s another thing: from actually exploring the computational universe, we have a huge amount of new intuition about what’s possible, even in very simple models. We might have thought that the kind of richness we know exists in physics would require some very elaborate underlying model. But what’s become clear is that that kind of richness can perfectly well emerge even from a very simple underlying model.
 What might the underlying model be like? I’m not going to discuss this in great detail here, but suffice it to say that I think the most important thing about the model is that it should have as little as possible built in. We shouldn’t have the hubris to think we know how the universe is constructed; we should just take a general type of model that’s as unstructured as possible, and do what we typically do in the computational universe: just search for a program that does what we want.
My favorite formulation for a model that’s as unstructured as possible is a network: just a collection of nodes with connections between them. It’s perfectly possible to formulate such a model as an algebraic-like structure, and probably many other kinds of things. But we can think of it as a network. And in the way I’ve imagined setting it up, it’s a network that’s somehow “underneath” space and time: every aspect of space and time as we know it must emerge from the actual behavior of the network.
Over the past decade or so there’s been increasing interest in things like loop quantum gravity and spin networks. They’re related to what I’ve been doing in the same way that they also involve networks. And maybe there’s some deeper relationship. But in their usual formulation, they’re much more mathematically elaborate.
From the point of view of the traditional methods of physics, this might seem like a good idea. But with the intuition we have from studying the computational universe—and using it for science and technology—it seems completely unnecessary. Yes, we don’t yet know the fundamental theory of physics. But it seems sensible to start with the simplest hypothesis. And that’s definitely something like a simple network of the kind I’ve studied.
At the outset, it’ll look pretty alien to people (including myself) trained in traditional theoretical physics. But some of what emerges isn’t so alien. A big result I found nearly 20 years ago (that still hasn’t been widely understood) is that when you look at a large enough network of the kind I studied you can show that its averaged behavior follows Einstein’s equations for gravity. In other words, without putting any fancy physics into the underlying model, it ends up automatically emerging. I think it’s pretty exciting.
People ask a lot about quantum mechanics. Yes, my underlying model doesn’t build in quantum mechanics (just as it doesn’t build in general relativity). Now, it’s a little difficult to pin down exactly what the essence of “being quantum mechanical” actually is. But there are some very suggestive signs that my simple networks actually end up showing what amounts to quantum behavior—just like in the physics we know.
OK, so how should one set about actually finding the fundamental theory of physics if it’s out there in the computational universe of possible programs? Well, the obvious thing is to just start searching for it, starting with the simplest programs.
 I’ve been doing this—more sporadically than I would like—for the past 15 years or so. And my main discovery so far is that it’s actually quite easy to find programs that aren’t obviously not our universe. There are plenty of programs where space or time are obviously completely different from the way they are in our universe, or there’s some other pathology. But it turns out it’s not so difficult to find candidate universes that aren’t obviously not our universe.
But we’re immediately bitten by computational irreducibility. We can simulate the candidate universe for billions of steps. But we don’t know what it’s going to do—and whether it’s going to grow up to be like our universe, or completely different.
It’s pretty unlikely that in looking at that tiny fragment of the very beginning of a universe we’re going to ever be able to see anything familiar, like a photon. And it’s not at all obvious that we’ll be able to construct any kind of descriptive theory, or effective physics. But in a sense the problem is bizarrely similar to the one we have even in systems like neural networks: there’s computation going on there, but can we identify “conceptual waypoints” from which we can build up a theory that we might understand?
It’s not at all clear our universe has to be understandable at that level, and it’s quite possible that for a very long time we’ll be left in the strange situation of thinking we might have “found our universe” out in the computational universe, but not being sure.
Of course, we might be lucky, and it might be possible to deduce an effective physics, and see that some little program that we found ends up reproducing our whole universe. It would be a remarkable moment for science. But it would immediately raise a host of new questions—like why this universe, and not another?
Box of a Trillion Souls
Right now us humans exist as biological systems. But in the future it’s certainly going to be technologically possible to reproduce all the processes in our brains in some purely digital—computational—form. So insofar as those processes represent “us”, we’re going to be able to be “virtualized” on pretty much any computational substrate. And in this case we might imagine that the whole future of a civilization could wind up in effect as a “box of a trillion souls”.
Inside that box there would be all kinds of computations going on, representing the thoughts and experiences of all those disembodied souls. Those computations would reflect the rich history of our civilization, and all the things that have happened to us. But at some level they wouldn’t be anything special.
It’s perhaps a bit disappointing, but the Principle of Computational Equivalence tells us that ultimately these computations will be no more sophisticated than the ones that go on in all sorts of other systems—even ones with simple rules, and no elaborate history of civilization. Yes, the details will reflect all that history. But in a sense without knowing what to look for—or what to care about—one won’t be able to tell that there’s anything special about it.
OK, but what about for the “souls” themselves? Will one be able to understand their behavior by seeing that they achieve certain purposes? Well, in our current biological existence, we have all sorts of constraints and features that give us goals and purposes. But in a virtualized “uploaded” form, most of these just go away.
I’ve thought quite a bit about how “human” purposes might evolve in such a situation, recognizing, of course, that in virtualized form there’s little difference between human and AI. The disappointing vision is that perhaps the future of our civilization consists in disembodied souls in effect “playing videogames” for the rest of eternity.
But what I’ve slowly realized is that it’s actually quite unrealistic to project our view of goals and purposes from our experience today into that future situation. Imagine talking to someone from a thousand years ago and trying to explain that people in the future would be walking on treadmills every day, or continually sending photographs to their friends. The point is that such activities don’t make sense until the cultural framework around them has developed.
It’s the same story yet again as with trying to characterize what’s interesting or what’s explainable. It relies on the development of a whole network of conceptual waypoints.
Can we imagine what the mathematics of 100 years from now will be like? It depends on concepts we don’t yet know. So similarly if we try to imagine human motivation in the future, it’s going to rely on concepts we don’t know. Our best description from today’s viewpoint might be that those disembodied souls are just “playing videogames”. But to them there might be a whole subtle motivation structure that they could only explain by rewinding all sorts of steps in history and cultural development.
By the way, if we know the fundamental theory of physics then in a sense we can make the virtualization complete, at least in principle: we can just run a simulation of the universe for those disembodied souls. Of course, if that’s what’s happening, then there’s no particular reason it has to be a simulation of our particular universe. It could as well be any universe from out in the computational universe.
Now, as I’ve mentioned, even in any given universe one will never in a sense run out of things to do, or discover. But I suppose I myself at least find it amusing to imagine that at some point those disembodied souls might get bored with just being in a simulated version of our physical universe—and might decide it’s more fun (whatever that means to them) to go out and explore the broader computational universe. Which would mean that in a sense the future of humanity would be an infinite voyage of discovery in the context of none other than A New Kind of Science!
The Economics of the Computational Universe
Long before we have to think about disembodied human souls, we’ll have to confront the issue of what humans should be doing in a world where more and more can be done automatically by AIs. Now in a sense this issue is nothing new: it’s just an extension of the long-running story of technology and automation. But somehow this time it feels different.
And I think the reason is in a sense just that there’s so much out there in the computational universe, that’s so easy to get to. Yes, we can build a machine that automates some particular task. We can even have a general-purpose computer that can be programmed to do a full range of different tasks. But even though these kinds of automation extend what we can do, it still feels like there’s effort that we have to put into them.
But the picture now is different—because in effect what we’re saying is that if we can just define the goal we want to achieve, then everything else will be automatic. All sorts of computation, and, yes, “thinking”, may have to be done, but the idea is that it’s just going to happen, without human effort.
At first, something seems wrong. How could we get all that benefit, without putting in more effort? It’s a bit like asking how nature could manage to make all the complexity it does—even though when we build artifacts, even with great effort, they end up far less complex. The answer, I think, is it’s mining the computational universe. And it’s exactly the same thing for us: by mining the computational universe, we can achieve essentially an unbounded level of automation.
If we look at the important resources in today’s world, many of them still depend on actual materials. And often these materials are literally mined from the Earth. Of course, there are accidents of geography and geology that determine by whom and where that mining can be done. And in the end there’s a limit (if often very large) to the amount of material that’ll ever be available.
But when it comes to the computational universe, there’s in a sense an inexhaustible supply of material—and it’s accessible to anyone. Yes, there are technical issues about how to “do the mining”, and there’s a whole stack of technology associated with doing it well. But the ultimate resource of the computational universe is a global and infinite one. There’s no scarcity and no reason to be “expensive”. One just has to understand that it’s there, and take advantage of it.
The Path to Computational Thinking
Probably the greatest intellectual shift of the past century has been the one towards the computational way of thinking about things. I’ve often said that if one picks almost any field “X”, from archaeology to zoology, then by now there either is, or soon will be, a field called “computational X”—and it’s going to be the future of the field.
I myself have been deeply involved in trying to enable such computational fields, in particular through the development of the Wolfram Language. But I’ve also been interested in what is essentially the meta problem: how should one teach abstract computational thinking, for example to kids? The Wolfram Language is certainly important as a practical tool. But what about the conceptual, theoretical foundations?
Well, that’s where A New Kind of Science comes in. Because at its core it’s discussing the pure abstract phenomenon of computation, independent of its applications to particular fields or tasks. It’s a bit like with elementary mathematics: there are things to teach and understand just to introduce the ideas of mathematical thinking, independent of their specific applications. And so it is too with the core of A New Kind of Science. There are things to learn about the computational universe that give intuition and introduce patterns of computational thinking—quite independent of detailed applications.
One can think of it as a kind of “pre computer science” , or “pre computational X”. Before one gets into discussing the specifics of particular computational processes, one can just study the simple but pure things one finds in the computational universe.
And, yes, even before kids learn to do arithmetic, it’s perfectly possible for them to fill out something like a cellular automaton coloring book—or to execute for themselves or on a computer a whole range of different simple programs. What does it teach? Well, it certainly teaches the idea that there can be definite rules or algorithms for things—and that if one follows them one can create useful and interesting results. And, yes, it helps that systems like cellular automata make obvious visual patterns, that for example one can even find in nature (say on mollusc shells).
As the world becomes more computational—and more things are done by AIs and by mining the computational universe—there’s going to an extremely high value not only in understanding computational thinking, but also in having the kind of intuition that develops from exploring the computational universe and that is, in a sense, the foundation for A New Kind of Science.
What’s Left to Figure Out?
My goal over the decade that I spent writing A New Kind of Science was, as much as possible, to answer all the first round of “obvious questions” about the computational universe. And looking back 15 years later I think that worked out pretty well. Indeed, today, when I wonder about something to do with the computational universe, I find it’s incredibly likely that somewhere in the main text or notes of the book I already said something about it.
But one of the biggest things that’s changed over the past 15 years is that I’ve gradually begun to understand more of the implications of what the book describes. There are lots of specific ideas and discoveries in the book. But in the longer term I think what’s most significant is how they serve as foundations, both practical and conceptual, for a whole range of new things that one can now understand and explore.
But even in terms of the basic science of the computational universe, there are certainly specific results one would still like to get. For example, it would be great to get more evidence for or against the Principle of Computational Equivalence, and its domain of applicability.
Like most general principles in science, the whole epistemological status of the Principle of Computational Equivalence is somewhat complicated. Is it like a mathematical theorem that can be proved? Is it like a law of nature that might (or might not) be true about the universe? Or is it like a definition, say of the very concept of computation? Well, much like, say, the Second Law of Thermodynamics or Evolution by Natural Selection, it’s a combination of these.
But one thing that’s significant is that it’s possible to get concrete evidence for (or against) the Principle of Computational Equivalence. The principle says that even systems with very simple rules should be capable of arbitrarily sophisticated computation—so that in particular they should be able to act as universal computers.
And indeed one of the results of the book is that this is true for one of the simplest possible cellular automata (rule 110). Five years after the book was published I decided to put up a prize for evidence about another case: the simplest conceivably universal Turing machine. And I was very pleased that in just a few months the prize was won, the Turing machine was proved universal, and there was another piece of evidence for the Principle of Computational Equivalence.
There’s a lot to do in developing the applications of A New Kind of Science. There are models to be made of all sorts of systems. There’s technology to be found. Art to be created. There’s also a lot to do in understanding the implications.
But it’s important not to forget the pure investigation of the computational universe. In the analogy of mathematics, there are applications to be pursued. But there’s also a “pure mathematics” that’s worth pursuing in its own right. And so it is with the computational universe: there’s a huge amount to explore just at an abstract level. And indeed (as the title of the book implies) there’s enough to define a whole new kind of science: a pure science of the computational universe. And it’s the opening of that new kind of science that I think is the core achievement of A New Kind of Science—and the one of which I am most proud.
For the 10th anniversary of A New Kind of Science, I wrote three posts:
 It’s Been 10 Years: What’s Happened with A New Kind of Science? 
 Living a Paradigm Shift: Looking Back on Reactions to A New Kind of Science  
 Looking to the Future of A New Kind of Science  
 The complete high-resolution A New Kind of Science is now available on the web. There are also a limited number of print copies of the book still available (all individually coded!).
A New Kind of Science: A 15-Year View—Stephen Wolfram Writings
https://writings.stephenwolfram.com/2017/05/a-new-kind-of-science-a-15-year-view/