[math-ph]

# Many-Worlds Interpretation of Quantum Mechanics:
A Paradoxical Picture

Amir Abbass Varshovi
ab.varshovi@sci.ui.ac.ir/amirabbassv<EMAIL_ADDRESS>Faculty of
Mathematics and Statistics, Department of Applied Mathematics and Computer
Science, University of Isfahan, Isfahan, IRAN.
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box: 19395-5746, Tehran, IRAN.

###### Abstract

Abstract: The many-worlds interpretation (MWI) of quantum mechanics is studied
from an unprecedented ontological perspective based on the reality of (semi-)
deterministic parallel worlds in the interpretation. It is demonstrated that
with thanks to the uncertainty principle there would be no consistent way to
specify the correct ontology of the Universe, hence the MWI is subject to an
inherent contradiction which claims that the world where we live in is unreal.

Keywords: Many-Worlds Interpretation of Quantum Mechanics, (Semi-)
Deterministic World, Fiducial World, Inclusive Consciousness.

##  I Introduction

About a century after its inception, quantum mechanics is still in a
perplexity to provide a precise comprehensive interpretation of its peculiar
features. Even though, the orthodox interpretation of quantum theory, the so
called Copenhagen interpretation, soon gained a worldwide acceptance for
describing the quantum behaviors of the objective world around us, this
quandary is still felt in the basics of quantum mechanics cabello . The most
profound consideration to this dilemma dates back to 1930s when the Einstein-
Podolsky-Rosen criticism of the Copenhagen interpretation of the theory, the
EPR paradox epr , revealed an unexpected aspect of quantum physics which
violates the main principle of special relativity allowing information to be
transmitted faster than light. This phenomenon that was the most serious
objection to the interpretation until then, was referred to as the
_entanglement_ effect by Schrodinger sch3 .

However, despite to the responses and the paraphrases of the founding fathers
of quantum mechanics bohr ; sch1 ; sch2 , the EPR paradox led to significant
implications to the interpretation and yield the birth of other intellectual
paradoxes in next years most of which were fundamentally rooted in the
elaborately planned Schrodinger’s cat experiment and its ontological
expression of reality.111This experiment is an attempt to explain the reality
of the counterintuitive nature of quantum mechanics by transferring the
superposition states to macroscopic objects sch3 . Here we should emphasize
that through this article by _reality_ we mean the _observed reality_ defined
based on the practical features of the pragmatic worldview in the literature
epr ; heisenberg-ph . Thus, hereby we hinder the subjective speculations with
unobserved effects, including the hidden variables and the QBism
interpretation of quantum mechanics. We acknowledge that via the Copenhagen
interpretation the observable reality undergoes an ontological revision in
favor of the Heisenberg’s uncertainty principle and its inevitable
consequence, i.e. the non-locality. (For a more detailed discussion on
ontological and epistemological perspectives of reality and the relevant
semantic disputes see also penrose2 ; dewitt .) Among them, as an immediate
generalization of the Schrodinger’s cat paradox to the consciousness problem,
the Wigner’s friend experiment wigner ; deutsch , has been considered
delicately in some extended scenarios leading to critical discoveries of
serious intellectual inconsistencies in the Copenhagen interpretation fr0 ;
sudbery ; fr ; brukner1 ; brukner2 ; laloe ; healey1 ; healey2 ; dieks ;
lombardi .222Even, revisiting the Schrodinger’s cat experiment itself, in some
definite viewpoints, has yield some paradoxical aspects of the Copenhagen
interpretation such as the problem of quantum suicide and immortality lewis0 ;
lewis . See also hobson  for more detailed discussions/suggestions and castel
for fatal features of the experiment. 333Other intellectual inconsistencies of
the Copenhagen interpretation are listed in diverse realms of quantum physics
such as elementary particle physics (the Hardy’s paradox) hardy1 ; hardy2 ;
hardy3 ; hardy4 , quantum cosmology cos1 ; cos2  (see also shestakova  for a
counterproposal viewpoint), the Hawking’s paradox hawking ; susskind  and
other inconsistencies in quantum viewpoints to general relativity pavsic ;
time ; camacho  (besides the well-known contradictions of quantum gravity).
For a more complete list of such paradoxes and their explanations see smolin ;
qmparadox .

Although the subsequent experiments confirmed the completeness of the
Copenhagen interpretation despite the EPR paradox claim aspe1 ; aspe2 , recent
developments of experimental physics have established some inconsistencies of
the interpretation by running quite complex empirical tests in last two
decades. As one of the pioneers, the Afshar’s experiment afshar1 ; afshar2
reported a controversial violation of the wave-particle duality in the early
years of the present century. A few years lated weak measurements hardy-exp1 ;
hardy-exp2 ; hardy-exp3 ; hardy-exp4  also revealed objective contradictions
in particle-anti-particle annihilation process due to the Hardy’s paradox
hardy1 ; hardy2 ; hardy3 ; hardy4 . Even more recently, some spectacular
experiments, fundamentally designed based on the extended versions of the
Wigner’s friend scenario, released fatal conflicts in basic foundations of the
Copenhagen interpretation proietti ; kok . All in all, these facts prove the
obligation of a basic reconsideration to the Copenhagen interpretation of
quantum mechanics.

Actually, after releasing the EPR paradox, the Copenhagen interpretation as
the only reliable acknowledged expression of quantum mechanics in its time,
was gradually distrusted in next years and encountered with diverse rival
interpretations which were proposed to explain the deceptive effects of the
theory from distinct outlooks and philosophies. Among them, the Many-Worlds
interpretation (MWI) of quantum mechanics, proposed by Hugh Everett in 1950s
everett1 ; everett2 ,444And was popularized by Bryce DeWitt during 1970s
everett4 ; dewitt-mwi . has widely been considered as the most reliable
alternative for the Copenhagen interpretation everett3 ; tegmark . Unlike the
Copenhagen interpretation which embraces a single world and its observable
reality, the MWI construes the Universe as an expanding cluster of infinitely
many increasingly divergent parallel worlds. It is, in principle, a
deterministic interpretation which includes the non-locality by introducing
parallel worlds to realize or to embody each possible history of outcomes of
quantum mechanical measurements everett1 .555In this sense, the _Universe_ is
sometimes referred to as the _Multiverse_. Also what we call _world_ here is
seldom addressed to as _universe_. Actually, through this paper we follow the
terminology of vaidman . In particular, the status of the wavefunction for the
MWI is ontological in contrast to that in the Copenhagen interpretation as
epistemic, summarizing information about the results of measurements becker .

Actually, in the MWI, despite of the Copenhagen interpretation, the central
role of observer in the process of quantum measurement is disclaimed by
replacing wavefunction collapse with quantum decoherence. On the other hand,
the observer’s role lies at the central part of all paradoxes listed above.
Hence, the MWI is considered as one of the most plausible adequate
alternatives to resolve the inconsistencies of the Copenhagen interpretation
such as those of the EPR paradox, the Schrödinger’s cat and the Wigner’s
friend thought experiments,666Including their extended scenarios, e.g. the
Frauchiger-Renner’s discussion fr0 ; fr . See also the Brukner’s brukner1 ;
brukner2  and the Healey’s healey1  expression of the experiment. the Hardy’s
paradox and even the wave-particle duality.777See mwi-epr  for the EPR
paradox. Also, see mwi-oxford  for more discussions about the Schrodinger’s
cat and the Wigner’s friend experiments..

However, despite of the proposals that put forward to assess the practical
accuracy of the MWI test1 ; test2 ; test3 , it does not seem to be testable
with current and any foreseeable future experimental capability deutsch ;
vaidman . In fact, all such tests rely on contrived conditions and idealized
circumstances which would be infeasible or exceedingly difficult to realize in
the real world. Also, as the most interesting candidate scenario to
distinguish the MWI from all single world-base interpretations of quantum
mechanics, the quantum suicide gedankenexperiment, would not also work in the
real world suicide . Therefore, the theoretical premises of testability of the
MWI are controversial even among the proponents of the interpretation dewitt .
Although it will threaten the refutability of the interpretation,888But,
however, there are other claims and counterproposals against this failure
vaidman . it is also seen as an opportunity for the MWI not to be criticized
or falsified practically.

On the other hand, in contrast to empirical confusion about the MWI, it has
already faced to a number of intellectual criticisms including the
philosophical objection due to the law of parsimony,999Also known as the
Occam’s razor. See vaidman  for more discussions. the preferred basis
ambiguity,101010See wallace2010  for the critique and its solution. the
ontological insufficiency of the wavefunction of the Universe,111111See bell ;
allorri  for the detail of the objection and vaidman  for the reply. and the
incoherence probability problem.121212See wallace2003  for the analysis of the
problem and its solution. However, it is purported by the supporters of the
MWI that all such critiques stem in incomplete viewpoints and
misunderstandings vaidman . Hence, it seems that the MWI can keep developing
increasingly without any serious experimental/intellectual obstacle. But,
actually, in contrast to such privileges, the MWI is still far from being a
widely accepted interpretation just like the Copenhagen interpretation is,
since there would be no easily reachable consensus on its ontological
perspective based on introducing infinitely many worlds that we do not see.

To overcome the dilemma arisen in understanding quantum mechanics via the
unprovable MWI, it is mandatory to figure out all its possible aspects with a
careful scrutiny of the very assumptions and a comprehensive contemplation of
the ontological results. Only then after we can argue and judge about the
possibility of replacing the Copenhagen interpretation with the MWI for
explaining quantum physics. It is the main objective of this paper to make
some troubling features of the MWI more apparent and to shed more light on its
reliability. Actually, through this research we will report an inherent
intellectual paradox in the very ontological foundations of the MWI by
enquiring: _Who can judge about the reality of other parallel worlds?_ But
here, the interrogative pronoun ” _Who_ ” refers to sentient beings of all
parallel worlds introduced in the MWI and the whole question enquires: _Whose
interpretation is correct?_ Despite of its philosophical statement we try to
give a mathematical response to the question. We will discover a fundamental
conflict among the ontological assumptions of the MWI leading to unreality of
the world where we live in.

##  II Many-Worlds Interpretation: Illustrative or Seductive

Despite its abilities to describe a significant portion of quantum mechanical
results, the MWI is substantially a paradoxical approach. In other words,
although the MWI is the second most popular and thought-provoking way to
interpret quantum mechanics tegmark , a careful investigation will find it to
include a contradiction in its basic assumptions. In fact, this inconsistency
is rooted in ascribing a real world to each possible history of measurement
outcomes. More precisely, this assumption leads to a serious problem for
defining the reference worlds whose sentient beings are eligible and capable
to judge about the ontological aspects of quantum physics. Since, in
principle, assuming the MWI there would be some parallel worlds whose
inhabitants will totally oppose to register the MWI for describing the
Universe. This, in principle, shows that the only way to accept the MWI is to
rely on the reference worlds. But, it is established here that assuming a
reference world (or a group of reference worlds) is theoretically impossible,
hence, the MWI is a paradoxical picture.

Therefore, based on its very assumptions, the MWI is not rationally
acceptable, unless it undergoes a fundamental revision within a firm
framework. Such a framework, as we see below, should include an overall
assessment of the Universe as the ensemble of all parallel worlds in the MWI
and their embraced registered established theories of physics. This, in its
turn may cause a statistical/stochastic investigation on various theories of
physics, with almost no limit for exotic laws of nature, to study the aspects
of reality and the promised comprehensive interpretation of quantum physics.
However, this extreme complexity shows that the MWI will have a long way to go
before it can be accepted theoretically.

To prove the claim mentioned above with details we have to consider two
following essential enquiries:

Q1- _Assume that the MWI is correct. Are there some parallel worlds whose
agents reject the existence of some other parallel worlds based on the history
of their measurement outcomes?_

Q2- _Assume that the MWI is correct. Are the whole ontologies of the possible
universes described by the sentient beings of all parallel worlds (based on
their observed reality) equivalent in every way?_ 131313Actually, the
_possible universe_ for any agent is the largest cluster of parallel worlds
each of which is feasible based on the laws of nature discovered from the
history of the agent’s measurement outcomes. For instance, for us, the
possible universe is equivalent to the _Universe_ as a whole, just as we
assume in the MWI. On the other hand, our possible universe excludes the
parallel worlds with no experience of Pauli’s exclusion principle. Upon the
laws of our theory of physics, such worlds are not real and do not belong to
the Universe.

We will prove that the enquiries Q1 and Q2 both do have positive responses,
leading us to the following paradoxical conclusion:

Conclusion: _Assuming the MWI as a correct theory, there would be a parallel
world whose agents acquire a correct complete description of the possible
universe and reject some parts of the ontological claims of the MWI. Hence,
some parallel worlds assumed in the MWI, including ours, are both real (by our
ontology) and unreal (based on the ontology of the agents) simultaneously!_

More precisely, the above conclusion asserts that the ontology of the MWI
rejects itself, hence an inherent conflict. Our arguments to prove the
mentioned conclusion are as following:

Q1: This inquiry can be rewritten within the following wording:

_Assuming the MWI as a correct theory, is there any agent in some parallel
world whose description of the possible universe (based on the reality
observed in her/his world) disclaims the existence of some other parallel
worlds and consequently rejects the ontology of the MWI?_

Reply: YES!

Reasoning: Actually, one can easily build up real worlds, based on the MWI
ontology, which reject some acknowledged quantum mechanical aspects
(principles) and embrace an entirely different theory of physics. In fact,
such peculiar physics may include semi-deterministic theories,141414By semi-
deterministic theories or semi-deterministic worlds we mean the parallel
worlds in which some sort of quantum mechanical measurements (specially, those
which are implemented in laboratories and preferably have no immediate results
on biological status of the world, if there is one) gain absolutely predicted
outcomes. For example, we can assume the world in which all measurements of
the electron spin of the superposition states lead to a single result, but the
rest experiments of quantum mechanics retain the usual probabilistic features.
deterministic theories,151515By deterministic theories or deterministic worlds
we mean the parallel worlds in which all quantum mechanical tests lead to
definite predicted results (from a moment onward and for all superposition
states). In fact, a deterministic world is defined by employing the inductive
approach to semi-deterministic worlds. deformed quantum theories,161616By
deformed quantum theories or deformed quantum worlds we mean the parallel
worlds with different probabilities of observations. For example, we may
consider a world in which the probability of observing
$+\frac{1}{2}\hbar$-spin electron, i.e. $\left|\uparrow\right\rangle$, is not
equal to $|a|^{2}$ in the superposition state
$\left|\psi\right\rangle=a\left|\uparrow\right\rangle+b\left|\downarrow\right\rangle$,
for each $a,b\in\mathbb{C}$ with $|a|^{2}+|b|^{2}=1$. This quantum theory,
although adheres to the uncertainty principle and may be described with (a
likely modified version of) the Schrodinger’s equation, should be equipped
with a distinct Hilbert space and its corresponding mathematics for electrons.
Also, in (FIG. 1) a deformed quantum theory (world) is depicted wherein the
probability of observing an upward-spin electron $\left|\uparrow\right\rangle$
(with $+\frac{1}{2}\hbar$ spin on the $z$-axis) is twice to that of the
downward-spin electron $\left|\downarrow\right\rangle$ (with
$-\frac{1}{2}\hbar$ spin, measured on the same axis) in the superposition
state
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|\uparrow\right\rangle+\frac{1}{\sqrt{2}}\left|\downarrow\right\rangle$
(the blue path in (FIG. 1)). However, despite to the different formulation of
such a theory, with thanks to the uncertainty principle, this world may
embrace the ontology of the MWI. and in general manually adjusted
laws.171717By manually adjusted laws or manually adjusted worlds we mean
parallel worlds in the MWI wherein the probabilities of measurement outcomes
have been adjusted manually to capture (almost) controlled laws of nature. For
example, we can assume a real world in which the electron spin of any
superposition state is $+\frac{1}{2}\hbar$ during autumn and winter and
$-\frac{1}{2}\hbar$ in spring and summer. All in all, according to the MWI
there do exist real parallel worlds in which some of the objective realities
of ours181818The possessive pronoun ”ours” (just like ”our” and ”us”) is used
in this article to address to the author’s and the readers’ ”world” and
”nature”, until the moment we can think and speculate about the MWI. are
concealed or eliminated from the cognition of their sentient beings.
Therefore, even if in such a parallel world there are evidences for existence
of other parallel worlds and the possible universe, its agents’ interpretation
of the corresponding ontology will not be compatible to that of ours.

Specifically, these incompatibilities occur in (semi-) deterministic worlds.
As described above, such worlds can be designed by arranging the recorded
history of outcomes of measurements to gain some controlled laws of nature
with certain predictability. An example of a semi-deterministic
world191919Actually, any parallel world which (from a definite moment onward)
recognizes a clear ordering in some sort of quantum mechanical tests (e.g.
measurement of the spin of electrons) is semi-deterministic. More precisely, a
semi-deterministic world is a parallel world which violates the Heisenberg’s
uncertainty principle for one or more quantum observables. is the _world of
upward-spin electrons_ wherein all superposition states of electron are
measured as $+\frac{1}{2}\hbar$-spin in experiments.202020Through this paper,
all spin measurements are assumed to be performed on the $z$-axis.
Practically, there would be no superposition amongst _up_ and _down_ spin
states for electrons,212121Obviously, assuming such a world is permitted in
the MWI. In particular, by the uncertainty principle the state vector of any
electron is actually (almost always) a non-trivial superposition of
$+\frac{1}{2}\hbar$-spin and $-\frac{1}{2}\hbar$-spin states. Thus, it can be
claimed that there would be no evidence for (non-entangled) electrons of
$-\frac{1}{2}\hbar$-spin in this world. (However, since the probability of
finding a pure state of $-\frac{1}{2}\hbar$-spin electron in nature is
ignorable, it would be neglected statistically and be reported as an
experimental error whenever observed.) and thus the spinor structure of
electron (i.e. the existence and the reality of electrons of
$-\frac{1}{2}\hbar$ spin) is ruled out in action (see (FIG. 1)).

Figure 1: The Many-Worlds Interpretation of the Universe: A schematic diagram
for the MWI of quantum mechanics of electron spins (measured on the $z$-axis)
and the corresponding Universe. The simplest example of the MWI generated by a
sequence of measurements on the superposition state
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|\uparrow\right\rangle+\frac{1}{\sqrt{2}}\left|\downarrow\right\rangle$.
The horizontal axis indicates the direction of time and the vertical axis
exhibits the amount of the measured electron spin. Any single vertex displays
a measurement in which the outcomes are drawn as two segments: Each segment
with positive slope correspond to observing a $+\frac{1}{2}\hbar$ spin
($\left|\uparrow\right\rangle$) and that with negative slopes is related to
$-\frac{1}{2}\hbar$ one ($\left|\downarrow\right\rangle$). Hence, any possible
right-going zigzag path depicts a parallel world in the MWI. The straight
paths seen at the extreme edges of the lattice (the red line and the purple
line) are actually illustrating two (semi-) deterministic worlds with the only
$+\frac{1}{2}\hbar$-spin (red) or $-\frac{1}{2}\hbar$-spin (purple) electrons.
The blue path also shows a non-deterministic world with a different theory of
quantum mechanics in which the possibility of observing $+\frac{1}{2}$-spin
electrons is twice to that of $-\frac{1}{2}$-spin ones.

More precisely, according to the MWI assumptions we can find real (semi-)
deterministic worlds with different objective features of reality, hence with
restricted deformed ontology of the possible universe.222222Even though this
can endanger the existence of any inhabitant sentient beings and generally of
life (as we acknowledge it) in such worlds, it is an inevitable consequence of
the MWI of quantum mechanics. For instance, the sentient beings of the world
of upward-spin electrons will not even think about the existence of parallel
worlds with downward electron spins, just like we do not consider a parallel
world of electrons with negative mass.232323We should note that assuming the
existence of such exotic worlds is out of the fundamentals of the MWI, wherein
the splitting of parallel worlds take place for all possible outcomes of
measurements, as we verify. Hence, assuming MWI, there would be parallel
worlds (e.g. (semi-) deterministic worlds) with exotic interpretations of the
possible universe (based on their cognition and upon their ontological
perspective) which decline the ontology of the MWI. Q.E.D.

Let us now discuss the next enquiry:

Q2: This question can be recounted as the following restatement:

_Assuming the MWI as a correct theory, is there any obstruction to define a
(practical or semantic) dissociation method on parallel worlds of the MWI (and
their inhabitant agents) to recognize the most comprehensive ontology and its
possible universe?_

Reply: YES!

Reasoning: In fact, we have to prove that there would be no well-defined way
to specify some definite fiducial worlds whose sentient inhabitants are the
only agents who are eligible for describing the Universe. This assertion is
the most fatal consequence of the assumptions of the MWI. More precisely, the
main difficulty with the MWI emerges just when one tries to compare the
awareness and the ontology of the inhabitants of faithful worlds (i.e. the
worlds in which the history of all quantum mechanical tests confirm the
Heisenberg’s uncertainty principle242424One should note that this definition
also contains some of the worlds with deformed versions of quantum mechanics.
More precisely, a faithful world is a parallel world which its agents may
embrace the MWI entirely, just as e.g. Hugh Everett do in our world.) and of
the agents who are resident in other parallel worlds. Actually, since
according to the assumptions of the MWI none of the parallel worlds is
substantially the superior, then there would be no predetermined
distinguishing criteria to prefer and to admit quantum mechanical description
of the possible universe (the MWI) and to discard that of the (semi-)
determinist agents.252525Perhaps one may speculate about the _measure of
existence_ as an appropriate criteria to recognize the fiducial worlds vaidman
; everett2 ; test2  by assuming; _A fiducial world would have the most measure
of existence_. But actually, this definition will not work properly, since
there are (semi-) deterministic worlds which their measure of existence is
equal to (or even is larger than) that of a faithful world. For example, all
parallel worlds in (FIG. 1) are of the same measure of existence. Moreover, if
we replace
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|\uparrow\right\rangle+\frac{1}{\sqrt{2}}\left|\downarrow\right\rangle$
with
$\left|\psi\right\rangle=\sqrt{\frac{2}{3}}\left|\uparrow\right\rangle+\frac{1}{\sqrt{3}}\left|\downarrow\right\rangle$
in (FIG. 1), then the (semi-) deterministic world depicted with the red line
at the top of the figure would have an absolutely larger measure of existence
rather than any faithful world.

In other words, upon its very assumptions, the MWI is not capable to
distinguish fiducial references amongst its claimed parallel worlds and hence
is not able to recognize the correct interpretation of the possible universe.
That is assuming any preference or priority for specifying some parallel
worlds would be an unprecedented hypothesis. Here, we show that this
assumption is also inherently inconsistent provided the MWI be correct.
Actually, the machinery we introduce here can be applied to any other
concept/characteristic which is defined upon the history of measurement
outcomes for separating parallel worlds. More precisely, our analysis here
demonstrates that the parallel worlds can never be distinguished by any
statistical subjective/objective characteristic.

At the first glance, one should note that ascribing any separating
characteristic to a group of specific real worlds and their inhabitants, is
theoretically equivalent to append a consciousness-like attribute to the
sentient beings who accomplish measurements in the Winger-von Neumann
interpretation of wave collapse wigner ; vonneumann . Just as consciousness
separates the observer from the cat in the Schrodinger’s cat experiment, the
inclusive consciousness, if exists, must separate the fiducial worlds from the
other ones in the MWI. Let us refer to this characteristic of the fiducial
worlds, as the _inclusive consciousness_. Obviously, since the MWI is assumed
to be a correct theory, the fiducial worlds who enjoy this property the most
must respect the uncertainty principle, hence being the faithful worlds.

Actually, as we show in the following, the inclusive consciousness is a
superfluous concept with no specific efficacy. To see this clearly we first
show that the inclusive consciousness has not to be specified to some definite
individuals, but it must be continuously distributed over all parallel worlds.
Actually, this is the immediate consequence of physical laws as statistical
consequences. Thereafter, we will mathematically prove that the distribution
is inevitably a constant function, which in its turn, leads to uselessness of
the concept, indicating all parallel worlds have the same amount of inclusive
consciousness and their sentient beings acquire enough eligibility to describe
the ontology of the possible universe. In fact, the monotony of the
distribution is inferred by considering the Universe of (FIG. 1) (as a simple
example of the MWI) and stems in the fact that both the set of faithful worlds
(fiducial worlds) and that of the (semi-) deterministic worlds are dense in
the Universe.

In the following we prove the above claims accordingly:

1- Inclusive Consciousness is a Continuous Distribution

To approve that the inclusive consciousness is a continuous distribution over
the Universe, one should first note that a fiducial world is a world whose
sentient inhabitants confirm the uncertainty principle of quantum mechanics
based on the recorded statistics (history) of all quantum mechanical
measurement outcomes. On the other hand, based on its statistical essence, the
uncertainty principle needs enough time for performing sufficient numbers of
experiments to be registered or dismissed in a parallel world.262626We should
note that this is a theoretical fact which does not depend on the extent of
inhabitants cognition and ingenuity. Then, the inclusive consciousness, arises
or disappears over time. Consequently, a semi-deterministic world, separated
from a faithful world (a fiducial world) at some specific branching point,
also carries inclusive consciousness and there would be a no definite time for
losing it thoroughly. Hence, a (semi-) deterministic world have some inclusive
consciousness depending on how much it is close (in some sense) to a faithful
world.

In particular, observing the only upward spins in a (semi-) deterministic
world could be statistically interpreted as a finite portion of a longer
sequence of observations which confirm the uncertainty principle as a whole.
In other words, at each finite specific moment none can talk about absolute
deterministic worlds in the Universe. The only thing we can probably comment
on is how much a world is statistically inclined to be (semi-) deterministic.
We assume that the more of this tendency a parallel world may have (upon the
history of measurement outcomes), the less inclusive consciousness it would
include. Actually, according to the statistical essence of the uncertainty
principle this is the only possible way to define inclusive consciousness in a
consistent disposition. Consequently, to introduce inclusive consciousness
correctly one has no choice except employing a continuous distribution over
the parallel worlds.

2- Inclusive Consciousness is a Constant Function

Let us redefine that the concept of inclusive consciousness so that it could
be evaluated with real numbers272727The formulation which we workout here is
too flexible to be generalized to any topological set such as $\mathbb{C}$,
$\mathbb{H}$, and $\mathbb{R}^{n}$, $n>1$. assigned to parallel worlds to
determine the credit rate or the amount of validity of their inhabitants’
understanding of the uncertainty principle and the MWI.282828We should note
that this is not a psychological assessment, but an evaluation based on the
statistics of the history of the outcomes of quantum mechanical tests.
Therefore, considering $U$ as the set of all parallel worlds, the inclusive
consciousness is evaluated with a continuous function, say $f:U\to\mathbb{R}$,
which takes the same value for faithful worlds. Moreover, for parallel worlds
$w_{1}$ and $w_{2}$, we must have $f(w_{2})>f(w_{1})$ if and only if the
ontology of the inhabitants of $w_{2}$ is more comprehensive and more accurate
with respect to that of the residents of $w_{1}$. Thus, we may assume
$f(w_{2})>f(w_{1})$ whenever $U_{1}\subset U_{2}\subseteq U$, wherein $U_{1}$
(resp. $U_{2}$) is the possible universe of the agents of $w_{1}$ (resp.
$w_{2}$).292929 For example, the world of upward-spin electrons (which its
sentient beings agree with us on all quantum concepts except the spin of
electrons), as a semi-deterministic world, encompasses more inclusive
consciousness with respect to a deterministic world (which its inhabitants do
not realize any evidence to figure out and to trust in quantum mechanics and
the uncertainty principle), and we hope that $f$ could recognize this
dissimilitude. For instance, if $w_{2}$ is a faithful world and $w_{1}$ is a
semi-deterministic world we should find:303030Similarly, if $w_{2}$ is a semi-
deterministic world and $w_{1}$ is a deterministic world once again we obtain
$f(w_{2})>f(w_{1})$.

$f(w_{2})>f(w_{1}).$ (II.1)

In the following we show that $f$ violates (II.1) for the Universe depicted in
(FIG. 1). In order to do this, we must first study $U$ of (FIG. 1) with some
more details. Actually, here $U$ can be considered as the Universe generated
by an agent (and its copies in parallel worlds) during the process of
measuring the spins of elements of an ensemble which includes infinitely many
copies of the superposition state
$\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left|\uparrow\right\rangle+\frac{1}{\sqrt{2}}\left|\downarrow\right\rangle$
of electron.

Initially, we claim that there exists an isomorphism between $U$ and $[0,1]$.
To see this let us assume that the ensemble elements are labeled with natural
numbers $\mathbb{N}$. Then, if we assign a $1$ to each $+\frac{1}{2}\hbar$
outcome, and a $0$ to any observed $-\frac{1}{2}\hbar$, each parallel world is
uniquely shown with a sequence of integers $1$ and $0$. For instance, the blue
path (world) in (FIG. 1) depicts $01101101100111010111\cdots$. Let us assume
these sequences as decimal digits in base two, e.g. the above blue world be
denoted as $0.01101101100111010111\cdots$ in base two. Actually, this will
assign a unique real number in $[0,1]$ to each parallel world.313131We note
that $0=0.000\cdots$ and $1=0.111\cdots$. In fact, the former is depicted with
the purple line and the later one is shown by the red line in (FIG. 1).
Conversely, each number of $[0,1]$ is also indicating a parallel world in
$U$.323232It is the immediate consequence of our assumption: the ensemble
contains infinitely many elements. Consequently, one simply finds; $U=[0,1]$.
Thus, from now on any parallel world is understood as a real number in
$[0,1]$, and the inclusive consciousness is considered to be evaluated with a
real function as; $f:[0,1]\to\mathbb{R}$. Although, this redefinition of $f$
may cause its continuity to be lost, but fortunately, it can be shown that $f$
is still a continuous function.

To study the continuity of $f$ let us compare the topology of $U$ and $[0,1]$.
Note that the closer the two points are to each other in $[0,1]$, their more
decimal digits will be equal to. On the other hand, the more equal decimal
digits two parallel worlds have, the longer common history (and statistics) of
experiments they have passed. Therefore, the standard topology of $[0,1]$
coincides with the overall understanding of the correlations of the parallel
worlds: _The distance of two points in $[0,1]$ reveals the disagreement of the
cognitions about the objective reality and the disparity of the ontologies of
the possible universe the agents of the two corresponding worlds may gain
based on their prior experiments_. Hence, the standard topology of $[0,1]$
respects the topological structure of $U$ from the ontological viewpoint. Due
to this property, once again we consider $f:[0,1]\to\mathbb{R}$ as a
continuous function: _The more compatible ontologies (interpretations of the
possible universe) the agents of two worlds may obtain, the closer evaluation
of inclusive consciousness they will have_.

A (semi-) deterministic world, as we consider here,333333Since we restrict
ourselves to investigate the Universe generated by observing the spin of
electrons and we ignore all other quantum mechanical observables, the semi-
deterministic worlds are in fact deterministic. is in fact, a parallel world
in which the electron spin measurements have absolutely predicted outcomes
after some specific moment. In general, from a definite moment onward,
observing any perceivable order in the history of the measured spins will
result in a deterministic world.343434Actually, if in such a parallel world
after a few number of first experiments a definite order is observed in the
outcomes of measurements of the spin of electrons, then the uncertainty
principle is excluded and we are encountered with a (semi-) deterministic
world. As a simple example, if the outcome of even tests is
$+\frac{1}{2}\hbar$-spin and the result of odd ones is $-\frac{1}{2}\hbar$
after a specific moment, then we have a (semi-) deterministic world. This
means that any deterministic world is actually shown with some $r\in[0,1]$
which after a number of its first decimal digits, its other decimal digits
follow a certain order expressed by repeating a fixed sequence over and over.
That is, $r$ has to be a rational number; $r\in\mathbb{Q}$ hosch .

On the other hand, in faithful worlds, with thanks to the uncertainty
principle, there must be no definite order in measurement outcomes. Thus, any
faithful world is shown with a real number $s\in[0,1]$ which its decimal
digits, will not show any order created by repeating some sequence, hence is
an irrational number; $s\in\mathbb{Q}^{\prime}$. Consequently, upon our
definition $f$ must take the same value for irrational numbers of $[0,1]$.

Now we are ready to study the analytic structure of $f$. Let us see what we
already know about it so far:

1) $f:[0,1]\to\mathbb{R}$ is a continuous function.

2) $f:[0,1]\to\mathbb{R}$ takes the same value for irrational numbers.
According to two above properties we have no choice but to assume $f$ as a
constant function, the fact which violates (II.1). This means that the
inclusive consciousness of all (semi-) deterministic words must be equal to
that of the faithful ones in the Universe of (FIG. 1). In general, by
employing the inductive process for the Universe of several quantum
observables we see that no matter how much of them are turned to be
predictable, the value of $f$ will still not make a difference.353535The
analysis method introduced here can be generalized to any numbers of quantum
variables by inductive process. To prove this claim, at the first step we have
to show that for any given quantum observable with finitely many possible
outcomes the inclusive consciousness is evaluated similarly with a continuous
function, i.e. $f:[0,1]\to\mathbb{R}$. Actually, in order to display any
parallel world in $U$ with a real number of [0,1] for an observable with $n$
possible outcomes it is sufficient to encode the observed results with
integers 0, 1, 2, $\cdots,n-1$, and write down the history of outcomes as
decimal numbers in base $n$, just as we did here for the spin of electron.
Therefore, $U$ and its topology is thoroughly characterized with $[0,1]$.
Moreover, one can simply generalize this formulation for the Universe
generated by several numbers of quantum observables. In fact, for $m$
different kinds of quantum mechanical tests the Universe $U$ is encoded with
points of $[0,1]^{m}$. Obviously, hereby any (semi-) deterministic world is
shown by a $m$-plet with at least one rational component. Once again we see
that both the set of (semi-) deterministic worlds and that of the faithful
ones are dense in $U$, and consequently, the inclusive consciousness must also
be uniformly distributed over $U$. That is, $f$ remains constant over all
possible parallel worlds in the entire Universe of the MWI. This proves that
the inclusive consciousness is an irrelevant null concept for the MWI, just
like the ether was included in classical electrodynamics. Hence, assuming the
MWI, all agents in all parallel worlds are equally competent from ontological
viewpoints. Q.E.D.

Conclusion: According to the positive responses to Q1 and Q2, we conclude that
some parallel worlds in the Universe should have paradoxical existence. Since,
on the one hand, upon the arguments to Q1, embracing the MWI leads to birth of
(semi-) deterministic worlds and their sentient beings interpretations of the
possible universe that reject the existence of some parallel worlds with
unreal (upon their cognition) features. Specially, the reality of the world
which we live in is itself rejected by the ontology of (semi-) determinist
agents. On the other hand, due to the argumentations of Q2, assuming the MWI,
all interpretations of the possible universe reported by the sentient beings
of different parallel worlds, based on their cognition and contemplation about
the statistics (history) of quantum mechanical observations, are equally
correct from the ontological perspective. Altogether, we conclude that some
parallel worlds must and must not exist simultaneously. Surprisingly, the
unreality of our world (from the viewpoint of (semi-) determinist agents) is a
true fact, just as we recognize it. Hence, the MWI includes paradoxical
propositions within its very assumptions.

##  III Summary and Discussion

In this paper we argued that the MWI is inherently a paradoxical picture
because of its exaggerated ontological perspective to the Universe. Actually,
considering any possible parallel world to be a real entity would lead to
theoretically incontrollable situations such as the violation of the laws of
nature as we verify as quantum physics in our world. The most fatal situation
arises when we consider the (semi-) deterministic worlds (which statistically
reject the Heisenberg’s uncertainty principle) as real. Since, it was shown
that there would be no well-defined way to discard their agents’ ontology as
an incorrect picture of the possible universe (even if the measure of
existence of such (semi-) deterministic parallel worlds are ignorable).
Therefore, there are parallel worlds which are both real (due to our ontology)
and unreal (via the ontology of agents in some (demi-) deterministic world)
simultaneously. Among them, the most controversial case, is our world which is
actually unreal from the ontological perspective of any (semi-) deterministic
world.

The above argument shows that the MWI is inherently a paradoxical picture of
quantum mechanics. To resolve this conflict one may speculate about a number
of thought-provoking proposals. As the first solution, one may think about the
_chance of detecting (semi-) deterministic worlds_ amongst the Universe, i.e.
_Since the chance of detecting a (semi-) deterministic world is fairly zero,
then we could ignore its agents’ ontology_. However, although the chance of
finding a (semi-) deterministic world among the Universe is ignorable, but
based on the ontological perspective of the MWI, such worlds exist actually
and are as real as ours as we perceive and cognize it. Thus, despite the
ontologies of (semi-) determinist agents are proposed less frequently,
according to the argument of enquiry Q2 they release true pictures of the
possible universe. Therefore, it seems that the chance of detection would not
resolve the discovered paradox properly.

Also one may contemplate about a proposal for _relativity in ontology_ of
parallel worlds, assuming no conflict in disparity of ontologies via
transferring from a parallel world to another one. That is, we should trust in
all possible universes proposed by the agents of the parallel worlds in the
MWI as relative ontologies. However, if we try to rely on this proposal, we
must in turn admit more complete universes with respect to what we consider in
the MWI, and hence embrace any illusive theory of nature (illusive world)
which is unreal based on the objective reality we cognize in our world. For
instance, we can assume the Pauli’s exclusion principle as a deterministic law
imposed to our world among a larger possible universe to turn us to semi-
determinist agents for a more extended theory of quantum mechanics. Thus,
since the mentioned theory does not include the Pauli’s exclusion principle,
there would be some real parallel worlds with no definite structures of atoms
and no acknowledged Mendeleev’s table. In principle, pursuing this proposal
would lead us to uncontrollable theories of exotic real worlds, an endless
diverging illlusive process.

In fact, the problems of the MWI has some more complicated features through
with the picture we presented here. As we argued, since in the MWI the
existence of any possible world is considered as a real fact, the established
laws of nature of any world may probably change from a definite moment onward
(after splitting from a branching point). Actually, this is the most troubling
situation when we assume the laws of physics of our world be suddenly broken
and replaced with some different laws of nature.363636Also, such a
catastrophic phenomenon would have happened in the past, and then we would
likely be encountered with a pseudoscientific theory of physics which can
confirm (if we insist more inquisitively) the mythological stories. All in
all, the MWI results in the fact that the recorded reports of observable
reality could be based on transient objective laws and unstable reality.
Consequently, there would be no guarantee that quantum mechanics become
suddenly extinct and be replaced with a new theory of nature. Particularly, it
must be a catastrophic end of physics. Thus, despite of its interesting
expressions of peculiar aspects of quantum mechanics, the MWI seems not to be
a correct interpretation of the theory. Therefore, although the Copenhagen
interpretation of quantum mechanics has been established to be inconsistent,
it cannot be safely replaced with the MWI.

##  IV Acknowledgments

The author says his special gratitude to S. Ziaee. Moreover, this article is
dedicated to medical staff from all around the world who are selflessly
fighting the pandemic of Covid-19. Also it should be noted that this research
was in part supported by a grant from IPM (No.99810422).

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Using Scratch to Teach Undergraduate Students' Skills on Artificial Intelligence

Julian Estevez,
Gorka Garate,
JM Lopez-Guede,
Manuel Graña

University of the Basque Country (UPV-EHU), Spain<EMAIL_ADDRESS>
This paper presents a educational workshop in Scratch that is proposed for the active participation of undergraduate students in contexts of Artificial Intelligence. The main objective of the activity is to demystify the complexity of Artificial Intelligence and its algorithms. For this purpose, students must realize simple exercises of clustering and two neural networks, in Scratch. The detailed methodology to get that is presented in the article.

rtificial Intelligence, Scratch, Neural Network

§ INTRODUCTION

Scientific Method is key in the development of technologically advanced communities [6], and it is of great importance that our students include it in their curricula. As many other competences, it is convenient that some knowledge of scientific  method is learned at the undergraduate stage, and any effort done in the Educational System to foster the learning of it undoubtedly pays.

For a better understanding of scientific method, the Educational Community widely recognizes that school curricula must move on from traditional expositive classes to more informal and collaborative contexts. The fact is, however, that active participation of the students is difficult to achieve at the classroom. One of the reasons should appeal to the interests of the students themselves; in order to achieve active involvement, the classes should make use of more attractive resources, such as fun, games, social interaction, observation of real problems, novelty, etc. This is specially clear in the case of undergraduate students. Surprisingly, there is a quite widespread mistrust of science in the post-truth era where we live. This makes it necessary to insist on a bigger effort to spread the benefits of science among younger students [1],[2].

The present article presents the design and development of several simple educational exercises to promote understanding and learning of Artificial Intelligence (AI) at schools with the usage of Scratch, which is a graphic programming environment. By allowing novices to build programs by snapping together graphical blocks that control the actions of different actors and algorithms make programming fairly easy for beginners.

AI is one of the technologies that will transform our society, economy and jobs in a greater extent along the next decades. Some of the most known examples of AI are driverless cars, chatbots, voice assistants, internet search engines, robot traders, etc. These systems can be embedded in physical machines or in software, and the promising capacities of both architectures makes it necessary for society and politicians to regulate the functions and limits of these devices [8]. Despite the myth of destructive AI represented in films such as Terminator or I, Robot, the truth is that nowadays most usual smart algorithms consist of a series of simple rules applied to large series of numbers, and the result of that is called Artificial Intelligence.

British Parliament and other institutions recommend the education from high school of Artificial Intelligence [28], regardless of the pace of development of this technology, in order to cope future technological and social challenges.

The main reason is to improve technological understanding, enabling people to navigate an increasingly digital world, and inform the debate around how AI should, and should not, be used.

Computational thinking is a problem-solving method that uses techniques typically practiced by computer-scientists and is increasingly being viewed as an important ingredient of STEM learning in primary and secondary education [13]. On the other hand, several studies in higher education research report on the low student pass rate in mathematics. Therefore, research into mathematics education has been greatly emphasized in the last decade [20]. Though education reform efforts have been made around the world, the trouble lies on the fact that most of schools are trying to prepare students for the future by continuing what was done in the past [13].

Thus, in this paper an educational proposal for teaching the basic mathematics behind simple AI algorithms is presented. The solution is developed in an open source platform, Scratch, and the main objective for the students is to be aware of the rules behind intelligent systems rather than learning or memorizing anything. The specific tasks to understand are automatic clustering, learning and prediction with AI. The software is designed for students of 16-18 years. The algorithms chosen for teaching are specifically designed or adapted to the mathematical background of the students. Moreover, the article will be publicly available in a web repository[<https://gitlab.com/juleste>].

The present article is divided in the following sections. In section <ref> a background for teaching AI is presented. In Section <ref> the mathematics that will be used in the software are described. In Section <ref> the methodology that teachers will use with the students is detailed. Finally, in Section <ref>, conclusions of the experiment are analyzed.

§ BACKGROUND

There is a wide consensus among computer scientist that it is quite difficult to teach the basics of AI [11]. This is due to the lack of a unified methodology and to the blend of many other disciplines which involve a wide range of skills, ranging from very applied to quite formal [12]. AI modeling, algorithms and applications may be taught using tools as simple as paper and pencil, traditional computer programming or hands-on-computer programming [10]. Several papers discuss the prerequisites needed to understand machine learning [19].

Some experts suggest that high school education reforms should encompass a drive on STEM skills and coding in schools; others support the idea of focusing on digital understanding, rather than in skills.

Nowadays, however, the tendency is to teach coding to students, mostly framed in robotics workshops [13, 16, 17]. Seymour Papert [14] laid much of the groundwork for using robots in the classroom in the 1970s. An argument for teaching to children and students using robots is that they see these machines as toys [18]. Studies show that robotics generates a high degree of student interest and engagement and promotes interest in maths and science careers.

[12] uses a Lego robot to make the students build the hardware and control the software that manages the sensors and the environment. [10] uses a low-cost robot platform to teach students how a neural network is trained and built for a navigation robotic problem. There is the possibility that students pay attention to the robotic exercise rather than to the AI learning fundamentals, and this is what [15] tries to settle down in another robotic navigation problem. Both authors specify which are the premises to be transmitted to the students about the objective of the activity. These premises are, first, underlining that the exercise is about Artificial Intelligence and not robotics; second, emphasizing that they are clearly defined open projects, with specific start and end points.

However, although robotics is recognized as a proper way for teaching computational-thinking (CT) skills to students [22], in this article we present a wider approach to AI, based on the teaching of some mathematics with the help of Scratch [21], which will let us introduce the students into more than one algorithm.

CT can be defined as the process of recognizing aspects of computation in the world that surrounds us and applying tools and techniques from Computer Science to understand and reason about both natural and artificial systems and processes [22, 23].

The software used to enhance CT on students should be easy for a beginner to overcome the difficulty of creating working programs, but it should also be powerful and extensive enough to satisfy the needs of more advanced programmers. Several programming tools fit these criteria in varying degrees: Scratch, Crickets, Karel the Robot, Alice, Game Maker, Kodu, Greenfoot and Agentcubes. All those tools are based on the Logo philosophy [14]. Graphical programming environments are relatively easy to use and allow early experiences to focus on designing and creating, avoiding issues of programming syntax.

From a pedagogical perspective, computational tools are capable of deepening the learning of mathematics and science contents [24, 25], and the reverse is also true [26]. Scratch is today one of the most popular of these programming environments, and it has proved to be very effective for the engagement and motivation of young students with no experience at programming [27].

§ CONTENT

This article will depict algorithms that can be practiced using Scratch in a workshop with students 16-18 years old. After the workshop, the students should be able to understand, play and eventually code these algorithms.

Specific algorithms are:

* K-means

* Neural network

And the exercises that students will have to do are described next.

The algorithm K-means, developed in 1967 by MacQueen [3], is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. K-means is a clustering algorithm which tries to show the students how items in a big dataset are automatically classified, even while new items are being added. The adopted technique for creating as many clusters as we want, is the minimum square error rule.

Starting from a given cloud of N points and a smaller cloud of K mass centers, the aim of this activity is that the student learns how to cluster the points into K groups, by programming an application in Scratch. In order to do the clustering, each point will belong to the cluster defined by the closest mass center. Finally, each of the K clusters will be colored with a different color (see Figure <ref>).

\begin{array}{c}
\fbox{\includegraphics[width=7cm]{Kmeans_initial_final.jpg}} \\

\end{array}$

Initial and final clouds of points

Neural network

The basic idea behind a neural network is to simulate lots of densely interconnected brain cells inside a computer so you can get it to learn things, recognize patterns, and make decisions in a human-like way. The main characteristic of this tool is that a neural network learns all by itself. The programmer just needs to design the physical structure (number of outputs, inputs, hidden layers) and set some very simple rules involving additions, multiplications and derivatives. They are based on perceptrons, which were developed in the 1950s and 1960s by the scientist Frank Rosenblatt [7], inspired by earlier work by Warren McCulloch and Walter Pitts [4]. It is important to note that neural networks are (generally) software simulations: they are made by programming very ordinary computers. Computer simulations are just collections of algebraic variables and mathematical equations linking.

Habitually, neural networks use backpropagation type algorithms which require the usage of derivatives. However, the target students of this exercise (16-18 years old) do not have this mathematical operation in their curricula yet. As a consequence, the neural networks have been created with a logic activation function, which are understandable by the students. Necessary formulas are presented to them without any previous mathematical demonstration.

Different exercises are developed with neural networks. First, a simple neural network with two inputs and an output neuron, is trained with an AND logic gate (see Figure <ref>). In each iteration, students will see the different weights that the neural network gets. Next, and OR logic gate will be used, and as a consequence, students will observe how adjusting parameters change with this new data. The mathematical description of this neural network is described in Subsection <ref>.

First neural network interface

As a second exercise of this type, a more complex neural network exercise is presented (Figure <ref>. Considering a 3-2-1 neural network (three inputs, a hidden layer with two neurons and an output) the students will train the neural network with AND and OR logic gates again. The detailed operations of this exercise appear in Subsection <ref>.

Second neural network interface

§.§.§ Simple Neural Network operation

The neuron obtains an output $Y_1$ from the two inputs $Input_1$ and $Input_2$ using the corresponding weights.

\begin{equation}
\label{eq:erroreq}
N_1 = Input_1 \cdot W_1 + Input_2  \cdot W_2
\end{equation}

The activation function at $N_1$ defined as shown next:

\begin{equation}
N_1 \begin{cases}
\geq 1 &\text{$Y_1$=1}\\\\
\text{else}   &\text{$Y_1$=0}
\end{cases}
\end{equation}

where $Y_1$ is the output of the neural network.

The error is defined as the difference between the desired output $Desired\_output$ and the obtained output.

\begin{equation}
\label{eq:erroreq}
Error = Desired\_output - Y_1
\end{equation}

Each time the function is executed, the algorithm updates the weights using backpropagation and gradient descent rule, until output $Y_1$ converges to the desired output $Desired\_output$.

The new value of the first weight will be the sum of its previous value and the product of the first input, the learning rate (LR) and the error.

\begin{equation}
\label{eq:W1eq}
W_{1,new} = W_1 + Input_1 \cdot LR \cdot Error
\end{equation}

Similarly, the new value of the second weight will be:

\begin{equation}
\label{eq:W2eq}
W_{2,new} = W_2 + Input_2 \cdot LR \cdot Error
\end{equation}

§.§.§ Complex Neural Network operation

There are three neurons: two at the hidden layer ($N_1$ and $N_2$) and one at the output ($N_3$). Their calculus depend on the inputs ($X_1$, $X_2$ and $X_3$) and weights ($W_1$, $W_2$, $W_3$, $W_4$, $W_5$) following next expressions:

\begin{equation} \label{eq:N1eq}
\begin{cases}
N_1=X_1 \cdot W_1+ X_2 \cdot W_2 \\
N_2=X_3 \cdot W_3\\
N_3=N_1 \cdot W_4 + N_2 \cdot W_5
\end{cases}
\end{equation}

There is an activation function at $N_3$ defined as shown next:

\begin{equation}
N_3 \begin{cases}
\geq 0 &\text{$N_3=1$}\\\\
\text{else}   &\text{$N_3=0$}
\end{cases}
\end{equation}

where $N_3$ is the output of the neural network.

Error is defined as:

\begin{equation}
\label{eq:erroreq}
Error = Desired\_output - N_3
\end{equation}

Employing again the backpropagation and gradient descent rule, the calculus for updating the weights result in next equations:

\begin{equation}
\begin{cases}
W_{1,new}=W_1 + LR \cdot Error \cdot W_4 \cdot X_1 \\
W_{2,new}=W_2 + LR \cdot Error \cdot W_4 \cdot X_2 \\
W_{3,new}=W_3 + LR \cdot Error \cdot W_5 \cdot X_3 \\
W_{4,new}=W_4 + LR \cdot Error \cdot N_1\\
W_{5,new}=W_5 + LR \cdot Error \cdot N_2
\end{cases}
\end{equation}

Once the values of the error and weights are calculated, the student has to store them on the corresponding neurons.

§ METHODOLOGY

The workshop of Artificial Intelligence will be based on the usage of the educational tool Scratch [21]. Scratch is a visual programming language, and its online community is targeted primarily at children and young students. Using Scratch, users can create online projects and develop them into almost anything by using a simple block-like interface. When they are finished, students can share their projects and discuss their creations with each other. Before the students start the exercises here proposed, it is convenient that they acquire some knowledge of this language.

In all algorithms presented in Section <ref>, the first task of the student is to fill in the white gaps left to him/her among the lines of code, marked with a comment. That is, the students do not need to create the algorithm or write the whole code themselves; the code will be provided for the most part.

Students will work in pairs and the workshop will be structured in the following steps:

* At the beginning, the teachers will give a short explanation of 15-20 minutes about AI and the objective of the workshop, with a twofold aim: first, to demystify Artificial Intelligence; second, to understand some simple mathematics underneath the computations.

* The students will work in couples. Teachers will provide them with some written theoretical background about the algorithms and the instructions for the exercises (K-means and neural networks). Moreover, teachers will explain in another 10 minutes of presentation the basics of the algorithms.

* Students will have one hour to finish the codes (20 minutes for K-means and 40 minutes for both parts of the neural networks) and eventually execute the applications to see whether they work properly. During this personal tasks teachers will be at hand ready to assist whenever necessary.

* Finally, at the end of the session, the teachers will provide the students with the finished proposed answers in Scratch software for the students to check them.

§.§ K-Means

The algorithm is developed using one main block and three function blocks of Scratch code. The function blocks are called ,  and .

The student must finish up the three function blocks: the block , block  and block .

The main block will create the mass centers and the cloud of points. The set of mass centers contains $K$ random points with $X$ coordinates between $\left( -230,230 \right)$ and $Y$ coordinates between $\left( -170,170 \right)$. These $K$ points will be the mass centers of the clusters (see Figure <ref>).

Main block to create the mass centers and the cloud of points

The first task is to finish the programming of the block  that will create the cloud of $N$ points. The cloud must contain $N$ points with $X$ coordinates between $\left( -230,230 \right)$ and $Y$ coordinates between $\left( -170,170 \right)$ (see Figure <ref>).

Task 1. Unfinished function to create the cloud of points

Block  stores the number of the mass center that will be assigned to each point. To do that, the student must code the calculation of the Euclidean distance from each of the points to each the mass centers in variable $A$. The program will then find the minimum distance of them all and fill up vector , which contains the numbers of the cluster to which each point belongs.

Finally, block  graphs the clouds of points and the cloud of centers of mass, each in a colour given by vector .

The algorithm here presented just tries to show automatic clustering, and not getting equivalent size clusters.

Task 2. Unfinished function to calculate the distances $A$

§.§ Neural Network: AND/OR logic gate

The algorithm is developed using a main block, which initializes the data, and two blocks,  and .

The interface used is described in Figure <ref>. The student must code the equations that define the function that performs the only neuron of the network. The equations to calculate is the update of $W_2$, as can be seen in Figure <ref>.

The students must train the neuron using two sets of data: one set for the AND logic gate and another for the OR logic gate.

§.§ Complex Neural Network Operation

This exercise is based on training a three inputs neural network with AND and OR logic gates. As an additional complexity comparing to previous exercise, this neural network is multilayered. It uses three neurons, ,  and , and .

The exercise is planned to fill the gaps of the following operations:

* Net calculus of $N_3$

* Update of $W_2$ and $W_5$

* Update of $B_2$

Task 3. Unfinished function to calculate the weight update

§ CONCLUSIONS

The paper presents teacher-guided, easy to implant activities that can be performed at schools using Scratch. Moreover, the operations have been adapted for 16-18 year-old students' mathematical background.

The tasks presented are scalable; students can delve into the maths involved in the mathematical iterations or into the Scratch code itself, or even propose new neural networks to deal with other problems.

The work can be extensible to students of different ages and more AI algorithms can be added to the system.

The easiness of the equations of K-means and neural networks permit their implementation in other formats, such as MS Excel, which in some occasions could be more familiar to students than Scratch.

Moreover, the students will realize that the mathematical knowledge acquired throughout the year will help them finishing the programming of automatic clustering and neuron networks, being able to train AND and OR logic gates.

As a future work, first, the authors should organize the AI workshop several times and measure the grade of the objective achievement in collaboration with pedagogical researchers. Secondly, more programming languages should be explored (GeoGebra, HTML, ShynnyApps) in order to implement more exercises, such as the usage of neural networks for data prediction.

§ ACKNOWLEDGEMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777720.

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