We assume here that you've already gotten [Schema and OCaml installed on your computer](/how_to_get_the_programming_languages_running_on_your_computer/). ## Programming in the pure untyped lambda calculus ## There are several ways to do this, and we're still thinking out loud in this space about which method we should recommend you use. 1. To get started, Chris has a nice [Lambda Tutorial](http://homepages.nyu.edu/~cb125/Lambda) webpage introducing the untyped lambda calculus. This page has embedded Javascript code that enables you to type lambda expressions into your web browser page and click a button to "execute" (that is, reduce or normalize) it. To do more than a few simple exercises, though, you'll need something more complex. 2. One option is to use a short Scheme macro, like the one [linked at the bottom of Chris' webpage](http://homepages.nyu.edu/~cb125/Lambda/lambda.scm). You can use this by loading into a Scheme interpreter (EXPLAIN HOW...) and then (STEP BY STEP...). Here's Chris' explanation of the macro: (define (reduce f) ; 1 ((lambda (value) (if (equal? value f) f (reduce value))) ; 2 (let r ((f f) (g ())) ; 3 (cond ((not (pair? f)) ; 4 (if (null? g) f (if (eq? f (car g)) (cadr g) (r f (caddr g))))) ; 5 ((and (pair? (car f)) (= 2 (length f)) (eq? 'lambda (caar f))) ; 6 (r (caddar f) (list (cadar f) (r (cadr f) g) g))) ; 7 ((and (not (null? g)) (= 3 (length f)) (eq? 'lambda (car f))) ; 8 (cons 'lambda (r (cdr f) (list (cadr f) (delay (cadr f)) g)))) ; 9 (else (map (lambda (x) (r x g)) f)))))) ;10 If you have a Scheme interpreter, you can call the function like this: (reduce '(((lambda x (lambda y (x y))) 2) 3)) ;Value: (2 3) (reduce '((lambda x (lambda y (x y))) 2)) ;Value: (lambda #[promise 2] (2 #[promise 2])) Comments: f is the form to be evaluated, and g is the local assignment function; g has the structure (variable value g2), where g2 contains the rest of the assignments. The named let function r executes one pass through a form. The arguments to r are a form f, and an assignment function g. Line 2: continue to process the form until there are no more conversions left. Line 4 (substitution): If f is atomic [or if it is a promise], check to see if matches any variable in g and if so replace it with the new value. Line 6 (beta reduction): if f has the form ((lambda variable body) argument), it is a lambda form being applied to an argument, so perform lambda conversion. Remember to evaluate the argument too! Line 8 (alpha reduction): if f has the form (lambda variable body), replace the variable and its free occurences in the body with a unique object to prevent accidental variable collision. [In this implementation a unique object is constructed by building a promise. Note that the identity of the original variable can be recovered if you ever care by forcing the promise.] Line 10: recurse down the subparts of f. 3. Oleg Kiselyov has a [richer lambda interpreter](http://okmij.org/ftp/Scheme/#lambda-calc) in Scheme. Here's how he describes it (I've made some trivial changes to the text): A practical Lambda-calculator in Scheme The code below implements a normal-order interpreter for the untyped lambda-calculus. The interpret permits "shortcuts" of terms. The shortcuts are not first class and do not alter the semantics of the lambda-calculus. Yet they make complex terms easier to define and apply. The code also includes a few convenience tools: tracing of all reduction, comparing two terms modulo alpha-renaming, etc. This calculator implements a normal-order evaluator for the untyped lambda-calculus with shortcuts. Shortcuts are distinguished constants that represent terms. An association between a shortcut symbol and a term must be declared before any term that contains the shortcut could be evaluated. The declaration of a shortcut does not cause the corresponding term to be evaluated. Therefore shortcut's term may contain other shortcuts -- or even yet to be defined ones. Shortcuts make programming in lambda-calculus remarkably more convenient. Besides terms to reduce, this lambda-calculator accepts a set of commands, which add even more convenience. Commands define new shortcuts, activate tracing of all reductions, compare terms modulo alpha-conversion, print all defined shortcuts and evaluation flags, etc. Terms to evaluate and commands are entered at a read-eval-print-loop (REPL) "prompt" -- or "included" from a file by a special command. Examples First we define a few shortcuts: (X Define %c0 (L s (L z z))) ; Church numeral 0 (X Define %succ (L n (L s (L z (s (n z z)))))) ; Successor (X Define* %c1 (%succ %c0)) (X Define* %c2 (%succ %c1)) (X Define %add (L m (L n (L s (L z (m s (n s z))))))) ; Add two numerals (%add %c1 %c2) REPL reduces the term and prints the answer: (L f (L x (f (f (f x))))). (X equal? (%succ %c0) %c1) (X equal?* (%succ %c0) %c1) The REPL executes the above commands and prints the answer: #f and #t, correspondingly. The second command reduces the terms before comparing them. See also . 4. Oleg also provides another lambda interpreter [written in Haskell](http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell). Jim converted this to OCaml and bundled it with a syntax extension that makes it easier to write pure untyped lambda expressions in OCaml. You don't have to know much OCaml yet to use it. Using it looks like this: let zero = << fun s z -> z >>;; let succ = << fun n s z -> s (n s z) >>;; let one = << $succ$ $zero$ >>;; let two = << $succ$ $one$ >>;; let add = << fun m n -> n $succ$ m >>;; (* or *) let add = << fun m n -> fun s z -> m s (n s z) >>;; church_to_int << $add$ $one$ $two$ >>;; - : int = 3 To install Jim's OCaml bundle, DO THIS... Some notes: * When you're talking to the interactive OCaml program, you have to finish complete statements with a ";;". Sometimes these aren't necessary, but rather than learn the rules yet about when you can get away without them, it's easiest to just use them consistently, like a period at the end of a sentence. * What's written betwen the `<<` and `>>` is parsed as an expression in the pure untyped lambda calculus. The stuff outside the angle brackets is regular OCaml syntax. Here you only need to use a very small part of that syntax: `let var = some_value;;` assigns a value to a variable, and `function_foo arg1 arg2` applies the specified function to the specified arguments. `church_to_int` is a function that takes a single argument --- the lambda expression that follows it, `<< $add$ $one$ $two$ >>` -- and, if that expression when fully reduced or "normalized" has the form of a "Church numeral", it converts it into an "int", which is OCaml's (and most language's) primitive way to represent small numbers. The line `- : int = 3` is OCaml telling you that the expression you just had it evaluate simplifies to a value whose type is "int" and which in particular is the int 3. * If you call `church_to_int` with a lambda expression that doesn't have the form of a Church numeral, it will complain. If you call it with something that's not even a lambda expression, it will complain in a different way. * The `$`s inside the `<<` and `>>` are essentially corner quotes. If we do this: `let a = << x >>;; let b = << a >>;; let c = << $a$ >>;;` then the OCaml variable `b` will have as its value an (atomic) lambda expression, consisting just of the variable `a` in the untyped lambda calculus. On the other hand, the OCaml variable `c` will have as its value a lambda expression consisting just of the variable `x`. That is, here the value of the OCaml variable `a` is spliced into the lambda expression `<< $a$ >>`. * The expression that's spliced in is done so as a single syntactic unit. In other words, the lambda expression `<< w x y z >>` is parsed via usual conventions as `<< (((w x) y) z) >>`. Here `<< x y >>` is not any single syntactic constituent. But if you do instead `let a = << x y >>;; let b = << w $a$ z >>`, then what you get *will* have `<< x y >>` as a constituent, and will be parsed as `<< ((w (x y)) z) >>`. * `<< fun x y -> something >>` is equivalent to `<< fun x -> fun y -> something >>`, which is parsed as `<< fun x -> (fun y -> (something)) >>` (everything to the right of the arrow as far as possible is considered together). At the moment, this only works for up to five variables, as in `<< fun x1 x2 x3 x4 x5 -> something >>`. * The `<< >>` and `$`-quotes aren't part of standard OCaml syntax, they're provided by this add-on bundle. For the most part it doesn't matter if other expressions are placed flush beside the `<<` and `>>`: you can do either `<< fun x -> x >>` or `<x>>`. But the `$`s *must* be separated from the `<<` and `>>` brackets with spaces or `(` `)`s. It's probably easiest to just always surround the `<<` and `>>` with spaces. 5. To play around with a **typed lambda calculus**, which we'll look at later in the course, have a look at the [Penn Lambda Calculator](http://www.ling.upenn.edu/lambda/). This requires installing Java, but provides a number of tools for evaluating lambda expressions and other linguistic forms. (Mac users will most likely already have Java installed.) ## Reading about Scheme ## [R5RS Scheme](http://people.csail.mit.edu/jaffer/r5rs_toc.html) ## Reading about OCaml ##