2 ## Programming in the pure untyped lambda calculus ##
4 There are several ways to do this.
6 1. The easiest is to use a JavaScript interpreter that Chris wrote. Go [here](/lambda-let.html) and follow the template:
8 let true = (\x (\y x)) in
9 let false = (\x (\y y)) in
10 let and = (\l (\r ((l r) false))) in
14 ((((and false) false) yes) no)
16 ((((and false) true) yes) no)
18 ((((and true) false) yes) no)
20 ((((and true) true) yes) no)
28 If you try to evaluate a non-terminating form, like `((\x (x x)) (\x (x x)))`, you'll probably have to force-quit your browser and start over. Anything you had earlier typed in the upper box will probably be lost.
30 Syntax: you have to fully specify parentheses and separate your lambdas. So for example, you can't write `(\x y. y)`; you have to write `(\x (\y y))`. The parser treats symbols that haven't yet been bound (as `yes` and `no` above) as free variables.
33 2. A bit more flexibility and robustness can be had by using an OCaml package. This is based on a library on [a Haskell library by Oleg Kiselyov](http://okmij.org/ftp/Computation/lambda-calc.html#lambda-calculator-haskell).
35 Jim converted this to OCaml and bundled it with a syntax extension that makes
36 it easier to write pure untyped lambda expressions in OCaml. You don't have to
37 know much OCaml yet to use it. Using it looks like this:
39 let zero = << fun s z -> z >>;;
40 let succ = << fun n s z -> s (n s z) >>;;
41 let one = << $succ$ $zero$ >>;;
42 let two = << $succ$ $one$ >>;;
43 let add = << fun m n -> n $succ$ m >>;;
45 let add = << fun m n -> fun s z -> m s (n s z) >>;;
48 pp FORMULA to print a formula, unreduced
49 pn FORMULA to print the normal form of a formula (when possible)
50 pi FORMULA to print the integer which FORMULA is a Church numeral for (when possible)
52 alpha_eq FORM1 FORM2 are FORM1 and FORM2 syntactically equivalent (up to alpha-conversion)?
53 this does not do reductions on the formulae
56 pi << $add$ $one$ $two$ >>;;
59 To install this package, here's what you need to do. I've tried to explain it in basic terms, but you do need some familiarity with your operating system: for instance, how to open a Terminal window, how to figure out what directory the Terminal is open to (use `pwd`); how to change directories (use `cd`); and so on.
63 We assume here that you've already [gotten OCaml installed on your computer](/how_to_get_the_programming_languages_running_on_your_computer/).
67 * 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.
69 * 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.
71 * 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.
73 * 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 `<< $a$ >>`.
75 * 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) >>`.
77 * `<< 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 >>`.
79 * The `<< >>` and `$`-quotes aren't part of standard OCaml syntax, they're provided by this add-on bundle. For the most part it doesn't matter if other expressions are placed flush beside the `<<` and `>>`: you can do either `<< fun x -> x >>` or `<<fun x->x>>`. But the `$`s *must* be separated from the `<<` and `>>` brackets with spaces or `(` `)`s. It's probably easiest to just always surround the `<<` and `>>` with spaces.
84 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.
86 1. To get started, Chris has a nice [Lambda Tutorial](http://homepages.nyu.edu/~cb125/Lambda)
87 webpage introducing the untyped lambda calculus. This page has embedded Javascript
88 code that enables you to type lambda expressions into your web browser page
89 and click a button to "execute" (that is, reduce or normalize) it.
91 To do more than a few simple exercises, though, you'll need something more complex.
93 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).
94 You can use this by loading into a Scheme interpreter (EXPLAIN HOW...) and then (STEP BY STEP...).
96 Here's Chris' explanation of the macro:
98 (define (reduce f) ; 1
99 ((lambda (value) (if (equal? value f) f (reduce value))) ; 2
100 (let r ((f f) (g ())) ; 3
101 (cond ((not (pair? f)) ; 4
102 (if (null? g) f (if (eq? f (car g)) (cadr g) (r f (caddr g))))) ; 5
103 ((and (pair? (car f)) (= 2 (length f)) (eq? 'lambda (caar f))) ; 6
104 (r (caddar f) (list (cadar f) (r (cadr f) g) g))) ; 7
105 ((and (not (null? g)) (= 3 (length f)) (eq? 'lambda (car f))) ; 8
106 (cons 'lambda (r (cdr f) (list (cadr f) (delay (cadr f)) g)))) ; 9
107 (else (map (lambda (x) (r x g)) f)))))) ;10
109 If you have a Scheme interpreter, you can call the function like this:
111 (reduce '(((lambda x (lambda y (x y))) 2) 3))
114 (reduce '((lambda x (lambda y (x y))) 2))
115 ;Value: (lambda #[promise 2] (2 #[promise 2]))
117 Comments: f is the form to be evaluated, and g is the local assignment
118 function; g has the structure (variable value g2), where g2 contains the rest
119 of the assignments. The named let function r executes one pass through a form.
120 The arguments to r are a form f, and an assignment function g. Line 2: continue
121 to process the form until there are no more conversions left. Line 4
122 (substitution): If f is atomic [or if it is a promise], check to see if matches
123 any variable in g and if so replace it with the new value. Line 6 (beta
124 reduction): if f has the form ((lambda variable body) argument), it is a lambda
125 form being applied to an argument, so perform lambda conversion. Remember to
126 evaluate the argument too! Line 8 (alpha reduction): if f has the form (lambda
127 variable body), replace the variable and its free occurences in the body with a
128 unique object to prevent accidental variable collision. [In this implementation
129 a unique object is constructed by building a promise. Note that the identity of
130 the original variable can be recovered if you ever care by forcing the
131 promise.] Line 10: recurse down the subparts of f.
134 3. Oleg Kiselyov has a [richer lambda interpreter](http://okmij.org/ftp/Scheme/#lambda-calc) in Scheme. Here's how he describes it
135 (I've made some trivial changes to the text):
137 A practical Lambda-calculator in Scheme
139 The code below implements a normal-order interpreter for the untyped
140 lambda-calculus. The interpret permits "shortcuts" of terms. The shortcuts are
141 not first class and do not alter the semantics of the lambda-calculus. Yet they
142 make complex terms easier to define and apply.
144 The code also includes a few convenience tools: tracing of all reduction,
145 comparing two terms modulo alpha-renaming, etc.
147 This calculator implements a normal-order evaluator for the untyped
148 lambda-calculus with shortcuts. Shortcuts are distinguished constants that
149 represent terms. An association between a shortcut symbol and a term must be
150 declared before any term that contains the shortcut could be evaluated. The
151 declaration of a shortcut does not cause the corresponding term to be
152 evaluated. Therefore shortcut's term may contain other shortcuts -- or even yet
153 to be defined ones. Shortcuts make programming in lambda-calculus remarkably
156 Besides terms to reduce, this lambda-calculator accepts a set of commands,
157 which add even more convenience. Commands define new shortcuts, activate
158 tracing of all reductions, compare terms modulo alpha-conversion, print all
159 defined shortcuts and evaluation flags, etc. Terms to evaluate and commands are
160 entered at a read-eval-print-loop (REPL) "prompt" -- or "included" from a file
161 by a special command.
165 First we define a few shortcuts:
167 (X Define %c0 (L s (L z z))) ; Church numeral 0
168 (X Define %succ (L n (L s (L z (s (n z z)))))) ; Successor
169 (X Define* %c1 (%succ %c0))
170 (X Define* %c2 (%succ %c1))
171 (X Define %add (L m (L n (L s (L z (m s (n s z))))))) ; Add two numerals
174 REPL reduces the term and prints the answer: (L f (L x (f (f (f x))))).
176 (X equal? (%succ %c0) %c1)
177 (X equal?* (%succ %c0) %c1)
179 The REPL executes the above commands and prints the answer: #f and #t,
180 correspondingly. The second command reduces the terms before comparing them.
182 See also <http://okmij.org/ftp/Computation/lambda-calc.html>.
185 5. To play around with a **typed lambda calculus**, which we'll look at later
186 in the course, have a look at the [Penn Lambda Calculator](http://www.ling.upenn.edu/lambda/).
187 This requires installing Java, but provides a number of tools for evaluating
188 lambda expressions and other linguistic forms. (Mac users will most likely
189 already have Java installed.)
191 ## Reading about Scheme ##
193 [R5RS Scheme](http://people.csail.mit.edu/jaffer/r5rs_toc.html)
195 ## Reading about OCaml ##