## Programming in the pure untyped lambda calculus ##
There are several ways to do this.
1. The easiest is to use a JavaScript interpreter that Chris wrote. Go [here](/lambda-let.html) and follow the template:
let true = (\x (\y x)) in
let false = (\x (\y y)) in
let and = (\l (\r ((l r) false))) in
(
((((and false) false) yes) no)
((((and false) true) yes) no)
((((and true) false) yes) no)
((((and true) true) yes) no)
)
will evaluate to:
(no no no yes)
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.
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.
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).
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) >>;;
.
(* now use:
pp FORMULA to print a formula, unreduced
pn FORMULA to print the normal form of a formula (when possible)
pi FORMULA to print the integer which FORMULA is a Church numeral for (when possible)
.
alpha_eq FORM1 FORM2 are FORM1 and FORM2 syntactically equivalent (up to alpha-conversion)?
this does not do reductions on the formulae
*)
.
pi << $add$ $one$ $two$ >>;;
- : int = 3
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.
INCLUDE INSTRUCTIONS
We assume here that you've already [gotten OCaml installed on your computer](/how_to_get_the_programming_languages_running_on_your_computer/).
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.