((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:
+
+ 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.
+
+ 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 f (L x x))) ; Church numeral 0
- (X Define %succ (L c (L f (L x (f (c f x)))))) ; Successor
+
+ (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 c (L d (L f (L x (c f (d f x))))))) ; Add two numerals
- (X Define %mul (L c (L d (L f (c (d f)))))) ; Multiplication
-
- (%add %c1 %c2)
+ (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.
+
+ 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 <http://okmij.org/ftp/Computation/lambda-calc.html>.
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 k = <<fun y _ -> y>>
- let id = <<fun x -> x>>
- let add = <<fun m n -> n $succ$ m>>
- let pred = <<fun n s z -> n (fun u v -> v (u s)) ($k$ z) $id$ >>;;
+ 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 `$`-quotes aren't part of standard OCaml syntax, they're provided by this add-on bundle. For the most part it doesn't matter if other expressions are placed flush beside the `<<` and `>>`: you can do either `<< fun x -> x >>` or `<<fun x->x>>`. But the `$`s *must* be separated from the `<<` and `>>` brackets with spaces or `(` `)`s. It's probably easiest to just always surround the `<<` and `>>` with spaces.
- church_to_int << $pred$ ($add$ $one$ ($succ$ $one$)) >>;;
- - : int = 2
5. To play around with a **typed lambda calculus**, which we'll look at later