Convert between lambda calculus and combinatory logic in Schönfinkel's BCIKS system
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λx.λy.(y x) → (C I)

(C I) → λx.λy.(y x)

Convert between lambda expressions and combinatory logic in Schönfinkel's BCIKS system.

BCIKS quick guide

  • B f g x = f (g x) (composition)
  • C f x y = f y x (order swap)
  • I x = x (identity)
  • K x y = x (drop arg)
  • S f g x = (f x) (g x) (duplicate arg)


On Racket, require ./cl.rkt, and on other scheme interpreters, load ./cl.scm. These provide the T, L, and P functions (described below), as well as the combinators B, C, I, K, S.

$ racket -i --require cl.rkt
$ mit-scheme -load cl.scm


Arguments must be given in curried form. The T function can parse lambda expressions and curried function applications (i.e. pairs of the form (a b)). The L function can only parse curried function applications. Any free variables will be left intact unless a variable name conflicts with a combinator (watch out for i!).

T: λ -> CL

To convert from expressions of lambda calculus to combinatory logic, use the T function:

> (T '(lambda (x) (lambda (y) (y x))))
; (c i)

> (T '(lambda (x) (lambda (y) ((+ (sin x)) (sin y)))))
; ((c ((b b) ((b +) sin))) sin)

L: CL -> λ

To convert from combinatory logic to expressions of lambda calculus, use the L function. The resulting lambda expression is in beta normal form.

> (L '(C I))
; (lambda (a) (lambda (b) (b a)))

> (L '(((C I) 3) sin))
; (sin 3)

> #| racket     |# (define f (eval (L '(C I)) (current-namespace)))
> #| mit-scheme |# (define f (eval (L '(C I)) user-initial-environment))
; f
> ((f 3) sin)
; .1411200080598672

> (T (L '(C I)))
; (c i)

P: λ -> uncurried and pretty

To uncurry and "prettify" a lambda expression, use the P function. This will minimize use of parentheses, spaces, and lambda literals (λaλb.b a -> λab.b a). It returns a string, not a quoted expression like the other functions, so it can use the unicode λ symbol rather than the verbose word lambda.

> (P '(lambda (x) (lambda (y) (lambda (z) ((((f (g x)) z) (lambda (w) ((+ y) w))) b)))))
; "λxyz.f (g x) z (λw.+ y w) b"

> (P (L '(C I)))
; "λab.b a"