Combinator

<theory> A function with no free variables.

A term is either a constant, a variable or of the form A B denoting the application of term A (a function of one argument) to term B.

Juxtaposition associates to the left in the absence of parentheses.

All combinators can be defined from two basic combinators – S and K.

These two and a third, I, are defined thus:

S f g x = f x (g x) K x y = x I x = x = S K K x

There is a simple translation between combinatory logic and lambda-calculus.

The size of equivalent expressions in the two languages are of the same order.

Other combinators were added by David Turner in 1979 when he used combinators to implement SASL:

B f g x = f (g x) C f g x = f x g S’ c f g x = c (f x) (g x) B* c f g x = c (f (g x)) C’ c f g x = c (f x) g

See fixed point combinator, curried function, supercombinators.