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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.