Combinator
In mathematics, a combinator is a function with no free variables.
In computer science, combinators have been used as a basis for the semantics of functional programming languages.
A term of a combinator is either a constant, a variable, or an application of the form (A B), denoting the application of term A (a function of one argument) to term B. Application associates to the left in the absence of parentheses. All combinators can be defined as a composition of two basic combinators - S and K. These two and a third, I, are defined thus:
There is a simple translation between combinatory logic and the lambda-calculus. However, combinatorial expressions are much larger than their lambda-calculus equivalents.
A combinator of particular interest is the Y combinator, which is a fixed point combinator and can be used to implement recursion.
Other combinators were added by David Turner[?] in 1979 when he used combinators to implement SASL[?]:
Some special-purpose computers have been built to perform combinator calculations by graph reduction[?]. Examples include the SKIM[?] ("S-K-I machine") computer, built at Cambridge University, and the multiprocessor GRIP[?] ("Graph Reduction In Parallel") computer, built at UCL.
See also:
The original version of article was based on materical from FOLDOC, used with permission.
(S f g x) = (f x (g x))
(K x y) = x
(I x) = x = (S K K x)
(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)