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.