A special case of a
combinator is the
Y combinator or
Y constructor, also sometimes known as "
fix". The Y combinator is a formula in
lambda calculus which allows the definition of
recursive functions in that formalism. The Y combinator is a
fixed point combinator that has the property that:
Y x = x (Y x)
Somewhat surprisingly, the Y combinator can be defined as the non-recursive lambda abstraction:
Y = λ h . (λ x . h (x x)) (λ x . h (x x))
See the lambda calculus article for a detailed explanation.
As an example, consider the factorial function. A single step in the recursion of the factorial function is
- H = (λf.λn.(ISZERO n) 1 (MULT n (f (PRED n))))
which is non-recursive. If the factorial function is like a chain (of factors), then the h function above joins two links. Then the factorial function is simply
- FACT = (Y H)
- FACT = (((λ h . (λ x . h (x x)) (λ x . h (x x))) (λf.λn.(ISZERO n) 1 (MULT n (f (PRED n)))))
The Y-constructor causes the H combinator to repeat itself indefinitely until it trips itself up with (ISZERO 0) = TRUE.
By the way, these equations are meta-equations; functions in lambda calculus are all anonymous. The function labels Y, H, FACT, PRED, MULT, ISZERO, 1, 0 (defined in the article for lambda calculus) are meta-labels, to which correspond meta-definitions and meta-equations, and with which a user can perform algebraic meta-substitutions. That is how mathematicians can prove properties of the lambda calculus. The equals sign as an assignment operation is not part of the lambda calculus.