Fixed point combinator

A fixed point combinator is a function which computes fixed pointss of other functions. A 'fixed point' of a function is a value left 'fixed' by that function; for example, 0 and 1 are fixed points of the squaring function. Formally, a value x is a fixed point of a function f if f(x) = x.

In certain formalizations of mathematics, such as the lambda calculus and combinatorial calculus, every function has a fixed point. In these formalizations, it is possible to produce a function, often denoted Y, which computes a fixed point of any function it is given. Y is a function with the property that f(Y(f)) = Y(f) for all functions f.

From a more practical point of view, fixed point combinators allow the definition of anonymous recursive functions. Somewhat surprisingly, they can be defined with non-recursive lambda abstractions.

One well-known fixed point combinator, discovered by Haskell B. Curry, is

Y = λf.(λx.(f (x x)) λx.(f (x x)))

and can be expressed in the SKI-calculus as

Y = S (K (S I I)) (S (S (K S) K) (K (S I I)))

The simplest fixed point combinator in the SK-calculus, found by John Tromp, is

Y = S S K (S (K (S S (S (S S K)))) K
Another common fixed point combinator is the Turing fixed-point combinator (named for its discoverer Alan Turing):

Θ = (λx.λy.(y (x x y)) λx.λy.(y (x x y)))

This combinator is of interest because a variation of it can be used with applicative-order reduction:

Θ_v = λh.(λx.(h λy.(y (x x y))) λx.(h λy.(y (x x y))))

Fixed point combinators are not especially rare. Here is one constructed by Jan Willem Klop:

Y_k = (L L L L L L L L L L L L L L L L L L L L L L L L L L L L)

where>

L = λabcdefghijklmnopqstuvwxyzr.(r (t h i s i s a f i x e d p o i n t c o m b i n a t o r))

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 fixed point combinator 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.