|
|
|
|
|
Fixed Point CombinatorA fixed point combinator is a function which computes fixed points 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: - Θv = 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: - Yk = (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. See also External link - http://okmij.org/ftp/Computation/fixed-point-combinators.html
- http://www.cs.brown.edu/courses/cs173/2002/Lectures/2002-10-28-lc.pdf
- http://www.mactech.com/articles/mactech/Vol.07/07.05/LambdaCalculus/
- http://www.csse.monash.edu.au/~lloyd/tildeFP/Lambda/Examples/Y/ (executable)
|
 |
|
| Copyright 2005-2009 OnPedia.com. All Rights Reserved |
|
|