fixed-point theorem

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fixed-point theorem A theorem concerning the existence and nature of fixed points used to give solutions to equations. A fixed point of a function f f : XX

is an element x such that f(x) = x. A least fixed point is one that, among all the fixed points of f, is lowest in some partial ordering that has been imposed on the elements of X. Specifically, if ← is a partial ordering of X then x is a least fixed point if for fixed point y we have xy.

The most often-cited form of fixed-point theorem to do with computing is due originally to S. C. Kleene, and originated in recursive function theory. It states that, subject to certain assumptions, notably that f is continuous, f has a least fixed point, xf, which moreover can be characterized as the limit of a sequence x0,x1,x2,… of approximations. This abstract fact is of great relevance to the semantics of programming languages, in particular in specifying the precise meaning of constructs like iteration, recursion, and recursive types using equations.