Editing Fixed-point combinator (section)

{{Short description|1=Higher-order function Y for which Y f = f (Y f)}}
In [[combinatory logic]] for [[computer science]], a '''fixed-point combinator''' (or '''fixpoint combinator''')<ref name="SPJ1987">{{cite book |last1=Peyton Jones |first1=Simon L. |author1-link=Simon Peyton Jones |year=1987 |title=The Implementation of Functional Programming |publisher=Prentice Hall International |url=https://www.microsoft.com/en-us/research/uploads/prod/1987/01/slpj-book-1987.pdf}}</ref>{{rp|p.26}} is a [[higher-order function]] (i.e., a [[Function (mathematics)|function]] which takes a function as [[Argument of a function|argument]]) that returns some ''[[Fixed point (mathematics)|fixed point]]'' (a value that is mapped to itself) of its argument function, if one exists.

Formally, if <math>\mathrm{fix}</math> is a fixed-point combinator and the function <math>f</math> has one or more fixed points, then <math>\mathrm{fix}\ f</math> is one of these fixed points, i.e.,
: <math>  \mathrm{fix}\ f\ = f\ (\mathrm{fix}\ f) .</math>

Fixed-point combinators can be defined in the [[lambda calculus]] and in [[functional programming]] languages, and provide a means to allow for [[recursion (computer science)|recursive definitions]].