Fixed-point combinator

In <a href="/facts/Combinatory_logic/D7NNo4xT">combinatory logic</a> for <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, a fixed-point combinator (or fixpoint combinator): p.26  is a <a href="/facts/Higher-order_function/VjnjGbpt">higher-order function</a> (i.e., a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> which takes a function as <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a>) that returns some <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> (a value that is mapped to itself) of its argument function, if one exists.
Formally, if 
 
 
 
 
 f
 i
 x
 
 
 
 {\displaystyle \mathrm {fix} }
 
 is a fixed-point combinator and the function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has one or more fixed points, then 
 
 
 
 
 f
 i
 x
 
  
 f
 
 
 {\displaystyle \mathrm {fix} \ f}
 
 is one of these fixed points, i.e.,

f
          i
          x
        
         
        f
         
        =
        f
         
        (
        
          f
          i
          x
        
         
        f
        )
        .
      
    
    {\displaystyle \mathrm {fix} \ f\ =f\ (\mathrm {fix} \ f).}

Fixed-point combinators can be defined in the <a href="/facts/Lambda_calculus/DaD3RuQ4">lambda calculus</a> and in <a href="/facts/Functional_programming/cdxqPEIM">functional programming</a> languages, and provide a means to allow for <a href="/facts/Recursion_(computer_science)/2uXo18Gv">recursive definitions</a>.

Fixed-point combinator open-in-new

Fixed-point combinator