## fix

1. (mathematics) The fixed point combinator. Called Y in combinatory logic. Fix is a higher-order function which returns a fixed point of its argument (which is a function).

fix :: (a -> a) -> a fix f = f (fix f)Which satisfies the equation

fix f = x such that f x = x.Somewhat surprisingly, fix can be defined as the non-recursive lambda abstraction:

fix = \ h . (\ x . h (x x)) (\ x . h (x x))Since this involves self-application, it has an infinite type. A function defined by

f x1 .. xN = Ecan be expressed as

f = fix (\ f . \ x1 ... \ xN . E) = (\ f . \ x1 ... \xN . E) (fix (\ f . \ x1 ... \ xN . E)) = let f = (fix (\ f . \ x1 ... \ xN . E)) in \ x1 ... \xN . EIf f does not occur free in E (i.e. it is not recursive) then this reduces to simply

f = \ x1 ... \ xN . EIn the case where N = 0 and f is free in E, this defines an infinite data object, e.g.

ones = fix (\ ones . 1 : ones) = (\ ones . 1 : ones) (fix (\ ones . 1 : ones)) = 1 : (fix (\ ones . 1 : ones)) = 1 : 1 : ...Fix f is also sometimes written as mu f where mu is the Greek letter or alternatively, if f = \ x . E, written as mu x . E.

Compare quine.

Last updated: 1995-04-13

2. bug fix.

Last updated: 1998-06-25

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