1. The fixed pointcombinator. 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 = E can 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 . E If f does not occur free in E (i.e. it is not recursive) then this reduces to simply f = x1 ... xN . E In 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. [Jargon File] (1995-04-13) 2. bug fix. (1998-06-25)
