FIX

fix

<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 = 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.

(13 Apr 1995)