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 = 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.
2. bug fix.