least fixed-pointcomputing dictionary

<mathematics> A function f may have many fixed-points (x such that f x = x). For example, any value is a fixed-point of the identity function, (\ x . x).

If f is recursive, we can represent it as

f = fix F

where F is some higher-order function and

fix F = F (fix F).

The standard denotational semantics of f is then given by the least fixed-point of F. This is the least upper bound of the infinite sequence (the ascending Kleene chain) obtained by repeatedly applying F to the totally undefined value, bottom. I.e.

fix F = LUB bottom, F bottom, F (F bottom), ....

The least fixed-point is guaranteed to exist for a continuous function over a cpo.

(01 Oct 2005)