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