<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 Fwhere 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.
(2005-04-12)
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