x . x).If f is recursive f = fix Fwhere F is some ">, we can represent it as f = fix Fwhere F is some higher-order function fix F = F (fix F).The standard "> 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 fix F = LUB {bottom, F bottom, F (F bottom), ...}.The least fixed point is guaranteed to exist for a {continuous} function over a "> 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)cpo.(2005-04-12)……"> least fixed point : FOLDOC
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least fixed point : FOLDOC
least fixed point
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.
(2005-04-12)

cpo.
(2005-04-12)



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