least fixed point
<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}.
(2005-04-12)