least fixed point
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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.