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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), ....