# least fixed point

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<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)

Nearby terms: leap second « learning curve « leased line « least fixed point » least recently used » least significant bit » least upper bound

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