Editing Least fixed point (section)

==Examples==
With the usual order on the [[real number]]s, the least fixed point of the real function ''f''(''x'') =&nbsp;''x''<sup>2</sup> is ''x''&nbsp;=&nbsp;0 (since the only other fixed point is 1 and 0&nbsp;<&nbsp;1). In contrast, ''f''(''x'') =&nbsp;''x''&nbsp;+&nbsp;1 has no fixed points at all, so has no least one, and ''f''(''x'') =&nbsp;''x'' has infinitely many fixed points, but has no least one.

Let <math>G = (V, A)</math> be a [[directed graph]] and <math>v</math> be a vertex. The [[set (mathematics)|set]] of vertices accessible from <math>v</math> can be defined as the  least fixed-point of the function <math>f: \wp(V) \to \wp(V)</math>, defined as <math>f(X) = \{ v \} \cup \{ x \in V: \text{ for some } w \in X \text{ there is an arc from } w \text{ to } x \} .</math>
The set of vertices which are co-accessible from <math>v</math> is defined by a similar least fix-point. The [[strongly connected component]] of <math>v</math> is the [[intersection (set theory)|intersection]] of those two least fixed-points.

Let <math>G = (V, \Sigma, R, S_0)</math> be a [[context-free grammar]]. The set <math>E</math> of symbols which produces the [[empty string]] <math>\varepsilon</math> can be obtained as the least fixed-point of the function <math>f: \wp(V) \to \wp(V)</math>, defined as <math>f ( X ) = \{ S \in  V: \; S \in X \text{ or } (S \to \varepsilon) \in R \text{ or } (S \to S^1 \dots S^n) \in R \text{ and } S^i \in X \text{, for all } i \}</math>, where <math>\wp(V)</math> denotes the [[power set]] of <math>V</math>.