Editing Possibility theory (section)

==Formalization of possibility==
For simplicity, assume that the [[universe of discourse]] Ω is a finite set. A possibility measure is a function <math>\Pi</math> from <math>2^\Omega</math> to [0,&nbsp;1] such that:

:Axiom 1: <math>\Pi(\varnothing) = 0</math>

:Axiom 2: <math>\Pi(\Omega) = 1</math>

:Axiom 3: <math>\Pi(U \cup V) = \max \left( \Pi(U), \Pi(V) \right)</math> for any [[disjoint sets|disjoint]] subsets <math>U</math> and <math>V</math>.<ref>Dubois, D.; Prade, H.: Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, 1988</ref>

It follows that, like probability on finite [[probability space]]s, the possibility measure is determined by its behavior on singletons:

:<math>\Pi(U) = \max_{\omega \in U} \Pi (\{\omega\}).</math>

Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω.

Axiom 2 could be interpreted as the assumption that the evidence from which <math>\Pi</math> was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1.

Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that:

:<math>\Pi(U \cup V) = \max \left( \Pi(U), \Pi(V) \right)</math> for ''any'' subsets <math>U</math> and <math>V</math>.

Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is [[Principle of compositionality|''compositional'']] with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally:

:<math>\Pi(U \cap V) \leq \min \left( \Pi(U), \Pi(V) \right) \leq \max \left( \Pi(U), \Pi(V) \right).</math>

When Ω is not finite, Axiom 3 can be replaced by:

:For all index sets <math>I</math>, if the subsets <math>U_{i,\, i \in I}</math> are [[pairwise disjoint]], <math>\Pi\left(\bigcup_{i \in I} U_i\right) = \sup_{i \in I}\Pi(U_i).</math>