P-(D): A Probabilistic Extension of (D) for Probabilistic Ontologies in the Semantic Web

Ontologies play a central role in the development of the semantic web, as they provide precise definitions of shared terms in web resources. One important web ontology language is DAML+OIL; it has a formal semantics and a reasoning support through a mapping to the expressive description logic $$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D) with the addition of inverse roles. In this paper, we present a probabilistic extension of $$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D), called P-$$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D), to allow for dealing with probabilistic ontologies in the semantic web. The description logic P-$$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D) is based on the notion of probabilistic lexicographic entailment from probabilistic default reasoning. It allows to express rich probabilistic knowledge about concepts and instances, as well as default knowledge about concepts.We also present sound and complete reasoning techniques for P-$$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D), which are based on reductions to classical reasoning in $$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D) and to linear programming, and which show in particular that reasoning in P-$$ \mathcal{S}\mathcal{H}\mathcal{O}\mathcal{Q} $$(D) is decidable.