Preferential Description Logics

Abstract

We extend the Description Logic $\mathcal{ALC}$ with a “typicality” operator T that allows us to reason about the prototypical properties and inheritance with exceptions. The resulting logic is called $\mathcal{ALC}+{\bf T}$. The typicality operator is intended to select the “most normal” or “most typical” instances of a concept. In our framework, knowledge bases may then contain, in addition to ordinary ABoxes and TBoxes, subsumption relations of the form “T(C) is subsumed by P”, expressing that typical C-members have the property P. The semantics of a typicality operator is defined by a set of postulates that are strongly related to Kraus-Lehmann-Magidor axioms of preferential logic P. We first show that T enjoys a simple semantics provided by ordinary structures equipped by a preference relation. This allows us to obtain a modal interpretation of the typicality operator. Using such a modal interpretation, we present a tableau calculus for deciding satisfiability of $\mathcal{ALC}+{\bf T}$ knowledge bases. Our calculus gives a nondeterministic-exponential time decision procedure for satisfiability of $\mathcal{ALC}+{\bf T}$. We then extend $\mathcal{ALC}+{\bf T}$ knowledge bases by a nonmonotonic completion that allows inferring defeasible properties of specific concept instances.