ALC + T: a Preferential Extension of Description Logics

We extend the Description Logic ALC with a "typicality" operator T that allows us to reason about the prototypical properties and inheritance with exceptions. The resulting logic is called ALC + 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 with a preference relation. This allows us to obtain a modal interpretation of the typicality operator. We show that the satisfiability of anALC+Tknowledge base is decidable and it is precisely EXPTIME. We then present a tableau calculus for deciding satisfiability of ALC + T knowledge bases. Our calculus gives a (suboptimal) nondeterministic-exponential time decision procedure for ALC + T. We finally discuss how to extend ALC + T in order to infer defeasible properties of (explicit or implicit) individuals. We propose two alternatives: (i) a nonmonotonic completion of a knowledge base; (ii) a "minimal model" semantics for ALC + T whose intuition is that minimal models are those that maximise typical instances of concepts.