Reasoning About Typicality in ALC and EL

In this work we summarize our recent results on extending Description Logics for reasoning about prototypical properties and inheritance with exceptions. First, we focus our attention on the logic ALC. We present a nonmonotonic logic ALC + Tmin, which is built upon a monotonic logic ALC +T obtained by adding a typicality operator T to ALC. The operator T is intended to select the “most normal” or “most typical” instances of a concept, so that knowledge bases may contain subsumption relations of the form“T(C) is subsumed by P”, expressing that typical C-members have the property P. In order to perform nonmonotonic inferences, we define a “minimal model” semantics: the intuition is that preferred, or minimal models are those that maximise typical instances of concepts. By means of ALC+Tmin we are able to infer defeasible properties of (explicit or implicit) individuals.We also show that the satisfiability of an ALC + T-knowledge base is in EXPTIME, whereas deciding query entailment in ALC + Tmin is in co-NExp^NP. We apply our approach based on the operator T also to the low complexity Description Logic EL+⊥. We propose an extension EL+⊥T and we show that the problem of entailment in EL+⊥T is in co-NP.