A NonMonotonic Description Logic for Reasoning About Typicality

In this paper we propose a non-monotonic extension of the Description Logic ALC for reasoning about prototypical properties and inheritance with exceptions. The resulting logic, called ALC + Tmin, is built upon a previously introduced (monotonic) logic ALC + T that is 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) D (“T(C) is subsumed by D”), expressing that typical C-members are instances of concept D. From a knowledge representation point of view, the monotonic logic ALC + T is too weak to perform inheritance reasoning. In ALC + Tmin, in order to perform nonmonotonic inferences, we define a “minimal model” semantics over ALC + T. The intuition is that preferred or minimal models are those that maximize typical instances of concepts. By means of ALC + Tmin we are able to infer defeasible properties of (explicit or implicit) individuals. We also present a tableau calculus for deciding ALC + Tmin entailment that allows to give a complexity upper bound for the logic, namely that query entailment is in CO-NEXP^NP.