A non-monotonic Description Logic for reasoning about typicality

Abstract

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