Second-order reasoning in description logics

Abstract

Description logics refer to a family of formalisms concentrated around concepts, roles and individuals. They belong to the most frequently used knowledge representation formalisms and provide a logical basis to a variety of well known paradigms. The main reasoning tasks considered in the area of description logics are those reducible to subsumption. On the other hand, any knowledge representation system should be equipped with a more advanced reasoning machinery. Therefore in the current paper we make a step towards integrating description logics with second-order reasoning. One of the important motivations behind introducing second-order formalism follows from the fact that many forms of commonsense and nonmonotonic reasoning used in AI can be modelled within the second-order logic. To achieve our goal we first extend description logics with a possibility to quantify over concepts. Since one of the main criticisms against the use of second-order formalisms is their complexity, we next propose second-order quantifier elimination techniques applicable to a large class of description logic formulas. Finally we show applications of the techniques, in particular in reasoning with circumscribed concepts and approximated terminological formulas.