Rational closure for all description logics

Abstract

Rational closure is one of the most extensively studied nonmonotonic extensions of description logics. Nonetheless, so far it has been investigated only for description logics that satisfy the disjoint model union property, or limited fragments that support nominals. In this paper we show that for sufficiently expressive description logics, the traditional correspondence between rational closure and ranked interpretations does not hold. Therefore, in order to extend rational closure to a wider class of description logics it is necessary to change the definition of rational closure, or alternatively abandon its standard semantics. Here we pursue the former approach, and introduce stable rational closure, based on stable rankings. The resulting nonmonotonic logic is a natural extension of the standard rational closure: First, its refined exceptionality criterion yields a closure that satisfies the KLM postulates. Second, when a knowledge base enjoys the disjoint model union property, then stable rational closure equals the old notion. In the other cases, stable rankings may raise the exceptionality level of some concepts. Stable rational closure has a model-theoretic semantics based on upward-closed models, that relax the canonical models adopted in the past, in order to deal with logics that do not satisfy the disjoint union model property. Unfortunately, stable rankings do not always exist, and are not necessarily unique. However, they can be effectively enumerated for all defeasible knowledge bases in SROIQ, using any algorithm for reasoning with ranked models.