Concept Model Semantics for DL Preferential Reasoning

The preferential and rational consequence relations first studied by Lehmann and colleagues play a central role in non-monotonic reasoning, not least because they provide the foundation for the determination of the important notion of rational closure. Although they can be applied directly to a large variety of logics, these constructions suffer from the limitation that they are largely propositional in nature. One of the main obstacles in moving beyond the propositional case has been the lack of a formal semantics which appropriately generalizes the preferential and ranked models of Lehmann et al. In this paper we propose a semantics to fill that gap for description logics, an important class of decidable fragments of first-order logic. Our semantics replaces the propositional valuations used in the models of Lehmann et al. with structures we refer to as concept models. We prove representation results for the description logic ALC for both preferential and rational consequence relations. We argue that our semantics paves the way for extending preferential and rational consequence, and therefore also rational closure, to a whole class of logics that have a semantics defined in terms of first-order relational structures.