Semantic characterization of rational closure: From propositional logic to description logics

In this paper we provide a semantic reconstruction of rational closure. We first consider rational closure as defined by Lehman and Magidor [33] for propositional logic, and we provide a semantic characterization based on a minimal models mechanism on rational models. Then we extend the whole formalism and semantics to Description Logics, by focusing our attention to the standard ALC: we first naturally adapt to Description Logics Lehman and Magidor's propositional rational closure, starting from an extension of ALC with a typicality operator T that selects the most typical instances of a concept C (hence T(C) stands for typical C). Then, for the Description Logics, we define a minimal model semantics for the logic ALC and we show that it provides a semantic characterization for the rational closure of a Knowledge base. We consider both the rational closure of the TBox and the rational closure of the ABox.