Logical Characterizations of Crisp Bisimulations in Fuzzy Description Logics

Fuzzy description logics (FDLs) are useful for dealing with fuzzy terminological knowledge for domains with linked data. Logical similarity or indiscernibility between individuals with respect to a given FDL is a fuzzy measure, which becomes crisp when the logic is extended with the Baaz projection operator. The measure is closely related to bisimulation. While logical indiscernibility is defined semantically, its corresponding notion based on bisimulation enables the computation. In this article, we study crisp bisimulations between fuzzy interpretations in FDLs with the Baaz projection operator under a general semantics based on an abstract algebra of fuzzy truth values. We define such bisimulations for a large class of FDLs with a rich set of well-known concept and role constructors, including qualified/unqualified number restrictions, nominals and the role constructors that correspond to the program constructors of propositional dynamic logic. We formulate and prove their logical characterizations, including the invariance of concepts under crisp bisimulations and the Hennessy–Milner property of crisp bisimulations. Such logical characterizations do not depend on a concrete semantics, such as the Gödel, Łukasiewicz, and product semantics. Based on crisp bisimulations, we also study indiscernibility of individuals in FDLs with the Baaz projection operator. An interesting consequence of our results states that, when restricting to the considered FDLs and image-finite fuzzy interpretations that are witnessed and modally saturated, indiscernibility of individuals is independent from the underlying algebra of fuzzy truth values in the case without number restrictions, and it is the same for both the Gödel and product semantics in the case with number restrictions.