Fuzzy Bisimulations in Fuzzy Description Logics Under the Gödel Semantics

Abstract

Description logics (DLs) are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects. Fuzzy DLs can be used for such domains when data and knowledge about them are vague. One of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is characterized by bisimilarity. The bisimilarity relation of an interpretation is the largest auto-bisimulation of that interpretation. In (fuzzy) DLs, it can be used for concept learning. In this paper, for the first time, we define fuzzy bisimulation and (crisp) bisimilarity for fuzzy DLs under the Godel semantics. The considered logics are fuzzy extensions of the DL \(\mathcal {ALC}_{reg}\) with additional features among inverse roles, nominals, qualified number restrictions, the universal role and local reflexivity of a role. We give results on invariance of concepts as well as conditional invariance of TBoxes and ABoxes for bisimilarity in fuzzy DLs under the Godel semantics. We also provide a theorem on the Hennessy-Milner property for fuzzy bisimulations in fuzzy DLs under the Godel semantics.