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Description logics (DLs) are a suitable formalism for representing knowledgeabout domains in which objects are described not only by attributes but also bybinary relations between objects. Fuzzy extensions of DLs can be used for suchdomains when data and knowledge about them are vague and imprecise. One of thepossible ways to specify classes of objects in such domains is to use conceptsin fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs ischaracterized by bisimilarity. The bisimilarity relation of an interpretationis the largest auto-bisimulation of that interpretation. In DLs and their fuzzyextensions, such equivalence relations can be used for concept learning. Inthis paper, we define and study fuzzy bisimulation and bisimilarity for fuzzyDLs under the G\"odel semantics, as well as crisp bisimulation and strongbisimilarity for such logics extended with involutive negation. The consideredlogics are fuzzy extensions of the DL $\mathcal{ALC}_{reg}$ (a variant of PDL)with additional features among inverse roles, nominals, (qualified orunqualified) number restrictions, the universal role, local reflexivity of arole and involutive negation. We formulate and prove results on invariance ofconcepts under fuzzy (resp. crisp) bisimulation, conditional invariance offuzzy TBoxex/ABoxes under bisimilarity (resp. strong bisimilarity), and theHennessy-Milner property of fuzzy (resp. crisp) bisimulation for fuzzy DLswithout (resp. with) involutive negation under the G\"odel semantics. Apartfrom these fundamental results, we also provide results on using fuzzybisimulation to separate the expressive powers of fuzzy DLs, as well as resultson using strong bisimilarity to minimize fuzzy interpretations.

Bisimulation and bisimilarity for fuzzy description logics under the G\"odel semantics