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There are two kinds of bisimulation, namely crisp and fuzzy, between fuzzystructures such as fuzzy automata, fuzzy labeled transition systems, fuzzyKripke models and fuzzy interpretations in description logics. Fuzzybisimulations between fuzzy automata over a complete residuated lattice havebeen introduced by \'Ciri\'c et al. in 2012. Logical characterizations of fuzzybisimulations between fuzzy Kripke models (respectively, fuzzy interpretationsin description logics) over the residuated lattice [0,1] with the G\"odelt-norm have been provided by Fan in 2015 (respectively, Nguyen et al. in 2020).There was the lack of logical characterizations of fuzzy bisimulations betweenfuzzy graph-based structures over a general residuated lattice, as well as overthe residuated lattice [0,1] with the {\L}ukasiewicz or product t-norm. In thisarticle, we provide and prove logical characterizations of fuzzy bisimulationsin fuzzy modal logics over residuated lattices. The considered logics are thefuzzy propositional dynamic logic and its fragments. Our logicalcharacterizations concern invariance of formulas under fuzzy bisimulations andthe Hennessy-Milner property of fuzzy bisimulations. They can be reformulatedfor other fuzzy structures such as fuzzy label transition systems and fuzzyinterpretations in description logics.

Logical Characterizations of Fuzzy Bisimulations in Fuzzy Modal Logics over Residuated Lattices