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

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