  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.
