Characterizing crisp simulations and crisp directed simulations between fuzzy labeled transition systems by using fuzzy modal logics

We formulate and prove logical characterizations of fuzzy simulations for fuzzy labeled transition systems under the Gödel semantics. Apart from the usual setting, we consider also the setting that allows graded modal operators, which are used to talk about the number of successors with a certain property of the current state. Our characterizations are formulated w.r.t. the existential fragment of fuzzy propositional dynamic logic and its extension with graded modal operators. The results concern fuzzy preservation of existential modal formulas under fuzzy simulations as well as the Hennessy-Milner property of fuzzy simulations.