Fuzzy directed simulations for fuzzy modal logics over residuated lattices

We introduce the notion of fuzzy directed simulation between fuzzy Kripke models over any linear and complete residuated lattice and investigate its fundamental properties. In particular, we prove that all positive formulas of the fuzzy modal logic $\mathit{fPDL}$ are preserved under fuzzy directed simulations and establish a Hennessy-Milner theorem for this notion. Furthermore, we present a method for computing the greatest fuzzy directed simulation between two finite fuzzy Kripke models and implement it for the case where the underlying residuated lattice is the G\"odel, product, or Lukasiewicz structure. Finally, we experimentally evaluate the performance of the implementation and present the obtained results.