We introduce the notion of fuzzy directed simulation between fuzzy Kripke models over linear and complete residuated lattices and investigate its fundamental properties. In particular, we prove that all positive formulas of a fuzzy extension of propositional dynamic logic 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 Gödel, product, or Lukasiewicz structure. Finally, we experimentally evaluate the performance of the implementation and present the obtained results.

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