Approximate Bisimulations for Fuzzy Automata Over Complete Heyting Algebras

Fuzzy Simulations and Bisimulations between Fuzzy Automata

“…In particular, if there exists a forward simulation (respectively, bisimulation) between fuzzy automata A and A ′ , then the fuzzy language recognized by A is crisply less than or equal (respectively, crisply equal) to the fuzzy language recognized by A ′ . In [27] Stanimirović et al introduced approximate simulations and bisimulations between fuzzy automata with the aim to fuzzify this phenomenon. These notions are also fuzzy relations.…”

“…The motivation of this work is to introduce novel notions of simulation and bisimulation between fuzzy automata that are more refined than the ones defined in [6] and [27].…”

“…We also prove that the notions of fuzzy simulation and bisimulation have the Hennessy-Milner properties (Theorems 3.14 and 4.13), which are a logical characterization of the greatest fuzzy simulation or bisimulation between two fuzzy automata. Our notions of fuzzy simulation and bisimulation for fuzzy automata are more refined than the notions of forward simulation and bisimulation introduced in [6] and the notions of approximate forward simulation and bisimulation introduced in [27] in the following aspects:…”

“…• While approximate forward simulations and bisimulations [27] are defined only for fuzzy automata over a complete Heyting algebra, fuzzy simulations and bisimulations are defined for fuzzy automata over any complete residuated lattice.…”

“…In particular, if there exists a (forward) bisimulation between two fuzzy automata, then the fuzzy languages recognized by them are crisply equal. Approximate simulations and bisimulations introduced by Stanimirović et al (2020) aim at fuzzifying this phenomenon. However, they are defined only for fuzzy automata over a complete Heyting algebra and do not give the exact relationship between states of the automata.…”