Fuzzy Simulations and Bisimulations between Fuzzy Automata

TL;DR

This paper introduces new fuzzy simulation and bisimulation concepts for fuzzy automata over complete residuated lattices, providing logical characterizations and showing they are more general and refined than previous notions.

Contribution

The paper presents novel fuzzy simulation and bisimulation definitions applicable to any complete residuated lattice, with properties and logical characterizations.

Findings

Fuzzy languages are preserved under fuzzy simulations.

Fuzzy automata are invariant under fuzzy bisimulations.

The new notions are more general than previous models.

Abstract

Simulations and bisimulations between two fuzzy automata over a complete residuated lattice were defined by \'Ciri\'c et al. (2012) as fuzzy relations between the sets of states of the automata. However, they act as a crisp relationship between the automata. 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 Stanimirovi\'c 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. In this article, we introduce and study fuzzy simulations and bisimulations between fuzzy automata over a complete residuated lattice. These notions are novel and have good properties. They are defined for fuzzy automata over any complete residuated lattice. We prove that the fuzzy language recognized by a fuzzy automaton is fuzzily preserved by fuzzy simulations and fuzzily invariant under fuzzy bisimulations. We also prove that the notions of fuzzy simulation and bisimulation have the Hennessy-Milner properties, which are a logical characterization of the greatest fuzzy simulation or bisimulation between two fuzzy automata. In addition, we provide results showing that our notions of fuzzy simulation and bisimulation are more general and refined than the notions of simulation and bisimulation introduced by \'Ciri\'c et al. and the notions of approximate simulation and bisimulation introduced by Stanimirovi\'c et al.