$\epsilon$-Bisimulation Relations for Fuzzy Automata

Abstract

This paper investigates the approximate bisimulation relations for fuzzy automata in order to study the approximate minimization problem of fuzzy automata. For a small positive real number $\epsilon$ , we introduce the notion of $\epsilon$ -bisimulation relations between two fuzzy automata, and prove that the behavior of a fuzzy automaton $A$ differs by $\epsilon$ from the behavior of a fuzzy automaton $B$ under a $\epsilon$ -bisimulation relation between them. Also, the notion of surjective functional $\epsilon$ -bisimulation relations between two fuzzy automata is defined. According to surjective functional $\epsilon$ -bisimulation relations, we discuss $\epsilon$ -bisimulation relations for a fuzzy automaton. A construction of aggregated fuzzy automaton by the given $\epsilon$ -bisimulation for a fuzzy automaton is given. Furthermore, we find that there might not exist the greatest $\epsilon$ -bisimulation relation for a fuzzy automaton, and we novelly give an effective algorithm to construct all maximal $\epsilon$ -bisimulation relations for the given fuzzy automaton. Finally, we point out that bisimulation relations for a fuzzy automaton are also $\epsilon$ -bisimulation relations, the conditions for the real number $\epsilon$ to ensure the existence of the greatest $\epsilon$ -bisimulation are also discussed.