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

中文翻译：

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$\epsilon$-模糊自动机的互模拟关系


本文研究模糊自动机的近似互模拟关系，以研究模糊自动机的近似最小化问题。对于一个小的正实数ε，我们引入了两个模糊自动机之间的ε互模拟关系的概念，并证明了在两个模糊自动机之间的ε互模拟关系下，模糊自动机A的行为与模糊自动机B的行为相差ε。他们。此外，还定义了两个模糊自动机之间的满射泛函ε-互模拟关系的概念。根据满射泛函ε-互模拟关系，我们讨论了模糊自动机的ε-互模拟关系。给出了通过模糊自动机的给定ε-互模拟构建聚合模糊自动机。此外，我们发现模糊自动机可能不存在最大ε互模拟关系，并且我们新颖地给出了一种有效的算法来构造给定模糊自动机的所有最大ε互模拟关系。最后指出模糊自动机的互模拟关系也是ε互模拟关系，并讨论了实数ε保证最大ε互模拟存在的条件。