Fuzzy ε-approximate regular languages and minimal deterministic fuzzy automata ε-accepting them

In this paper, for a real number epsilon is an element of [0, 1], we put forward the notion of fuzzy epsilon-approximate regular languages and study their properties, especially the Pumping lemma in the context of this kind of languages. By comparing sets of fuzzy epsilon-approximate regular languages under the order of set inclusion, we get an (infinite) hierarchy of fuzzy languages, the smallest is the set of fuzzy regular languages and the biggest is the set of fuzzy languages, and the sets of epsilon-approximate regular languages are different for different epsilon is an element of (0, 1/2). We also investigate whether operations closed in the set of fuzzy regular languages are still closed in the set of fuzzy epsilon-approximate regular languages. Furthermore, for a fuzzy epsilon-approximate regular language f, epsilon-approximate equivalence relations for fare characterizedin order to construct deterministic fuzzy automata epsilon-accepting f. If a fuzzy epsilon-approximate regular language fis also a fuzzy regular language and accepted by a given accessible deterministic fuzzy automaton A, then we give a polynomial-time algorithm to construct at least one minimal deterministic fuzzy automaton epsilon-accepting f by means of A. Finally, we point out that the number of states of minimal deterministic fuzzy automata epsilon-accepting a fuzzy regular language is smaller than or equal to that of minimal deterministic fuzzy automata accepting this language. (C) 2020 Elsevier B.V. All rights reserved.