In this paper, for a real number ϵ ∈ [ 0 , 1 ] , we put forward the notion of fuzzy ϵ-approximate regular languages and study their properties, especially the Pumping lemma in the context of this kind of languages. By comparing sets of fuzzy ϵ-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 ϵ-approximate regular languages are different for different ϵ ∈ ( 0 , 1 2 ) . We also investigate whether operations closed in the set of fuzzy regular languages are still closed in the set of fuzzy ϵ-approximate regular languages. Furthermore, for a fuzzy ϵ-approximate regular language f, ϵ-approximate equivalence relations for f are characterized in order to construct deterministic fuzzy automata ϵ-accepting f. If a fuzzy ϵ-approximate regular language f is 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 ϵ-accepting f by means of A. Finally, we point out that the number of states of minimal deterministic fuzzy automata ϵ-accepting a fuzzy regular language is smaller than or equal to that of minimal deterministic fuzzy automata accepting this language.
