A generalization of Myhill–Nerode theorem for fuzzy languages

Abstract

The well-known Myhill–Nerode Theorem provides a necessary and sufficient condition for a language to be regular. In the context of fuzzy languages and automata theory, Myhill–Nerode type theorems have been proved for fuzzy languages with finite range. This paper introduces a new right equivalence relation on the free monoid of an alphabet based on the notion of factorization of fuzzy languages. The index of this relation for a fuzzy language with infinite range can be finite. This fact allows us to generalize the Myhill–Nerode Theorem for any kind of fuzzy languages. In this paper is proved that the following two conditions are mutually equivalent for a given fuzzy language X: (i) there exists a factorization such that the right equivalence relation of X (defined via the factorization) has a finite index; (ii) the fuzzy language X is recognized by a fuzzy deterministic finite automaton.