Myhill–Nerode type theory for fuzzy languages and automata

The Myhill–Nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid. In this paper we develop a general Myhill–Nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1, which are needed to take crisp languages in consideration. We establish connections between extensionality of fuzzy languages w.r.t. right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids, and we prove the Myhill–Nerode type theorem for fuzzy languages. We also prove that each fuzzy language possess a minimal deterministic automaton recognizing it, we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers. In the second part of the paper we introduce and study Nerodeʹs and Myhillʹs automata assigned to a fuzzy automaton with membership values in a complete residuated lattice. The obtained results establish nice relationships between fuzzy languages, fuzzy automata and deterministic automata.