Myhill--Nerode type theory for fuzzy languages and automata

  • Authors:

  • Jelena Ignjatović;Miroslav Ćirić;Stojan Bogdanović;Tatjana Petković

  • Affiliations:

  • Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P.O. Box 224, 18000 Niš, Serbia;Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P.O. Box 224, 18000 Niš, Serbia;Faculty of Economics, University of Niš, Trg Kralja Aleksandra 11, 18000 Niš, Serbia;Nokia, Joensuunkatu 7, FIN-24100 Salo, Finland

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2010

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Abstract

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 Nerode's and Myhill's 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.