Construction of fuzzy automata from fuzzy regular expressions

  • Authors:

  • Aleksandar Stamenkovi;Miroslav Ćiri

  • Affiliations:

  • University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia;University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2012

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Abstract

Li and Pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice-ordered monoid, and generalize the well-known results of the classical theory of formal languages. In particular, they have shown that a fuzzy language over an integral lattice-ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton. However, they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression. In this paper we provide such an efficient method. Transforming scalars appearing in a fuzzy regular expression @ainto letters of the new extended alphabet, we convert the fuzzy regular expression @ato an ordinary regular expression @a"R. Then, starting from an arbitrary nondeterministic finite automaton A that recognizes the language @?@a"R@? represented by the regular expression @a"R, we construct fuzzy finite automata A"@a and A"@a^r with the same or even less number of states than the automaton A, which recognize the fuzzy language @?@a@?represented by the fuzzy regular expression @a. The starting nondeterministic finite automaton A can be obtained from @a"R using any of the well-known constructions for converting regular expressions to nondeterministic finite automata, such as Glushkov-McNaughton-Yamada's position automaton, Brzozowski's derivative automaton, Antimirov's partial derivative automaton, or Ilie-Yu's follow automaton.