Construction of fuzzy automata from fuzzy regular expressions

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 α into letters of the new extended alphabet, we convert the fuzzy regular expression α to an ordinary regular expression αR. Then, starting from an arbitrary nondeterministic finite automaton A that recognizes the language ‖αR‖ represented by the regular expression αR, we construct fuzzy finite automata Aα and with the same or even less number of states than the automaton A, which recognize the fuzzy language ‖α‖ represented by the fuzzy regular expression α. The starting nondeterministic finite automaton A can be obtained from αR 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.