Construction of fuzzy automata from fuzzy regular expressions

Li and Pedrycz [Y. M. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids, Fuzzy Sets and Systems 156 (2005) 68--92] 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 effective method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression. In this paper we provide such an effective method. Transforming scalars appearing in a fuzzy regular expression {\alpha} into letters of the new extended alphabet, we convert the fuzzy regular expression {\alpha} to an ordinary regular expression {\alpha}_{R}. Then, starting from an arbitrary nondeterministic finite automaton A that recognizes the language ||{\alpha}_R|| represented by the regular expression {\alpha}_R, we construct fuzzy finite automata A_{\alpha} and A_{\alpha}^r with the same or even less number of states than the automaton A, which recognize the fuzzy language ||{\alpha}|| represented by the fuzzy regular expression {\alpha}. The starting nondeterministic finite automaton A can be obtained from {\alpha}_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.