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 equivalentways to represent fuzzy languageswithmembership values in a lattice-orderedmonoid, 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 expressionα 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 automataAα andA r α with the same or even lessnumber of states than the automatonA , 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.