  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.

Formal Languages and Automata Theory