Construction of fuzzy automata from fuzzy regular expressions

TL;DR

This paper presents an effective method to construct fuzzy finite automata from fuzzy regular expressions by transforming them into ordinary regular expressions and leveraging existing automaton construction techniques.

Contribution

It introduces a novel procedure to convert fuzzy regular expressions into fuzzy automata, improving upon previous theoretical results by providing a practical construction method.

Findings

Constructed fuzzy automata have fewer or equal states compared to initial automata.

Method applies standard automaton conversion techniques to fuzzy expressions.

Ensures recognition of the same fuzzy language as the original expression.

Abstract

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.