Canonization of max-min fuzzy automata

Abstract

In this paper, we propose a canonization method for fuzzy automata, i.e., a determinization method that is able to return a minimal fuzzy deterministic automaton equivalent to the original fuzzy automaton. The canonization method is derived from the well-known Brzozowski's algorithm for ordinary nondeterministic automata. For a given fuzzy automaton A, we prove that the construction Mˆ(r(N(r(A)))) returns a minimal fuzzy deterministic automaton equivalent to A. In that construction, r(.) represents the reversal of a fuzzy automaton, N(.) is the determinization of a fuzzy automaton based on fuzzy accessible subset construction, and Mˆ(.) is the determinization of a fuzzy automaton via factorization of fuzzy states which also includes a simple reduction of a particular case of proportional fuzzy states. The method is accomplished for fuzzy automata with membership values over the Gödel structure (also called max-min fuzzy automata). These fuzzy automata are always determinizable and have been proved useful in practical applications.