Conditions for Minimal Fuzzy Deterministic Finite Automata via Brzozowski's Procedure

This paper deals with the application of Brzozowski's minimization procedure to fuzzy finite automata with truth-values in a complete residuated (zero-divisor-free) lattice. For a given fuzzy finite automaton A, the procedure computes the automaton d(r(d(r(A))), where d(A) is a (fuzzy) determinization of A, and r(A) is the reverse automaton of A. It is observed that the size of the resulting automaton is strongly dependent on the determinization method used in the procedure. Consequently, this paper studies necessary and sufficient conditions for obtaining minimal fuzzy deterministic finite automata via Brzozowski's procedure. The study is accomplished for determinization methods based on a generalized notion of accessible fuzzy subset construction. The obtained conditions determine that Brzozowski's procedure returns a minimal fuzzy deterministic finite automaton when the determinization method does not produce proportional fuzzy states. This is the reason to obtain minimal fuzzy deterministic finite automata using Brzozowski's procedure for fuzzy finite automata with truth-values in the product-structure when the determinization method is the method based on factorization of fuzzy states.