Decidability of minimization of fuzzy automata

Abstract

State minimization is a fundamental problem in automata theory. The problem is also of great importance in the study of fuzzy automata. However, most work in the literature considered only state reduction of fuzzy automata, whereas the state minimization problem is almost untouched for fuzzy automata. Thus in this paper we focus on the latter problem. Formally, the decision version of the minimization problem of fuzzy automata is as follows: \begin{itemize} \item Given a fuzzy automaton $\mathcal{A}$ and a natural number $k$, that is, a pair $\langle \mathcal{A}, k\rangle$, is there a $k$-state fuzzy automaton equivalent to $\mathcal{A}$? \end{itemize} We prove for the first time that the above problem is decidable for fuzzy automata over totally ordered lattices. To this end, we first give the concept of systems of fuzzy polynomial equations and then present a procedure to solve these systems. Afterwards, we apply the solvability of a system of fuzzy polynomial equations to the minimization problem mentioned above, obtaining the decidability. Finally, we point out that the above problem is at least as hard as PSAPCE-complete.