  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.
