Decidability of the minimization of fuzzy tree automata with membership values in complete lattices

Abstract

The objective of this paper is to introduce a method for constructing a minimal lattice-valued tree automaton with membership values in a totally ordered lattice (in short $$\mathcal {LTA}),$$ based on the solvability of a system of fuzzy polynomial equations. Since the minimization problem strongly depends on the equivalence problem, at first, the equivalence problem is examined. For this purpose, the notion of h-equivalence is defined, and a necessary and sufficient condition for the equivalence between two $$\mathcal {LTA}s$$ is provided. It is shown that the equivalence problem of $$\mathcal {LTA}s$$ is decidable. In the minimization problem, the following question is replied: given an $$\mathcal {LTA}$$ $$\mathbb {A}$$ and a positive integer k, is there an $$\mathcal {LTA}$$ with k states equivalent to $$\mathbb {A}?$$ Decidability of the minimization problem is demonstrated, and an approach to return a minimal $$\mathcal {LTA}$$ equivalent to the original one is presented. Also, the time complexity of the proposed algorithm is analyzed. Finally, some examples are presented to clarify the minimization problem.