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

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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.

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  1. Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

    Maryam Ghorani & Somaye Moghari

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  1. Maryam Ghorani
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  2. Somaye Moghari
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Correspondence to Maryam Ghorani.

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Ghorani, M., Moghari, S. Decidability of the minimization of fuzzy tree automata with membership values in complete lattices. J. Appl. Math. Comput. 68, 461–478 (2022). https://doi.org/10.1007/s12190-021-01529-6

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