No abstract

This paper presents a contribution to the problem of similarity-based minimization of deterministic fuzzy finite tree automata (DFFTA). The main question is: how to minimize the number of states of a complete and reduced DFFTA such that the languages of the original automaton and the minimized one be similar but not necessarily equal? Based on extended concepts of fuzzy distance and similarity measures on L-fuzzy sets, we introduce the notion of similarity-based minimal (s-minimal) DFFTA which approximately accepts a fuzzy tree language. Then, a solution for handeling the trade-off between the amount of reduction and the quality of preserving the behavior of system is presented. The paper deals with fuzzy tree automata over complete lattices, but identical results can also be obtained in a more general context for fuzzy tree automata over complete residuated lattices, lattice-ordered monoids, and even for weighted automata over commutative semirings.

Until now, some methods for minimizing deterministic fuzzy finite tree automata (DFFTA) and weighted tree automata have been established by researchers. Those methods are language preserving, but the behavior of original automata and minimized one may be different. This paper, considers both language preserving and behavior preserving in minimization process. We drive Myhill-Nerode kind theorems corresponding to each proposed method and introduce PTIME algorithms for behaviorally and linguistically minimization. The proposed minimization algorithms are based on two main steps. The first step includes finding dependency between equivalency of states, according to the set of transition rules of DFFTA, and making merging dependency graph (MDG). The second step is refinement of MDG and making minimization equivalency set (MES). Additionally, behavior preserving minimization of DFFTA requires a preprocessing for modifying fuzzy membership grade of rules and final states, which is called normalization.

No abstract

No abstract