Similarity-based minimization of fuzzy tree automata

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.

Synthesizing fuzzy tree automata

“…The bisimulation relation models equivalence between states of nondeterministic automata [45]. The equivalence relation between states of an automaton is done to minimize the automaton, and the equivalence relation between states of two automata is usually done for comparing their behaviors [13,14,32].…”

“…Our definitions in this section, are borrowed from [7,32,33]. The set of natural numbers (including zero) is denoted by N. A Σ-alphabet is a finite and nonempty set of symbols.…”

“…As well, Fuzzy Tree Automata (FTA) [9,13,14,31] deal with the logic of computation by processing tree structures and calculating the grade of acceptance for each input tree based on its structural characteristics. A Fuzzy Tree Language (FTL) is a fuzzy set of trees in which an FTA recognize them [4,9,32,33,36].…”

“…An FTA calculates the membership grade of trees to its language by a nonlinear mapping which is known as the behavior of the automaton [31]. The analysis of the behavior of automata [3,32,33] involves the analysis of states of automata and how these states change. On the other hand, modeling an FTL is the task of reasoning states and transition rules of an FTA which accepts it.…”

State hyperstructures of tree automata based on lattice-valued logic

“…Bozapalidis and Bozapalidoy [3] showed that linear tree homomorphisms preserve syntactic recognizability. More results on fuzzy tree automata can be found in [24,26,39] and references therein. Since, fuzzy tree automata take values in the unit interval [0, 1], to enhance the processing ability of fuzzy tree automata, Ghorani and Zahedi [20][21][22], Ghorani et al [23] and Ghorani [18,19] extended the membership value to a more general algebraic structure and considered tree automata based on complete residuated lattice-valued logic.…”