Reduction of fuzzy automata by means of fuzzy quasi-orders

Abstract

In our recent paper we have established close relationships between state reduction of a fuzzy recognizer and resolution of a particular system of fuzzy relation equations. In that paper we have also studied reductions by means of those solutions which are fuzzy equivalences. In this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasiorders. Generally, fuzzy quasi-orders and fuzzy equivalences are equally good in the state reduction, but we show that right and left invariant fuzzy quasi-orders give better reductions than right and left invariant fuzzy equivalences. We also show that alternate reductions by means of fuzzy quasi-orders give better results than alternate reductions by means of fuzzy equivalences. Furthermore we study a more general type of fuzzy quasi-orders, weakly right and left invariant ones, and we show that they are closely related to determinization of fuzzy recognizers. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.

Fuzzy relation equations and subsystems of fuzzy transition systems

“…For each a 2 A, the R-afterset of a is the fuzzy subset aR 2 L A defined by (aR)(b) = R(a,b), for any b 2 A, and the R-foreset of a is the fuzzy subset Ra 2 L A defined by (Ra)(b) = R(b,a), for any b 2 A (cf. [2,9,25,54,87]). Clearly, R is a symmetric fuzzy relation if and only if aR = Ra, for each a 2 A.…”

“…[74,75,[93][94][95][96][97]). From a different point of view, fuzzy automata with membership values in a complete residuated lattice were studied by Ignjatović , Ć irić and their coworkers [19][20][21][22]39,41,42,44,86,87]. Fuzzy automata taking membership values in a lattice-ordered monoid were investigated by Li and others [55,56,58,60], fuzzy automata over other types of lattices were the subject of [3,31,57,59,50,51,[68][69][70][71], and automata which generalize fuzzy automata over any type of lattices, as well as weighted automata over semirings, have been studied recently in [16,30,46].…”

“…Recently, fuzzy relation equations and inequalities have been very successfully applied in the theory of fuzzy automata. In [21,22,87] they were used in the reduction of the number of states of fuzzy finite automata, and in [19,20] (see also [17]) they were employed in the study of simulation, bisimulation and equivalence of fuzzy automata. Here, fuzzy relation equations and inequalities are used in the study of subsystems of fuzzy transition systems (by which we mean fuzzy automata without fixed fuzzy sets of initial and terminal states).…”

“…Systems of fuzzy relation inequalities that we use to define subsystems, reverse subsystems and double subsystems of a fuzzy transition system are closely related to the so-called weakly linear systems of fuzzy relation inequalities, which have been recently studied in [40,43,45], and in [19][20][21][22]87] have been used in the study of fuzzy automata. These systems have similar forms, they consist of inequalities defined by residuated functions, and consequently, closures and openings related to subsystems, reverse subsystems and double subsystems can also be computed using the methodology developed in [40,43,45].…”

Bisimulations in fuzzy social network analysis

“…Such an approach has previously been shown to be very efficient in solving some fundamental problems of the theory of fuzzy automata, such as the reduction of the number of states and the problems of equivalence, simulation and bisimulation (cf. [7,8,9,10,17,25]). …”