Weakly linear systems of fuzzy relation inequalities and their applications: A brief survey

Abstract

Weakly linear systems of fuzzy relation inequalities and equations have recently emerged from research in the theory of fuzzy automata. From the general aspect of the theory of fuzzy relation equations and inequalities homogeneous and heterogeneous weakly linear systems have been discussed in two recent papers. Here we give a brief overview of the main results from these two papers, as well as from a series of papers on applications of weakly linear systems in the state reduction of fuzzy automata, the study of simulation, bisimulation and equivalence of fuzzy automata, and in the social network analysis. Especially, we present algorithms for computing the greatest solutions to weakly linear systems.

Regular fuzzy equivalences and regular fuzzy quasi-orders

“…We consider a two-mode fuzzy network -an ordered triple A = (A, B, R), where A and B are non-empty sets and R is a fuzzy relation between A and B, and define a pair of regular fuzzy equivalences on A as a pair (E, F ) of fuzzy equivalences on A and B, respectively, satisfying E • R = R • F . Similar fuzzy relation equations and inequalities have been recently extensively studied by Ćirić, Ignjatović and others in [13,14,15,16,17,24,25,26,27], where algorithms for computing their greatest solutions have been provided. Using the general ideas presented in these studies and of the well known Paige-Tarjan partition refinement algorithm [29], here we develop efficient procedures for computing the greatest pairs of regular fuzzy equivalences and regular fuzzy-quasi orders on twomode fuzzy networks.…”

“…The first generalization, that they called the regular similarity, specifies the degree of similarity between entities in the social network, whereas the second generalization, called the generalized regular equivalence, determines the crisp partition of the entities in a fuzzy social network. The first concept has been discussed from another aspect in [11,24,25], where the name the regular fuzzy equivalence has been introduced.…”

Bisimulations in fuzzy social network analysis

“…[5,13,14,15,17,18,19]). In the study of fuzzy social networks conducted in the listed papers positional analysis and regular fuzzy equivalences had a central place.…”

“…Regular fuzzy equivalences have been first studied by Fan et al [14,15], where they were called regular similarities, and from a similar point of view they have been recently investigated in [13]. From a different point of view regular fuzzy equivalences have been studied in [5,17,18,19], and there it has been shown that the greatest regular fuzzy equivalence on a fuzzy network can be computed as the greatest solution to a particular system of fuzzy relation equations. 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.…”

“…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]). …”