Fuzzy relation equations and reduction of fuzzy automata

Abstract

We show that the state reduction problem for fuzzy automata is related to the problem of finding a solution to a particular system of fuzzy relation equations in the set of all fuzzy equivalences on its set of states. This system may consist of infinitely many equations, and finding its non-trivial solutions may be a very difficult task. For that reason we aim our attention to some instances of this system which consist of finitely many equations and are easier to solve. First, we study right invariant fuzzy equivalences, and their duals, the left invariant ones. We prove that each fuzzy automaton possesses the greatest right (resp. left) invariant fuzzy equivalence, which provides the best reduction by means of fuzzy equivalences of this type, and we give an effective procedure for computing this fuzzy equivalence, which works if the underlying structure of truth values is a locally finite residuated lattice. Moreover, we show that even better reductions can be achieved alternating reductions by means of right and left invariant fuzzy equivalences. We also study strongly right and left invariant fuzzy equivalences, which give worse reductions than right and left invariant ones, but whose computing is much easier. We give an effective procedure for computing the greatest strongly right (resp. left) invariant fuzzy equivalence, which is applicable to fuzzy automata over an arbitrary complete residuated lattice.

Formal power series and regular operations on fuzzy languages

“…Note that the obtained results also hold for fuzzy languages and automata over a complete residuated lattice, which have been intensively studied in recent papers [7,8,10,11,21,[24][25][26][27]. Let us also note that fuzzy languages over a quantale L have a natural interpretation as power series on the underlying free monoid X Ã , with coefficients in the semiring reduct of L, but this approach completely differs than the approach proposed here.…”

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.…”

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]). …”