Miroslav Ćirić

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.

Myhill--Nerode type theory for fuzzy languages and automata

The Myhill-Nerode theory is a branch of the algebraic theory of languages and automata in which formal languages and deterministic automata are studied through right congruences and congruences on a free monoid. In this paper we develop a general Myhill-Nerode type theory for fuzzy languages with membership values in an arbitrary set with two distinguished elements 0 and 1, which are needed to take crisp languages in consideration. We establish connections between extensionality of fuzzy languages w.r.t. right congruences and congruences on a free monoid and recognition of fuzzy languages by deterministic automata and monoids, and we prove the Myhill-Nerode type theorem for fuzzy languages. We also prove that each fuzzy language possess a minimal deterministic automaton recognizing it, we give a construction of this automaton using the concept of a derivative automaton of a fuzzy language and we give a method for minimization of deterministic fuzzy recognizers. In the second part of the paper we introduce and study Nerodes and Myhills automata assigned to a fuzzy automaton with membership values in a complete residuated lattice. The obtained results establish nice relationships between fuzzy languages, fuzzy automata and deterministic automata.

Fuzzy relation equations and reduction of fuzzy automata

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.

Fuzzy equivalence relations and their equivalence classes

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semi-partitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered complete Heyting algebra, we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it. Most of the presented results are new, but some of them are generalizations of known results given in a way which simplifies and clarifies them.

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 automaton 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 quasi-orders. 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 automatons. We also demonstrate some applications of weakly left invariant fuzzy quasi-orders in conflict analysis of fuzzy discrete event systems.

Uniform fuzzy relations and fuzzy functions

In this paper we introduce and study the concepts of a uniform fuzzy relation and a (partially) uniform F-function. We give various characterizations and constructions of uniform fuzzy relations and uniform F-functions, we show that the usual composition of fuzzy relations is not convenient for F-functions, so we introduce another kind of composition, and we establish a mutual correspondence between uniform F-functions and fuzzy equivalences. We also give some applications of uniform fuzzy relations in approximate reasoning, especially in fuzzy control, and we show that uniform fuzzy relations are closely related to the defuzzification problem.

Factorization of fuzzy automata

We show that the size reduction problem for fuzzy automata is related to the problem of solving a particular system of fuzzy relation equations. This system consists of infinitely many equations, and finding its general solution is a very difficult task, so we first consider one of its special cases, a finite system whose solutions, called right invariant fuzzy equivalences, are common generalizations of recently studied right invariant or well-behaved equivalences on NFAs, and congruences on fuzzy automata. We give a procedure for constructing the greatest right invariant fuzzy equivalence contained in a given fuzzy equivalence, whichwork if the underlying structure of truth values is a locally finite residuated lattice.

Formal power series and regular operations on fuzzy languages

In this paper we study formal power series over a quantale with coefficients in the algebra of all languages over a given alphabet, and representation of fuzzy languages by these formal power series. This representation generalizes the well-known representation of fuzzy languages by their cut and kernel languages. We show that regular operations on fuzzy languages can be represented by regular operations on power series which are defined by means of operations on ordinary languages. We use power series in study of fuzzy languages which are recognized by fuzzy finite automata and deterministic finite automata, and we study closure properties of the set of polynomials and the set of polynomials with regular coefficients under regular operations on power series.

Fuzzy homomorphisms of algebras

In this paper we consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. In particular, we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations, called uniform fuzzy relational morphisms, and those which are partially uniform F-functions, called fuzzy homomorphisms. Both uniform fuzzy relations and partially uniform F-functions were introduced in a recent paper by us. Uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras. We give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms, we establish certain relationships between them and fuzzy congruences, and we prove homomorphism and isomorphism theorems concerning them. We also point to some applications of uniform fuzzy relational morphisms.

Decompositions of semigroups induced by identities

In this paper we consider decompositions of semigroups induced by identities. Here we give some new characterizations of a semilattice of Archimedean semigroups and, using this, we describe all identities which induce decompositions into a semilattice of Archimedean semigroups. Also, we give a solution for one problem ofШеврин andСуханов [27].