Fuzzy relation equations and subsystems of fuzzy transition systems

Abstract

In this paper we study subsystems, reverse subsystems and double subsystems of a fuzzy transition system. We characterize them in terms of fuzzy relation inequalities and equations, as eigen fuzzy sets of the fuzzy quasi-order Q_δ and the fuzzy equivalence E_δ generated by fuzzy transition relations, and as linear combinations of aftersets and foresets of Q_δ and equivalence classes of E_δ. We also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition system $\mathcal{T}$ form both closure and opening systems in the lattice of fuzzy subsets of A, where A is the set of states of $\mathcal{T}$, and we provide efficient procedures for computing related closures and openings of an arbitrary fuzzy subset of A. These procedures boil down to computing the fuzzy quasi-order Q_δ or the fuzzy equivalence E_δ, which can be efficiently computed using the well-known algorithms for computing the transitive closure of a fuzzy relation.

Introduction

From the very beginning of the theory of fuzzy sets, fuzzy automata and languages are studied as a means for bridging the gap between the precision of computer languages and vagueness and imprecision, which are frequently encountered in the study of natural languages. Study of fuzzy automata and languages was initiated in 1960s by Santos [82], [83], [84], Wee [91], Wee and Fu [92], and Lee and Zadeh [52]. From late 1960s until early 2000s mainly fuzzy automata and languages with membership values in the Gödel structure have been considered (cf., e.g., [32], [36], [64]). The idea of studying fuzzy automata with membership values in some structured abstract set comes back to Wechler [90], and in recent years researcher’s attention has been aimed mostly to fuzzy automata with membership values in complete residuated lattices, lattice-ordered monoids, and other kinds of lattices. Fuzzy automata taking membership values in a complete residuated lattice were first studied by Qiu [72], [73], where some basic concepts were discussed, and later, Qiu and his coworkers have carried out extensive research of these fuzzy automata (cf. [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, Ćirić 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]. During the decades, fuzzy automata and languages have gained wide field of application, including lexical analysis, description of natural and programming languages, learning systems, control systems, neural networks, knowledge representation, clinical monitoring, pattern recognition, error correction, databases, discrete event systems, and many other areas (cf., e.g., [32], [36], [49], [64], [70]).

On the other hand, fuzzy relation equations and inequalities were first studied by Sanchez, who used them in medical research (cf. [77], [78], [79], [80]). Later they found a much wider field of application, and nowadays they are used in fuzzy control, discrete dynamic systems, knowledge engineering, identification of fuzzy systems, prediction of fuzzy systems, decision-making, fuzzy information retrieval, fuzzy pattern recognition, image compression and reconstruction, and in many other areas (cf., e.g., [25], [29], [32], [33], [49], [67], [70]). 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).

A common problem that arises in many applications of transition systems and automata is to identify those sets of states that are closed under all transition relations, i.e., sets of states which with each of its states also contain all the states that are accessible from it. Here such sets of states are called subsystems. In the case of deterministic transition systems, if they are regarded as unary algebras, subsystems are precisely subalgebras of these algebras. Also, it is often needed to identify sets of states which are closed under reverse transition relations, i.e., sets of states which with each of its states also contain all the states that are coaccessible to it (c.f., e.g. [11]). We call such sets of states reverse subsystems. Sets of states that are both subsystems and reverse subsystems, called double subsystems, are actually components of decompositions of a transition system into a disjoint union of smaller transition systems with the property that there are no transitions between states from different components. Such decompositions are known as direct sum decompositions and are used when we want to detect and eliminate useless states and transitions of a transition system. In particular, the minimal nonempty double subsystems are the components of the greatest direct sum decomposition of a transition system (cf. [12], [13], [15]).

Subsystems of a fuzzy transition system are defined as fuzzy subsets of its set of states which are closed under all fuzzy transition relations. They were first introduced and studied by Malik et al. [61], who described some of their fundamental properties (see also [64]). Later, subsystems were investigated by Das [23], who showed that they form a topological closure system and identified a number of their topological properties. From a topological point of view, subsystems were also discussed in [85]. All the mentioned papers dealt with fuzzy transition systems over the Gödel structure. Here we study subsystems in a more general context, for fuzzy transition systems over a complete residuated lattice. We also introduce and examine reverse subsystems, which are defined as those fuzzy subsets of the set of states which are closed under reverse transition relations, and double subsystems, which are defined as fuzzy subsets of the set of states which are both subsystems and reverse subsystems. We show that all three types of subsystems can be considered as solutions to some particular systems of fuzzy relation inequalities and equations. Especially important role in our research play the fuzzy quasi-order Q_δ and fuzzy equivalence E_δgenerated by fuzzy transition relations, which can be efficiently computed using the well-known algorithms for computing the transitive closure of a fuzzy relation. In particular, we characterize subsystems, reverse subsystems and double subsystems respectively as eigen fuzzy sets of $Q_{\delta }\text{,}{Q}_{\delta }^{-1}$ and E_δ (in the sense of Sanchez [81]), and we also characterize them as linear combinations of aftersets and foresets of Q_δ and equivalence classes of E_δ (Theorem 4.3, Theorem 4.7, Theorem 4.10). We also show that subsystems, reverse subsystems and double subsystems of a fuzzy transition system $\mathcal{T}$ form both closure and opening systems in the lattice of fuzzy subsets of A, where A is the set of states of $\mathcal{T}$ (Proposition 5.1), and we provide efficient procedures for computing related closures and openings of an arbitrary fuzzy subset of A (Theorem 5.2). These procedures simply boil down to computing the fuzzy quasi-order Q_δ or the fuzzy equivalence E_δ.

The results obtained here generalize the results on subsystems of fuzzy transition systems over the Gödels structure from [23], [61], [64], [85], and the results from [12], [13], [14], [15] on subsystems, reverse subsystems, double subsystems and direct sum decompositions of ordinary transition systems. Moreover, they are closely related to results from [8], [10] on closure and opening operators defined by fuzzy relations. Compared with these papers, the main advantages of our approach are as follows. We study not only the subsystems and the dual concept of reverse subsystems, but also double subsystems that have been discussed only in the context of ordinary transition systems, in relation to study of direct sum decompositions of transition systems [12], [13], [14], [15]. In contrast to the previous articles, where only closures related to subsystems have been studied, here we study both closures and openings related to subsystems, reverse subsystems and double subsystems, and moreover, we deal with a more general structure of truth values. Das [23] proposed a method for computing closures related to subsystems, but his approach requires computation of all composite fuzzy transition relations, whose number can be exponential in the number of states, and for some structures of truth values this number may even be infinite. This makes his approach computationally inefficient. Our approach does not require computation of composite fuzzy transition relations, instead we have to compute only the union of basic fuzzy transition relations and its reflexive–transitive closure, and then to compute the composition of the resulting fuzzy relation and a fuzzy set. Computationally the most demanding part of this procedure is to compute the transitive closure of a fuzzy relation. When transitivity is defined by a triangular norm on the real unit interval, the transitive closure can be computed using some of the algorithms provided in [26], [27], [53], [66], [88]. Some of them have an impressive complexity of O(n³) [66], or even O(n²) [27], for the min-transitive closure, where n is the number of elements of the underlying set, i.e., in our case, the number of states of the considered fuzzy transition system. Hence, our algorithms can achieve the same overall complexity.

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]. However, this methodology is based on an iterative procedure which does not necessarily finishes in a finite number of iterations, depending on certain local properties of the complete residuated lattice that is used as the structure of truth values. In contrast, the methodology developed here provides the results for each complete residuated lattice, regardless of its local properties.

The structure of the paper is as follows. In Section 2 we introduce basic notions and notation concerning complete residuated lattices, fuzzy sets, fuzzy relations, fuzzy transition systems and fuzzy automata. In Section 3 we present the basic properties and give the construction of the fuzzy quasi-order and fuzzy equivalence generated by fuzzy transition relations of a fuzzy transition system. Section 4 contains our main results characterizing subsystems, reverse subsystems and double subsystems of a fuzzy transition system. Then in Section 5 we show that subsystems, reverse subsystems and double subsystems form both closure and opening systems and describe the corresponding closure and opening operators.