Fuzzy neighborhood operators and derived fuzzy coverings
Introduction
Rough set theory (RST) was originally proposed by Pawlak [34], [35] as a valid means of knowledge discovery and information processing, in which the fundamental tools consist of relations that represent information systems or decision tables. In rough set theory, two main factors affect the description capability of information systems or decision tables: set approximation and knowledge reduction. On the one hand, given a subset of the universe, two definable sets called lower and upper approximations are induced to approximate the subset. On the other hand, under the condition of keeping the set approximations unchanged, knowledge reduction is conducted to remove the redundant attributes from the information system or decision table to acquire some simpler rules than the original information system or decision table.
In the Pawlak's rough set model, the relationships of objects were built on equivalence relations. All equivalence classes form a partition of a universe of discourse. However, an equivalence relation imposes restrictions and limitations on many applications [6], [18], [21], [26], [49]. Hence, many extensions have been made in recent years by replacing equivalence relation or partition with notions such as binary relations [12], [25], [43], [44], [58], [59], neighborhood systems and Boolean algebras [2], [50], [60], and coverings of the universe of discourse [3], [37], [38]. Based on the notion of covering, Pomykala [37], [38] obtained two pairs of dual approximation operators. Yao [60] further examined these approximation operators by the concepts of neighborhood and granularity. Such undertakings have stimulated more researches in this area [4], [22], [27], [28], [62], [65], [66], [67], [68], [69], [70]. Over the past 30 years, RST has indeed become a topic of great interest to researchers and has been applied to many domains.
However, RST is designed to process qualitative (discrete) data, and it faces great limitations when dealing with real-valued data sets since the values of attributes in databases could be both symbolic and real-valued [19]. Fuzzy set theory (FST) [64] is favorable to overcome these limitations, as it can deal effectively with vague concepts and graded indiscernibility. Nowadays, rough set theory and fuzzy set theory are two main tools used to process uncertainty and incomplete information in the information systems. The two theories are related but distinct and complementary [36], [61]. In the past twenty years, the research on the connection between rough sets and fuzzy sets has attracted considerable attention. Intentions on combining rough set theory and fuzzy set theory can be found in different mathematical fields [33], [48], [61]. Dubois and Prade firstly proposed the concept of fuzzy rough sets [10], which combined these two theories and influenced numerous authors who used different fuzzy logical connectives and fuzzy relations to define fuzzy rough set models. Then more scholars generalized the fuzzy rough sets by using various methods [1], [7], [13], [14], [15], [16], [17], [30], [31], [32], [39], [40], [45], [51], [52], [53], [63]. The most common fuzzy rough set models are obtained by replacing the crisp binary relations and the crisp subsets with the fuzzy relations and the fuzzy subsets on the universe respectively.
As well as RST, some researchers have tried to generalize the fuzzy rough set based on fuzzy relation by using the concept of fuzzy covering. De Cock et al. [5] defined fuzzy rough sets based on the R-foresets of all objects in a universe of discourse with respect to (w.r.t.) a fuzzy binary relation. When R is a serial fuzzy relation, the family of all R-foresets forms a fuzzy covering of the universe of discourse. Analogously, Deng [9] examined the issue with fuzzy relations induced by a fuzzy covering. Li and Ma [23], on the other hand, constructed two pairs of fuzzy rough approximation operators based on fuzzy coverings, the standard min operator , and the Kleene–Dienes implicator . It should be noted that fuzzy coverings in the models proposed by Deng [9] and De Cock et al. [5] are induced from fuzzy relations. So, they are not fuzzy coverings in the general sense. Although fuzzy coverings are used by Li and Ma [23] in their models, they only employed two special logical operators i.e., the standard min operator and the Kleene–Dienes implicator. Thus, it is necessary to construct more general fuzzy rough set models based on fuzzy coverings. Recently, an excellent introduction to the topic of fuzzy covering-based rough set is due to some scholars [11], [24], [42], [46], [56], [57], which can be regarded as a bridge linking covering-based rough set theory and fuzzy rough set theory. The original definition of fuzzy covering is described as follows (see [11], [24]).
Let U be an arbitrary universal set, and be the fuzzy power set of U. We call , with , a fuzzy covering of U, if for each .
Based on the original definition of fuzzy covering, Ma [29] defined the definition of fuzzy β-covering by replacing 1 with a parameter , i.e., we call , with , a fuzzy β-covering of U, if for each . Meanwhile, Ma constructed two fuzzy covering-based rough set models by using the definition of fuzzy β-neighborhood, which was defined by Moreover, Ma generalized two fuzzy covering-based rough set models over fuzzy lattice in the sense of [47], that is, complete and completely distributive lattices with an involutive negation. Based on Ma's work, Yang and Hu studied some new properties of fuzzy β-covering and fuzzy β-neighborhood in [55]. Especially, the collection of fuzzy sets ( in Section 5), named fuzzy β-neighborhood family, is also a fuzzy β-covering of U. Then Yang and Hu gave a necessary and sufficient condition for . On the other hand, the interdependency of the fuzzy covering-based approximation operators defined by Ma was studied in [55]. In [27], the concept of a complementary neighborhood and some types of covering-based rough set models were proposed from the viewpoint of application. Similarly, by introducing the notion of a fuzzy complementary β-neighborhood, three new fuzzy covering-based rough set models were defined in [55]. In order to describe an object, we need only the essential characteristics related to the object, rather than all the characteristics of this object. That is the purpose of the minimal description [3]. The notion of minimal description plays a key role in the studies of covering-based rough sets. Correspondingly, Yang and Hu [54] firstly defined fuzzy β-minimal description in fuzzy β-covering approximation space and gave some new characterizations of fuzzy covering-based rough sets in terms of fuzzy β-minimal description. The definition of fuzzy β-minimal description is described as follows. Analogously, this paper proposes the notion of fuzzy β-maximal description in a fuzzy β-covering approximation space as follows. In [8], for a fuzzy covering of U, the authors defined a collection of fuzzy sets . Meanwhile, they proposed the notions of fuzzy minimal description of an object and fuzzy maximal description of an object as follows. Based on the definitions of fuzzy covering and fuzzy β-covering, we have that a fuzzy covering of U is a fuzzy β-covering of U for any and a fuzzy 1-covering of U is a fuzzy covering of U. In other words, fuzzy covering defined in [8] is a special case of fuzzy β-covering. For a giving fuzzy covering of U and , it is easy to find that , and for any . It shows that the fuzzy minimal and maximal descriptions which were defined in [8] are different from the fuzzy β-minimal and β-maximal descriptions which are defined in this paper. Especially, there exists an inclusion relation among the elements of fuzzy minimal description or fuzzy maximal description. More precisely, see the example in the following Remark 1. The definitions of fuzzy β-minimal and β-maximal descriptions based on fuzzy β-covering not only can be seen as the generalizations of minimal and maximal descriptions in the fuzzy setting, but also can be seen as the supplement of fuzzy minimal and maximal descriptions which were defined in [8]. Then we study some basic properties of the fuzzy β-minimal and β-maximal descriptions. Meanwhile, the interdependency of the β-neighborhood system, fuzzy β-minimal and β-maximal descriptions are investigated in this paper.
On the other hand, D'eer et al. [8] have extended the definitions of four covering-based neighborhood operators as well as six derived coverings discussed by Yao and Yao in the fuzzy setting. As we point out that fuzzy β-covering can be seen as a generalization of fuzzy covering. Then we consider the following two aspects:
- (1)
We extend the definitions of four covering-based neighborhood operators as well as six derived coverings discussed by Yao and Yao in the fuzzy setting from the viewpoint of fuzzy β-covering. In fact, if a fuzzy β-covering is a crisp covering, then the four types of fuzzy neighborhood operators defined in this paper coincide with the crisp neighborhood operators discussed by Yao and Yao. More precisely, see Remark 2, Remark 4, Remark 6, Remark 8.
- (2)
We study the relationship between the fuzzy neighborhood operators which were defined by D'eer et al. and the fuzzy neighborhood operators based on a fuzzy β-covering in this paper. Since the definitions of fuzzy minimal and maximal descriptions defined in [8] are different from the fuzzy β-minimal and β-maximal descriptions which are defined in this paper. Then the four types of fuzzy neighborhood operators defined in this paper are different from the four types of fuzzy neighborhood operators defined by D'eer et al. We know that a fuzzy 1-covering is a fuzzy covering. Especially, for a giving fuzzy β-covering of U with , if , then is a fuzzy covering, and for any ; if and satisfies (NP), then and for any . More precisely, see Remark 3, Remark 5, Remark 7, Remark 9.
The remainder of this paper is organized as follows. In Section 2, some preliminary definitions used in this paper are introduced. In Section 3, we introduce the notions of β-neighborhood system, fuzzy β-minimal description and fuzzy β-maximal description and study relationship among these three definitions. Especially, the interdependency of these three notions are investigated in this section, respectively. In Section 4, we define four types of fuzzy neighborhood operators by using the notions such as β-neighborhood system, fuzzy β-minimal description and fuzzy β-maximal description. In Section 5, six types of fuzzy β-coverings can be derived by using the four types of fuzzy neighborhood operators defined in Section 4 and their properties are considered. Section 6 concludes this paper.
Section snippets
Preliminaries
In this section, we review some notions in covering-based rough sets and fuzzy set theory used in this paper.
A neighborhood operator [62] on the universe U is a mapping , where represents the collection of subsets of U. In general, we assume that a neighborhood operator N is reflexive, i.e., for each . Moreover, a neighborhood operator is called symmetric if for all , and it is called transitive if for all .
In the following
Fuzzy β-minimal and β-maximal descriptions
In this section, we introduce the notions of fuzzy β-minimal description, fuzzy β-maximal description and β-neighborhood system in fuzzy β-covering approximation space. Moreover, we investigate the relationship among these notions. Some properties of these notions are also studied. At the end of this section, the conditions under which two fuzzy β-coverings generate the same fuzzy β-minimal description, fuzzy β-maximal description and β-neighborhood system for any are proposed.
Four types of fuzzy neighborhood operators based on a fuzzy β-covering
In this section, we discuss the notion of fuzzy neighborhood operators in rough set theory by using the fuzzy extensions such as β-neighborhood system, fuzzy β-minimal description and fuzzy β-maximal description. Moreover, we discuss which properties the different fuzzy neighborhood operators fulfill. In the most general setting, a fuzzy neighborhood operator in the context of rough sets is defined as follows.
Definition 23 ([8]) A fuzzy neighborhood operator is a mapping .
This means that a fuzzy
Fuzzy β-coverings derived from fuzzy neighborhood operators
Given a covering C, Yao and Yao [62] proposed six coverings derived from C. Similarly, we introduce fuzzy extensions of the derived coverings , , , , and in this section by giving a fuzzy β-covering . Furthermore, relationship among fuzzy β-coverings from fuzzy neighborhood operators are investigated.
Definition 30 Let be a fuzzy β-covering approximation space. Then we define the following families of fuzzy sets: , ,
Conclusions
In this paper, we have defined four types of fuzzy neighborhood operators based on a fuzzy β-covering by introducing the notions such as β-neighborhood system, fuzzy β-minimal description and fuzzy β-maximal description. Furthermore, six types of fuzzy β-coverings have been proposed by using these fuzzy neighborhood operators and their properties have been studied. Main conclusions in this paper and continuous work to do are listed as follows.
(1) By introducing the notions of β-neighborhood
Acknowledgements
The authors thank the anonymous referees and the editor for their valuable comments and suggestions in improving this paper. This research was supported by the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038) and the National Water Pollution Control and Treatment Science and Technology Major Project of China (Grant no. 2017ZX07108-001).
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