Covering Approximation Operators Research Articles

Soft rough fuzzy sets based on covering

<abstract><p>Soft rough fuzzy sets ($ \mathcal SRFSs $) represent a powerful paradigm that integrates soft computing, rough set theory, and fuzzy logic. This research aimed to comprehensively investigate the various dimensions of $ \mathcal SRFSs $ within the domain of approximation structures. The study encompassed a wide spectrum of concepts, ranging from covering approximation structures and soft rough coverings to soft neighborhoods, fuzzy covering approximation operators, and soft fuzzy covering approximation operators. We introduced three models of $ \mathcal SRFSs $ based on covering via the core of soft neighborhood. We discussed and analyzed our models' characteristics and properties. The relations between our models for soft fuzzy covering sets and Zhan's model for soft rough fuzzy covering were presented.</p></abstract>

Relationships between two pairs of covering approximation operators and belief structures

Relationships between two pairs of covering approximation operators and belief structures

Interpretations of belief functions in approximation operators by covering

The rough set theory and the evidence theory are two important methods used to deal with uncertainty. The relationships between the rough set theory and the evidence theory have been discussed. In covering rough set theory, several pairs of covering approximation operators are characterized by belief and plausibility functions. The purpose of this paper is to review and examine interpretations of belief functions in covering approximation operators. Firstly, properties of the belief structures induced by two pairs of covering approximation operators are presented. Then, for a belief structure with the properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions induced by one of the two pairs of covering approximation operators. Moreover, two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators are presented.

Numerical Characterizations of Topological Reductions of Covering Information Systems in Evidence Theory

The reductions of covering information systems in terms of covering approximation operators are one of the most important applications of covering rough set theory. Based on the connections between the theory of topology and the covering rough set theory, two kinds of topological reductions of covering information systems are discussed in this paper, which are characterized by the belief and plausibility functions from the evidence theory. The topological spaces by two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. Then, using plausibility function values of the sets in the partitions, the definitions of significance and relative significance of coverings are presented. Hence, topological reduction algorithms based on the evidence theory are proposed in covering information systems, and an example is adopted to illustrate the validity of the algorithms.

Multigranulation single valued neutrosophic covering-based rough sets and their applications to multi-criteria group decision making

In this paper, three types of (philosophical, optimistic and pessimistic) multigranulation single valued neutrosophic (SVN) covering-based rough set models are presented, and these three models are applied to the problem of multi-criteria group decision making (MCGDM).Firstly, a type of SVN covering-based rough set model is proposed.Based on this rough set model, three types of multigranulation SVN covering-based rough set models under the concept of multigranulation SVN $beta$-covering approximation space are proposed, where $beta$ is a SVN number.Moreover, the connections among these four models are investigated.Secondly, some conditions under which different multigranulation SVN $beta$-covering approximation spaces induced the same multigranulation SVN covering approximation operators are presented.Finally, three novel methods are presented to MCGDM problems under the multigranulation SVN covering-based rough set models.Furthermore, the proposed MCGDM methods are compared with other methods through a problem about paper defect diagnosis.

A New Type of Single Valued Neutrosophic Covering Rough Set Model

Recently, various types of single valued neutrosophic (SVN) rough set models were presented based on the same inclusion relation. However, there is another SVN inclusion relation in SVN sets. In this paper, we propose a new type of SVN covering rough set model based on the new inclusion relation. Furthermore, the graph and matrix representations of the new SVN covering approximation operators are presented. Firstly, the notion of SVN β 2 -covering approximation space is proposed, which is decided by the new inclusion relation. Then, a type of SVN covering rough set model under the SVN β 2 -covering approximation space is presented. Moreover, there is a corresponding SVN relation rough set model based on a SVN relation induced by the SVN β 2 -covering, and two conditions under which the SVN β 2 -covering can induce a symmetric SVN relation are presented. Thirdly, the graph and matrix representations of the new SVN covering rough set model are investigated. Finally, we propose a novel method for decision making (DM) problems in paper defect diagnosis under the new SVN covering rough set model.

Fixed points of covering upper and lower approximation operators

Pawlak’s rough set and its extension, covering-based rough set, are important techniques for reasoning in incomplete information systems. In this paper, by studying some results about the Feynman paths, we show that the family of all fixed points of covering upper and lower approximation operators is an atomic frame and a complete lattice, respectively. Then, we find a relation between some major causal operators of relativity theory and covering approximation operators. As a result of this connection, we introduce a Feynman index to classify space–times.

A generalized modal logic in causal structures

Recently, a comparison between causality in the theory of relativity and in distributed computing systems as causal nets has been carried out. In this paper, the theory of covering-based rough sets as an appropriate and comprehensive framework is applied to study causality in both branches. As a result, by future and past covering approximation operators, we propose a generalized modal logic on causal nets.

Two Types of Single Valued Neutrosophic Covering Rough Sets and an Application to Decision Making

In this paper, to combine single valued neutrosophic sets (SVNSs) with covering-based rough sets, we propose two types of single valued neutrosophic (SVN) covering rough set models. Furthermore, a corresponding application to the problem of decision making is presented. Firstly, the notion of SVN β -covering approximation space is proposed, and some concepts and properties in it are investigated. Secondly, based on SVN β -covering approximation spaces, two types of SVN covering rough set models are proposed. Then, some properties and the matrix representations of the newly defined SVN covering approximation operators are investigated. Finally, we propose a novel method to decision making (DM) problems based on one of the SVN covering rough set models. Moreover, the proposed DM method is compared with other methods in an example.

Topological structures of a type of granule based covering rough sets

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

Characteristic matrices of compound operations of coverings and their relationships with rough sets

As the matrix can compactly represent numeric data, simplify problem formulation and reduce time complexity, it has many applications in most of the scientific fields. For this purpose, some types of generalized rough sets have been connected with matrices. However, covering-based rough sets which play an important role in data mining and machine learning are seldom connected with matrices. In this paper, we define three composition operations of coverings and study their characteristic matrices; Moreover, the relationships between the characteristic matrices and covering approximation operators are investigated. First, for a covering, an existing matrix representation of indiscernible neighborhoods called the type-1 characteristic matrix of the covering is recalled and a new matrix representation of neighborhoods called the type-2 characteristic matrix of the covering is proposed. Second, considering the importance of knowledge fusion and decomposition, we define three types of composition operations of coverings. Specifically, their type-1 and type-2 characteristic matrices are studied. Finally, we also explore the representable properties of covering approximation operators with respect to any covering generated by each composition operation. It is interesting to find that three types of approximation operators, which are induced by each type of composition operation of coverings, can be expressed as the Boolean product of a coefficient matrix and a characteristic vector. These interesting results suggest the potential for studying covering-based rough sets by matrix approaches.

Boundary region-based rough sets and uncertainty measures in the approximation space

The approximation operators play an important role in rough set theory, which are mainly defined by means of neighborhood systems. In this paper, firstly, we try to propose a class of novel definitions of the approximation operators via a predefined boundary region based on a binary relation. Then we compare the proposed concepts with the originals, the necessary and sufficient conditions of their equipollence are investigated. Secondly, we give the definitions of boundary region based on a covering. By employing the boundary, a class of novel definitions of the approximation operators based on a covering are proposed. It is shown that the proposed operators are equivalent to a class of covering approximation operators introduced by Zakowski. Thirdly, the relationship between general binary relations and coverings based approximation operators is investigated with the aid of the novel boundary region. Finally, the more reasonable characterizations of the accuracy and the roughness are proposed by employing the boundary operators. Meanwhile, we study uncertainty measures of approximation spaces based on a partition and a covering.

A rough set approach to the characterization of transversal matroids

Rough sets are efficient for data pre-processing during data mining. However, some important problems such as attribute reduction in rough sets are NP-hard and the algorithms required to solve them are mostly greedy ones. The transversal matroid is an important part of matroid theory, which provides well-established platforms for greedy algorithms. In this study, we investigate transversal matroids using the rough set approach. First, we construct a covering induced by a family of subsets and we propose the approximation operators and upper approximation number based on this covering. We present a sufficient condition under which a subset is a partial transversal, and also a necessary condition. Furthermore, we characterize the transversal matroid with the covering-based approximation operator and construct some types of circuits. Second, we explore the relationships between closure operators in transversal matroids and upper approximation operators based on the covering induced by a family of subsets. Finally, we study two types of axiomatic characterizations of the covering approximation operators based on the set theory and matroid theory, respectively. These results provide more methods for investigating the combination of transversal matroids with rough sets. • construct a covering relative to transversal matroids, establish covering rough sets. • characterize transversal matroids with covering rough set. • construct two types of circuits of transversal matroids by using covering rough sets. • propose conditions under which a subset is a partial transversal by using rough sets. • build axiomatic characterizations of approximation operators from two viewpoints.

Connectedness of graphs and its application to connected matroids through covering-based rough sets

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and covering-based rough sets provide a systematic approach to this type of representation. In this paper, we study the connectedness of graphs through covering-based rough sets and apply it to connected matroids. First, we present an approach to inducing a covering by a graph, and then study the connectedness of the graph from the viewpoint of the covering approximation operators. Second, we construct a graph from a matroid, and find the matroid and the graph have the same connectedness, which makes us to use covering-based rough sets to study connected matroids. In summary, this paper provides a new approach to studying graph theory and matroid theory.

Relationships between generalized rough sets based on covering and reflexive neighborhood system

Rough set theory is an important tool to deal with uncertainty, granularity and incompleteness of knowledge. The covering-based rough sets is one of the most important extensions of the classical Pawlak rough sets. In covering-based rough set theory, many types of rough sets were already defined in the literature. To find out the relationships among different types of covering-based rough sets and give an axiomatic definition is a very important issue. The generalized rough sets in neighborhood system is another important extension of the classical Pawlak rough sets. In this paper, we investigate relationships between covering-based rough sets and generalized rough sets in neighborhood system. Then, twenty-three types of the element based, the granule based and the subsystem based definitions of covering approximation operators are unified under the framework of generalized approximation operators in neighborhood systems. Moreover, properties of the generalized approximation operators in neighborhood systems are presented.

On measurements of covering rough sets based on granules and evidence theory

Covering rough sets generalize classical rough sets by relaxing partitions of the universe to coverings. The existing research of covering rough sets mainly concentrates on comparing of different kinds of approximation operators and designing algorithms for attribute reductions of decision systems. Less efforts have been made in measurements of covering rough sets. To enrich this field, in this paper we investigate the measurements of covering rough sets based on granules and evidence theory, which can be used to solve more complex practical problems. Firstly, it is shown that the most common sixteen pairs of covering approximation operators can be grouped into five pairs of dual covering approximation operators and six pairs of non-dual covering approximation operators according to definitions based on element, granule, and subsystem. Secondly, we define the rough degree and precision for the purpose of characterizing the uncertainty of these summarized covering approximation operators under a unified framework. The underlying uncertainty of covering rough sets results from the roughness of granulation of coverings. Finally, we investigate the relationships between the covering approximation operators and evidence theory. It shows that some kinds of covering approximation operators can be characterized by belief and plausibility functions, while others do not share this property. The sufficient conditions under which covering approximation operators can be measured by belief and plausibility functions are further explored, and the corresponding counterexamples are given to illustrate those that cannot be measured. Based on above discussions, we develop a basic framework of numerical characterization on measurements of covering rough sets with granules and evidence theory.

On the Connectivity of Wireless Network Systems and an Application in Teacher-Student Interactive Platforms

A wireless network system is a pair(U;B), whereBis a family of some base stations andUis a set of their users. To investigate the connectivity of wireless network systems, this paper takes covering approximation spaces as mathematical models of wireless network systems. With the help of covering approximation operators, this paper characterizes the connectivity of covering approximation spaces by their definable subsets. Furthermore, it is obtained that a wireless network system is connected if and only if the relevant covering approximation space has no nonempty definable proper subset. As an application of this result, the connectivity of a teacher-student interactive platform is discussed, which is established in the School of Mathematical Sciences of Soochow University. This application further demonstrates the usefulness of rough set theory in pedagogy and makes it possible to research education by logical methods and mathematical methods.

Characteristic matrix of covering and its application to Boolean matrix decomposition

Covering-based rough sets provide an efficient means of dealing with covering data, which occur widely in practical applications. Boolean matrix decomposition has frequently been applied to data mining and machine learning. In this paper, three types of existing covering approximation operators are represented by Boolean matrices, and then used in Boolean matrix decomposition. First, we define two characteristic matrices of a covering. Through these Boolean characteristic matrices, three types of existing covering approximation operator are concisely and equivalently represented. Second, these operator representations are applied to Boolean matrix decomposition, which has a close relationship with nonnegative matrix factorization, a popular and efficient technique for machine learning. We provide a sufficient and necessary condition for a square Boolean matrix to decompose into the Boolean product of another matrix and its transpose. We then develop an algorithm for this Boolean matrix decomposition. Finally, these three covering approximation operators are axiomatized using Boolean matrices. This work presents an interesting viewpoint from which to investigate covering-based rough set theory and its applications.

Multigranulation rough sets: From partition to covering

The classical multigranulation rough set (MGRS) theory offers a formal theoretical framework for solving the complex problem under multigranulation environment. However, it is noticeable that MGRS theory cannot be applied in multi-source information systems with a covering environment in the real world. To address this issue, we firstly present in this paper three types of covering based multigranulation rough sets, in which set approximations are defined by different covering approximation operators. Then, by using two different approximation strategies, i.e., seeking common reserving difference and seeking common rejecting difference, two kinds of covering based multigranulation rough set are presented, namely, a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets. Finally, we develop some properties and several uncertainty measures of the covering based multigranulation rough sets. These results will enrich the MGRS theory and enlarge its application scope.

Topological characterizations of covering for special covering-based upper approximation operators

The relationships among properties of covering approximation operators and their corresponding coverings have attracted intensive research in recent years. In particular, those among topological properties have drawn special attention because of their important applications in rough set theory. In this paper, we give topological characterizations of covering C for covering-based upper approximation operators FH, SH, TH and RH to be closure operators. We also give intuitive characterizations of covering C, and describe covering-based approximation space (U,C) as certain types of information exchange systems when SH or RH is a closure operator. By applying our new characterizations, we give inequalities about the relationship between the number of members in C and the number of elements in U, and discuss relationships among conditions for different covering based upper approximation operators to be closure operators. To the best of our knowledge, it is the first time that such characterizations, descriptions, inequalities and discussions are systematically considered in the literature of rough set theory. Furthermore, in this paper we also give several characterizations of unary coverings, an important type of coverings in studying relationships among basic concepts in covering-based rough sets, by the relationships among different types of covering-base approximation operators.