Covering Approximation Spaces Research Articles

Continuous Mapping of Covering Approximation Spaces and Topologies Induced by Arbitrary Covering Relations

In rough set theory, there are many covering approximation spaces, so how to classify covering approximation spaces has become a hot issue. In this paper, we propose the concepts of a covering approximation T1-space, F-symmetry, covering rough continuous mapping, and covering rough homeomorphism mapping, and we obtain some interesting results. We have used the above definitions and results to classify covering approximation spaces. Finally, we find a new method for constructing topologies, obtain some properties, and provide an example to illustrate our method’s similarities and differences with other construction methods.

Reductions of a covering approximation space from algebraic points of view

We first make a great improvement on the early definition of a reducible element of a covering approximation space and formulate the definition of a reduction of the covering. Then the crucial links between the two concepts are established and a method for obtaining an optimal reduction of the covering is proposed. More importantly, we introduce the representation matrix for a finite covering approximation space and define a new type of operation on matrices. To obtain a reduction of the covering, we need only to deal with the representation matrix, so that the reduction of the covering can be executed on computers. The concept of the product of two covering approximation spaces is defined in this paper and we explore the connections between the reductions of the covering of the product space and those of the coverings of the factor spaces. Furthermore, we deal with the connections from the standpoint of algebra. A new type of matrix product is defined and the property of the product is explored. Drawing on the product, we can research the reduction of the product space.

Grained matrix and complementary matrix: Novel methods for computing information descriptions in covering approximation spaces

As two important tools of information descriptions, minimal description (MinD) and maximal description (MaxD) are used to eliminate redundant information when describing an object in covering-based rough sets (CbRSs). Hence, there are many key issues (such as reductions, neighborhoods and approximations) have to do with them in CbRSs. However, it is complicated and time-consuming to use set approaches to compute them in a large cardinal covering approximation space. So the matrix approaches have been used to calculate them by which computers can implement the corresponding calculations easily. Based on the previous research work and inspired by the demand of knowledge discovery under large scale covering information systems, we propose two new matrix methods to calculate MinD and MaxD in CbRSs in this paper. Firstly, a note on the existing methods (matrix approach-1 and approach-2) is presented. Secondly, a new matrix, called grained matrix, is presented. After combining grained matrices and the mean of “sum”, the new matrix approach-1 is presented to calculate MinD and MaxD. Thirdly, three new matrix operations and the notion of complementary matrix are given which can simplify the calculation of computers. Based on new operations, we present the new matrix approach-2 to compute MinD and MaxD. Finally, the computational efficiency of the presented approaches are shown and compared with other methods by experiments on several UCI data sets.

Invariance of separation in covering approximation spaces

<abstract> In this paper, the invariance of separation in covering approximation spaces are discussed. This paper proves that some separations in covering approximation spaces are invariant to reducts of coverings, invariant to covering approximation subspaces and invariant under <italic>CAP</italic>-transformations of covering approximation spaces. These results deepen and enrich theory of separations in covering approximation spaces, which is helpful to give further researches and applications of Pawlak rough set theory in information sciences. </abstract>

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.

Fuzzy β-covering approximation spaces

All fuzzy covering-based rough set models are constructed under a corresponding fuzzy covering approximation space (FCAS). The fuzzy β-covering approximation space (β-FCAS) is a generalization of the FCAS through replacing the value 1 with a parameter β. In other words, the β-FCAS is the basis of studying fuzzy covering-based rough sets and their applications. Therefore, it is necessary to study some questions in the fuzzy covering approximation space, such as problems of reduction, relationships among some basic concepts and relationships between two fuzzy covering approximation spaces. In this article, we investigate the questions in the β-FCAS further. Firstly, the definitions of I-irreducible element and I-reduct are presented, which can be seen as the complement of the existing notions. Then, relationships among some concepts in the β-FCAS are investigated, such as the relationship between fuzzy β-minimal description and β-reduct, and the relationship between fuzzy β-covering and its I-reduct. Thirdly, inspired by some concepts in the β-FCAS, we present some new concepts between two β-FCASs and their properties. Based on these new concepts, we present some conditions such that two fuzzy β-coverings have the same reduct (or I-reduct). Finally, based on the results above, seven derived β-FCASs are investigated further and a corresponding lattice of them is proposed.

Knowledge reduction methods of covering approximate spaces based on concept lattice

Both rough sets and concept lattices, which are two complementary tools in data analysis, are analyzed based on binary relations. The relations between rough sets and concept lattices are important research topic. In this paper, the methods of union reduction and intersection reduction in covering approximation spaces based on concept lattice are discussed, and the relations between union reduction of covering approximation spaces and concept lattices reduction are investigated. We also discuss the relations of element characteristics between covering approximation spaces and the concept lattices. Meanwhile, the connections between reduction of a covering approximation space and that of its compliment space are revealed. The research results establish a bridge between the rough sets and concept lattices and help one to gain much more insights into the two theories.

Maps between covering approximation spaces and the product space of two covering approximation spaces

Let (V,D) be a covering approximation space and let f:U⟶V be a surjective map, where U is a nonempty set. We can achieve a covering C of U by defining C={f−1(D):D∈D}. This is a natural way to construct a covering for U and this motivates the research of covering rough sets from the mapping point of view. In this paper, we have shown that the inverse image of a neighborhood under f is a neighborhood and the inverse image of the closure of a point under f is again the closure of the corresponding point. Thus some approximation operators on the two covering approximation spaces must be related to each other. Some theorems and propositions with respect to the relationships between the approximation operators defined on U and the corresponding ones defined on V have been obtained. The purpose is to emphasize the importance of mappings between two covering approximation spaces. Finally, given two covering approximation spaces (U1,A) and (U2,B), we define the product covering for U1×U2 as {A×B|A∈A,B∈B}. This is a method to construct a new covering approximation space out of existing ones. Then we research the connections between the neighborhoods and closures of U1×U2 and the corresponding ones of factor spaces. In the last part of this paper, we explore the connections between the approximation operators defined on U1×U2 and the corresponding ones defined on factor spaces.

Decision reducts and bireducts in a covering approximation space and their relationship to set definability

In this paper, we discuss the relationship between different types of reduction and set definability. We recall the definition of a decision reduct, a γ-decision reduct, a decision bireduct and a γ-decision bireduct in a Pawlak approximation space and the notion of set definability both in a Pawlak and a covering approximation space. We extend the notion of discernibility between objects in a Pawlak approximation space to a covering approximation space. Moreover, we introduce the definition of a decision reduct, a γ-decision reduct, a decision bireduct and a γ-decision bireduct in a covering approximation space. In addition, we study interrelationships between the four types of reduction, the correspondence with positive regions and the relationship to set definability in Pawlak and covering approximation spaces.

Roughness measures of locally finite covering rough sets

The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC-, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC-space (U,C) and a nonempty set X(⊆U). Using a reduction of a given LFC-space (U,C), the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC-spaces (U,C), where each of the cardinalities of the sets U, C, and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC-spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X(⊆U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness.

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.

Two Types of Intuitionistic Fuzzy Covering Rough Sets and an Application to Multiple Criteria Group Decision Making

Intuitionistic fuzzy rough sets are constructed by combining intuitionistic fuzzy sets with rough sets. Recently, Huang et al. proposed the definition of an intuitionistic fuzzy (IF) β -covering and an IF covering rough set model. In this paper, some properties of IF β -covering approximation spaces and the IF covering rough set model are investigated further. Moreover, we present a novel methodology to the problem of multiple criteria group decision making. Firstly, some new notions and properties of IF β -covering approximation spaces are proposed. Secondly, we study the characterizations of Huang et al.’s IF covering rough set model and present a new IF covering rough set model for crisp sets in an IF environment. The relationships between these two IF covering rough set models and some other rough set models are investigated. Finally, based on the IF covering rough set model, Huang et al. also defined an optimistic multi-granulation IF rough set model. We present a novel method to multiple criteria group decision making problems under the optimistic multi-granulation IF rough set model.

Dicovering approximation spaces and definability

In this work, we define a category Cap of covering approximation spaces whose morphisms are functions satisfying a refinement property. We give the relations among Cap, and the category Top of topological spaces and continuous functions, and the category Rere of reflexive approximation spaces and the relation preserving functions. Further, we discuss the textural versions diCap, dfDitop and diRere of these categories. Then we study the definability in Cap with respect to five covering-based approximation operators. In particular, it is observed that via the morphisms of Cap, we may get more information about the subsets of the universe.

A unified framework for characterizing rough sets with evidence theory in various approximation spaces

Different rough set models may be numerically characterized with evidence theory by adopting different methodologies. That is to say, the evidence theory-based characteristics of rough sets vary with different approximation spaces and different rough set approximations. To address this issue, this paper proposes a theoretic framework within which rough set theory can be unifiedly characterized by evidence theory no matter what types of rough set approximations and approximation spaces they are. The main works are presented as follows. First, we declare a principle that in any given approximation space, a belief structure can be constructed, based on which the lower and upper approximation operators can be measured by the belief and plausibility functions if and only if a basic condition is satisfied. Second, in a given decision approximation space, a belief structure is also derived, based on which the lower and upper approximation operators can be always measured by the belief and plausibility functions without any additional condition. As theoretical applications, we employ the new principles to examine the evidence theory-based characteristics of four types of substantial rough sets, including the covering rough sets in covering approximation spaces, the decision-theoretic rough sets in Pawlak approximation spaces, the Pawlak rough sets in Pawlak decision approximation spaces, and the multigranulation rough sets in Pawlak decision approximation spaces, respectively. The proposed framework is applicable to various approximation spaces and various rough set approximations, and hence can make us clear the basic principles for relating rough set theory with evidence theory.

On completeness of interactive student networks

The completeness of interactive student networks are investigated in this paper, which can be used to improve middle school students’ independent environment for their study outside class. This paper takes covering approximation spaces as mathematical models of interactive student networks and characterizes completeness of interactive student networks by connectivity of covering approximation spaces. Further, this paper gives a simpler and practical method to check completeness of interactive student networks. Consequently, a way to communication among students is explored and applied in education.

Covering-Based Rough Single Valued Neutrosophic Sets

Abstract: Rough sets theory is a powerful tool to deal with uncertainty and incompleteness of knowledge in information systems. Wang et al. proposed single valued neutrosophic sets as an extension of intuitionistic fuzzy sets to deal with real-world problems. In this paper, we propose the covering-based rough single valued neutrosophic sets by combining covering-based rough sets and single valued neutrosophic sets. Firstly, three types of covering-based rough single valued neutrosophic sets models are built and the properties of lower/upper approximation operators are explored. Secondly, the lower/upper approximations in two different covering approximation spaces are studied. The sufficient and necessary condition for generating the same lower/upper approximations from two different covering approximation spaces is discussed. Moreover, the relations of the three models are discussed and the equivalence conditions for three models are given.

On twelve types of covering-based rough sets.

Covering approximation spaces are a generalization of equivalence-based rough set theories. In this paper, we will consider twelve types of covering based approximation operators by combining four types of covering lower approximation operators and three types of covering upper approximation operators. Then, we will study the properties of these new pairs and show they have most of the common properties among existing covering approximation pairs. Finally, the relation between these new pairs is studied.

Data volume reduction in covering approximation spaces with respect to twenty-two types of covering based rough sets

In this paper, we investigate whether consistent mappings can be used as homomorphism mappings between a covering based approximation space and its image with respect to twenty-two pairs of covering upper and lower approximation operators. We also consider the problem of constructing such mappings and minimizing them. In addition, we investigate the problem of reducing the data volume using consistent mappings as well as the maximum amount of their compressibility. We also apply our algorithms against several datasets.

Knowledge reduction of dynamic covering decision information systems caused by variations of attribute values

In practical situations, it is time-consuming to conduct knowledge reduction of dynamic covering decision information systems caused by variations of attribute values with the non-incremental approaches. In this paper, motivated by the need for knowledge reduction of dynamic covering decision information systems, we introduce incremental approaches to computing the type-1 and type-2 characteristic matrices for constructing the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces caused by revising attribute attributes. We also employ several examples to explain how to compute the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Then we propose the incremental algorithms for computing the second and sixth lower and upper approximations of sets and employ experimental results to illustrate the incremental algorithms are effective to calculate the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Finally, we give two examples to show how to conduct knowledge reduction of dynamic covering decision information systems caused by altering attribute values.

Characteristic matrixes-based knowledge reduction in dynamic covering decision information systems

In practical situations, dynamic covering decision information systems that change over time are of interest because databases of this kind are frequently encountered. Incremental approaches are effective in performing dynamic learning tasks because they can make the best use of previous knowledge. In this paper, motivated by the need for knowledge reduction of dynamic covering decision information systems due to variations in the object sets, we present incremental approaches for computing type-1 and type-2 characteristic matrixes of dynamic coverings. We update the characteristic matrixes with regard to two aspects: immigration and emigration of objects. Then, we provide incremental algorithms to compute the second and sixth lower and upper approximations of sets in the dynamic covering approximation spaces. The experimental results confirm that the computational complexity of constructing approximations of concepts is significantly reduced using the incremental approaches. Finally, we perform knowledge reduction of dynamic covering decision information systems by using the incremental approaches.