Elsevier

Knowledge-Based Systems

Volume 60, April 2014, Pages 28-34
Knowledge-Based Systems

The relationship among three types of rough approximation pairs

https://doi.org/10.1016/j.knosys.2014.01.001Get rights and content

Abstract

The aim of the paper is to clarify mathematical relationship among three types of rough approximation pairs. We propose subsystem-based definition of generalized rough sets and consider three different types of generalized rough approximation pairs from the viewpoint of three definitions, namely, the element, granule, and subsystem-based definitions. This paper investigates these three types of generalized rough sets with respect to arbitrary binary relations. The topologies induced by granule-based generalized rough sets are introduced. Finally, we give a characterization of granule-based generalized rough sets.

Introduction

Rough set theory, proposed by Pawlak [18], [19] in 1982, is an extension of set theory for the analysis of a vague and inexact description of objects. In recent years, there has been an internationally increasing interests in rough set theory and its applications [2], [3], [4], [6], [7], [9], [20], [21], [24], [25], [26].

In real world databases, data sets usually take on variant forms, Pawlak rough approximations are based on equivalence relations, this requirement is not satisfied in many situations. So from the angle of applications, it is found to be more important to look into situations where the binary relation may not be an equivalence relation but only an arbitrary. To address this issue, many researchers have recently proposed several data processing methods using generalized rough set models [1], [5], [11], [20], [22]. For this purpose, Pawlak rough approximations have been extended to the arbitrary binary relation based rough set models [9], [13], [27], the fuzzy relation based rough set models [17], [29], and the covering-based rough set models [14], [36]. Much significant research has been done in this direction. Recently, Samanta and Chakraborty [23] give an extensive survey of various generalized approaches to the approximation pairs of a set. Skowron and Rauszer [24] proposed the concepts of discernibility matrix and discernibility function and applied them to attribute reduction in information systems.

It is well-known that Pawlak rough approximations can be presented in many equivalent forms. For instance, Yao and Yao [33] considered three different equivalent forms according to the element, granule, and subsystem-based definitions. In their opinion, the three equivalent forms are the most commonly used ones in rough set literature and offer different interpretations of rough set approximations. Moreover, the three equivalent forms represent three research directions and have different application fields. Now many new applications are found by using a general binary relation in place of an equivalence relation. For example, incomplete information systems cannot be handled with Pawlak rough sets. So several generalizations [2], [3], [4] were proposed to solve this problem, one of approaches is to relax an equivalence relation to a general binary relation. Generally, if we use a general binary relation in place of an equivalence relation in the three forms, we can propose three types of generalized rough approximation pairs from the viewpoint of elements, granules, and subsystems, respectively, however, they are no longer equivalent. Naturally, we need to study these three different types of generalized rough approximation pairs. In the past, three types of rough approximations were studied independently, the relationship among them is not considered. The relationship is useful for theoretical respects. For instance, we will show in this paper that R is the composition of R-1 and R, similarly, R̲ is the composition of R-1 and R̲. These results tell us that we do not need to study independently approximation pair (R̲,R). Although we cannot guarantee that they are equivalent, there is a close relationship among them. Mathematically, it is an interesting problem to consider the relationship among these three different types of generalized rough sets. The aim of the paper is to compare these three types of generalized rough sets and to establish the relationship among element, granule, and subsystem-based definitions, which may help to develop methods for applications.

Yao [30] proposed an element-based definition of generalized rough sets, and this type of generalized rough sets is the most commonly used one in rough set theory. Many interesting properties and applications of this type of generalized rough sets were derived. According to the equivalent representation formula of Pawlak rough sets, Zhang et al. [35] proposed granule-based definition of generalized rough sets. Although two different definitions of approximation pairs have been presented, the relationship between them have not been analyzed adequately. In addition, subsystem-based form of Pawlak rough sets is an important interpretation of rough sets, along this line of consideration, we propose subsystem-based definition of generalized rough sets in this paper. Although theoretic and application results of subsystem-based definition is not enough in contrast with exist element and granule-based definitions. This type of generalized rough sets can hopefully extend the application fields of Pawlak rough sets.

The remainder of the paper is organized as follows. Section 2 reviews three different equivalent forms of Pawlak rough sets and their three generalizations by extending the rough sets to arbitrary binary relations. We give basic properties of these types of generalized rough sets. Section 3 discusses the relationship and difference between element and granule-based definition of generalized rough sets. Section 4 considers the relationship among three types of rough upper approximations. Section 5 studies the topologies induced by binary relations. Section 6 gives a characterization of granule-based generalized rough approximations. Finally, Section 7 concludes the paper.

Section snippets

Three types of generalized rough sets and their properties

In this section, we recall three equivalent forms of Pawlak rough sets [18], [19] as well as their generalization based on arbitrary binary relations.

Let U be any given universal set. Recall that a binary relation, R, on U is an equivalence relation if it is reflexive, symmetric, and transitive. Furthermore, every equivalence relation induces a partition and vice versa. If R is an equivalence relation on U, we use [x]R to denote an equivalence class in R containing an element, x, in U and use U/

Relationship between pairs (R̲,R) and (R̲,R)

In this section, we study the relationship between two types of approximation pairs, (R̲,R) and (R̲,R). The following proposition gives the relationship between operator pairs, (R̲,R) and (R̲,R).

Proposition 3.1

Suppose that U is any given universal set and R is an arbitrary binary relation on U. Then for each subset, XU,

  • (1)

    RX=R-1(RX)=R-1RX [15], where R-1 denotes the inverse of R and R-1R denotes the composition of R-1 and R.

  • (2)

    R̲X=R-1(R̲X).

Proof

(2) For each subset XU,R̲X={rR(x)|rR(x)X}={rR(x)|xR̲X

Relationship among pairs (R̲,R),(R̲,R) and (R̲,R)

In this section, we consider relationship among approximation pairs (R̲,R), (R̲,R) and (R̲,R). By Proposition 3.1 and Proposition 2.3, we have obtained R̲=R̲=R-1R̲. Thus, we only consider the relationship among R, R, and R.

Proposition 4.1

Let R be a binary relation on universal set U. Then, for each XU,XRX if and only if RXRX.

Proof

Suppose XRX. Since RXθR, by the definition of R, we have RXRX.

Conversely, if RXRX, then XRXRX, and XRX. 

Corollary 4.1

Let R be a reflexive

A topology induced by an approximation pair (R̲,R)

There exists a close relationship between topologies and rough sets [28]. This section considers the topology induced by an approximate pair (R̲,R). For a binary relation, R, on U, it is well-known that the upper approximation operator R satisfies the Kuratowski closure axioms [8], [16] if and only if R is reflexive and transitive. Now we study the conditions under which R satisfies the closure axioms.

Proposition 5.1

Let R be a binary relation on U. Then R satisfies closure axioms if and only if R-1R

The characterization of granule-based generalized rough sets

In this section, we give a characterization of granule-based generalized rough sets. Suppose that U is a universal set and P(U) is the power set of U. Consider a set-theoretical operator, f:P(U)P(U), we define the dual and inverse operators of f as follows.f̲:P(U)P(U),f̲(X)=(f(X)),where X is the complement of X in U, andf-1:P(U)P(U),f-1(X)={x|f({x})X}for allXP(U).

Proposition 6.1

Let U be any given universal set and f:P(U)P(U) be an operator. Suppose that f satisfies the following conditions:

  • (1)

    f()=

Conclusions

This paper is a preliminary attempt to clarify the relationship among three types of rough approximation pairs. For arbitrary relations, we propose the definition of generalized rough approximation pair based on subsystem. Pawlak rough approximations, based on equivalence relations, can be represented by at least three equivalent forms, and different forms offer different interpretations of rough set approximation pairs. According to this three forms, three corresponding generalized rough

Acknowledgment

The author would like to thank the anonymous referees for their helpful comments and suggestions which greatly improved the exposition of the paper. This work is partially supported by the National Natural Science Foundation of China (Nos. 61272031 and 60973148).

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