The relationship among three types of rough approximation pairs
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 is the composition of and , similarly, is the composition of and . These results tell us that we do not need to study independently approximation pair . 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 to denote an equivalence class in R containing an element, x, in U and use
Relationship between pairs and
In this section, we study the relationship between two types of approximation pairs, and . The following proposition gives the relationship between operator pairs, and . Proposition 3.1 Suppose that U is any given universal set and R is an arbitrary binary relation on U. Then for each subset, , [15], where denotes the inverse of R and denotes the composition of and R. .
Proof
(2) For each subset ,
Relationship among pairs and
In this section, we consider relationship among approximation pairs , and . By Proposition 3.1 and Proposition 2.3, we have obtained . Thus, we only consider the relationship among , , and . Proposition 4.1 Let R be a binary relation on universal set U. Then, for each if and only if . Proof Suppose . Since , by the definition of , we have . Conversely, if , then , and . □ Corollary 4.1 Let R be a reflexive
A topology induced by an approximation pair
There exists a close relationship between topologies and rough sets [28]. This section considers the topology induced by an approximate pair . For a binary relation, R, on U, it is well-known that the upper approximation operator satisfies the Kuratowski closure axioms [8], [16] if and only if R is reflexive and transitive. Now we study the conditions under which satisfies the closure axioms. Proposition 5.1 Let R be a binary relation on U. Then satisfies closure axioms if and only if
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 is the power set of U. Consider a set-theoretical operator, , we define the dual and inverse operators of f as follows.where is the complement of X in U, and Proposition 6.1 Let U be any given universal set and be an operator. Suppose that f satisfies the following conditions:
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).
References (36)
- et al.
Generalization of Pawlak’s rough approximation spaces by using -open sets
Int. J. Approx. Reason.
(2012) - et al.
Approximations and uncertainty measures in incomplete information systems
Inform. Sci.
(2012) - et al.
Axiomatic systems for rough set-valued homomorphisms of associative rings
Int. J. Approx. Reason.
(2013) - et al.
A sequential pattern mining algorithm using rough set theory
Int. J. Approx. Reason.
(2011) On the structure of generalized rough sets
Inform. Sci.
(2006)- et al.
Rough set theory for topological spaces
Int. J. Approx. Reason.
(2005) - et al.
Matroidal approaches to rough sets via closure operators
Int. J. Approx. Reason.
(2012) Using one axiom to characterize rough set and fuzzy rough set approximations
Inform. Sci.
(2013)- et al.
Invertible approximation operators of generalized rough sets and fuzzy rough sets
Inform. Sci.
(2010) - et al.
An axiomatic characterization of a fuzzy generalization of rough sets
Inform. Sci.
(2004)
Rough operations on Boolean algebras
Inform. Sci.
Constructive and axiomatic approaches of fuzzy approximation operators
Inform. Sci.
Two views of the theory of rough sets in finite universes
Int. J. Approx. Reason.
Constructive and algebraic methods of theory of rough sets
Inform. Sci.
A comparative study of fuzzy sets and rough sets
Inform. Sci.
Covering based rough set approximations
Inform. Sci.
Algebraic structures for rough sets
Trans. Rough Sets II, Lect. Notes Comput. Sci.
Three-valued logics for incomplete information and epistemic logic
JELIA
Cited by (16)
Reduction approaches for fuzzy coverings
2023, Fuzzy Sets and SystemsCitation Excerpt :For a covering, Zhu and Wang [9] proposed different CBRS models and studied their properties. We considered the connection of these different CBRS models [10]. Ma [11,12] considered such CBRSs from a viewpoint of neighborhoods and we [13] studied the relationship among these CBRSs.
The lattice and matroid representations of definable sets in generalized rough sets based on relations
2019, Information SciencesCitation Excerpt :Rough set theory, which was first formulated by Pawlak [17] in the early 1980s, has been successfully applied to many fields, such as, in artificial intelligence, computer science, decision theory, expert systems, knowledge representation, pattern recognition, etc. In order to deal with complex practical problems, Pawlak rough set model is extended, and various generalized rough set models have been established, for example, generalized rough set based on relation [19,21,32,33,39,40], covering rough set model [5,10,20,26,35,37,41–44], rough set models in multigranulation spaces [13,14,22,30,36,38,46], etc. Moreover, many mathematical structures have been used to study rough set theory, such as, topological structures [9,19,41,47], lattice structures [3,15,18,31] and matroidal structures [8,11,12,23,24,26–29].
Textures and Rough Sets
2017, Handbook of Neural ComputationKnowledge reduction of dynamic covering decision information systems when varying covering cardinalities
2016, Information SciencesCitation Excerpt :Since covering rough set theory was introduced by Zakowski in 1983, it has become a powerful mathematical tool for studying knowledge reduction of covering information systems [2,6–9,12,13,25,28,31–33,38,39,43,45,46,49,54,58,62–65,68,75–77].
Constructive methods of rough approximation operators and multigranulation rough sets
2016, Knowledge-Based SystemsCitation Excerpt :The Pawlak’s rough set model [17] is based on equivalence relations, it has been generalized to arbitrary binary relations based rough sets, tolerance or similarity relations based rough sets, fuzzy rough sets and intuitionistic fuzzy rough sets (see [4,5,44]), etc. Moreover, as one of generalized models, covering rough sets has attracted much attention and induced lots of interesting results [13,19,31,34,38,40,46]. For the above rough set models, we usually only consider a single approximation space.
Fast approach to knowledge acquisition in covering information systems using matrix operations
2015, Knowledge-Based SystemsCitation Excerpt :Tn address this issue, many scholars have combined the ideas of rough sets and fuzzy sets, and developed new theoretical mechanisms for data analysis, such as fuzzy rough sets [8,10], rough fuzzy sets [2,8] and bipolar fuzzy rough sets [28,29]. In fact, the neighborhood relations [23,41], tolerance relations [7,31], similarity relations [6,15] and fuzzy relations [4,27] all generate coverings of the universe. Therefore, these kinds of rough sets can be categorized into the so-called covering rough sets.