Abstract

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper discusses five types of existing neighborhood-based generalized rough sets. The concepts of minimal neighborhood description and maximal neighborhood description of an element are defined, and by means of the two concepts, the properties and structures of the third and the fourth types of neighborhood-based rough sets are deeply explored. Furthermore, we systematically study the covering reduction of the third and the fourth types of neighborhood-based rough sets in terms of the two concepts. Finally, two open problems proposed by Yun et al. (2011) are solved.

1. Introduction

Rough set theory was first proposed by Pawlak [1] for dealing with vagueness and granularity in information systems. It has been successfully applied to process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis, and other fields [2–10]. The further investigation into rough set theory and its extension will find new applications and new theories [11].

The classical rough set theory is based on equivalence relation. However, equivalence relation imposes restrictions and limitations on many applications [12–15]. Zakowski then established the covering-based rough set theory by exploiting coverings of a universe [16]. The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. For covering models, two important theoretical issues must be explored. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attribute reduct. The concept of attribute reduct can be viewed as the strongest and the most important result in rough set theory to distinguish itself from other theories. However, the current processes covering generalized rough sets mainly focus on constructing approximation operations [16–24]. Little attention has been paid to attribute reduction of covering generalized rough sets [14, 19, 25]. In this paper, five types of special covering generalized rough sets, that is, neighborhood-based generalized rough sets [24, 26, 27] are elaborated, and the covering reduction is also examined and discussed in detail.

Zhu and Wang investigated the covering reduction of the first type of generalized rough sets [14]. Yang and Li constructed a unified reduction theory for the first, the second, and the fifth types of generalized rough sets [25]. This paper establishes the reduction theory for the third and the fourth types of neighborhood-based generalized rough sets in terms of the new concepts defined by us. This newly proposed theory can reduce redundant elements in a covering and then find the minimal coverings that induce the same neighborhood-based lower and upper approximation.

The remainder of this paper is organized as follows. In Section 2, we review the relevant concepts and properties of generalized rough sets. Section 3 defines the concepts of minimal neighborhood description and maximal neighborhood description related to an element, and the new characterizations of the third and the fourth types of neighborhood-based rough sets are given by means of the two concepts. In Section 4, we study the reduction issues of the third and the fourth types of neighborhood-based rough sets. In Section 5, two open problems proposed by Yun et al. in [28] are solved. This paper concludes in Section 6.

2. Preliminaries

In this section, we will briefly review basic concepts and results of the generalized rough sets. Let be a nonempty set and . In this paper, we denote by the complement of .

Definition 2.1 ([17] Covering). Let be a universe of discourse and a family of subsets of . If no subsets in is empty, and , is called a covering of .

It is clear that a partition of is a covering of , so the concept of a covering is an extension of the concept of a partition. In the following discussion, unless stated to the contrary, the coverings are considered to be finite, that is, coverings consist of a finite number of sets in them.

Definition 2.2 ([17] Covering approximation space). Let be a nonempty set and a covering of . The pair is called a covering approximation space.

Definition 2.3 ([17], Neighborhood). Let be a covering of and . is called the neighborhood of . Generally, we omit the subscript when there is no confusion.

By the above Definition, it is easy to see that for all , and for all , .

In this paper, we consider five pairs of dual approximation operators defined by means of neighborhoods.

Definition 2.4 ([24] Neighborhood-based approximation operations). Let be a covering approximation space. The five types of neighborhood-based approximation operations are defined as follows: for any , (1), ,(2), ,(3), for all ,(4)for all , ,(5) for all , ,
, , , , and are called the first, the second, the third, the fourth, and the fifth neighborhood-based lower approximation operations with respect to , respectively. , , , , and are called the first, the second, the third, the fourth, and the fifth neighborhood-based upper approximation operations with respect to , respectively.

This paper is concerned with the list of five definitions of approximations (Definition 2.4). In fact, the above definition can be extended. For definitions of dual approximations and many other approximations look at [29].

Note 1. In [24], is denoted by , that is, . This is not accurate. For example, let , , , and . Clearly, is a covering of and and . Taking , since and , it follows that . However, it is easy to see that . Hence, . This contradicts the fact that and are dual with each other. In above definition, we denote by .

3. Minimal Neighborhood Description and Maximal Neighborhood Description

In this section, we define the concepts of minimal neighborhood description and maximal neighborhood description of an element. And we show that the two notions play essential roles in the studies of neighborhood-based rough sets.

Now we give the definitions of minimal neighborhood description and maximal neighborhood description related to an element.

Definition 3.1 (Minimal neighborhood description). Let be a covering approximation space and . The family of sets is called the minimal neighborhood description of the element . When there is no confusion, we omit the subscript .

By above definition, it is easy to see that every element in is a minimal neighborhood contained in .

Definition 3.2 (Maximal neighborhood description). Let be a covering approximation space and . The family of sets is called the maximal neighborhood description of the element . When there is no confusion, we omit the subscript .

By above definition, it is easy to see that every element in is a maximal neighborhood containing .

In order to describe an object, we need only the essential characteristics related to this object, not all the characteristics for this object. That are the purposes of the minimal neighborhood description and the maximal neighborhood description concepts.

For better understanding of Definitions 3.1 and 3.2, we illustrate them by the following example.

Example 3.3. Let , , , , , and . Clearly, is a covering of . It is easy to check that , , , and . By Definitions 3.1 and 3.2, we can get that , , and , , and .

Remark 3.4. Based on the above analysis, we know that every element in is a neighborhood of covering approximation space . Hence, for convenience, in this paper, we may use for all to express any element belonging to . Similarly, we may use for all to express any element belonging to .

3.1. The Third Type of Neighborhood-Based Rough Sets and the Minimal Neighborhood Description

In the following, we will employ the concept of minimal neighborhood description to characterize the third type of neighborhood-based rough sets. Firstly, we introduce a lemma.

Lemma 3.5. Let be a covering approximation space and . Then for all, there exists such that .

Proof. Since is a finite covering of , it follows from Definition 2.3 that the set has only finite elements. We will use this fact to prove the lemma.
Let . Then . Assume that for all , . Then , hence by Definition 3.1, , . By and the assumption, we have . Clearly , so again by Definition 3.1, , . Hence . By the assumption, . Clearly, , so again by Definition 3.1, , . Continue in this way, we have an infinite sequence in such that . But it is impossible since the set has only finite elements. This completes the proof.

Remark 3.6. Since for all , , it follows from Lemma 3.5 that there exists such that . This implies that for all , .

Theorem 3.7. Let be a covering approximation space. Then for , , .

Proof. Let . We first show that . For all , by the part (3) of Definition 2.4, we have that , . By Lemma 3.5, there exists such that . This implies that . It follows that . Thus . On the other hand, by Definitions 3.1 and 2.4, it is obvious that . Hence .
We have proved that . By Definition 2.4, we know that and are dual with each other. Thus for all , that is, for all .

The above theorem establishes the relationship between the third type of neighborhood-based rough sets and the notion of minimal neighborhood description. In order to study further the third type of neighborhood-based rough sets, we will explore the properties of minimal neighborhood description.

Proposition 3.8. Let be a covering approximation space and . Then for all , .

Proof. For all , by Definition 3.1, it is clear that . Thus by Definition 2.3, we conclude that .

Proposition 3.9. Let be a covering approximation space, and . Then for all , .

Proof. Let . Then by Definition 2.3, we have that . By and Definition 3.1, it is clear that .

The above proposition shows that every element in is a minimal one.

Proposition 3.10. Let be a covering approximation space, and . If for , then .

Proof. Let and . Suppose that . Then by Definition 3.1, we have that there exists such that . Thus . By and Proposition 3.9, this implies that , which contradicts the fact that . Hence .

Proposition 3.11. Let be a covering approximation space and . Then for all , .

Proof. Let . Clearly, , it follows from Proposition 3.10 that and so . On the other hand, for all , by Proposition 3.8, we conclude that and so . It follows from Proposition 3.9 that and thus . Hence . In summary, .

In the following, we will use the above properties to study the third type of neighborhood-based generalized rough sets.

The following example shows that two distinct coverings can generate the same neighborhood-based lower and upper approximation (the third type of neighborhood-based rough sets).

Example 3.12 (Two different coverings generate the same the third type of neighborhood-based rough sets). Let , , , , , , and . Clearly, and are two different coverings of . Then by Definition 2.3, we can get that , and , , . By Definition 3.1, it is easy to check that and . Hence , , and . Thus by Theorem 3.7, it is easy to see that for all , , and .

Now we present the conditions under which two coverings generate the same the third type of neighborhood-based rough sets.

Theorem 3.13. Let and be two coverings of a nonempty set . Then for all , and if and only if for all , .

In order to prove the theorem, we first introduce a lemma.

Lemma 3.14. Let be a covering approximation space. Then ,

Proof. By Theorem 3.7, we have that . By Proposition 3.9, we conclude that , and . Thus .

Now we prove our theorem.

Proof. The sufficiency follows directly from Theorem 3.7.
Conversely, let . For all .(i)We first show that . By Proposition 3.11, we have that . Thus by Lemma 3.14, . Applying the condition , , we can have that . Thus . It follows from Lemma 3.14 that . Therefore, .(ii)We will show that . For all , by Proposition 3.9, we have that . By Lemma 3.14, this implies that . In addition, applying the condition , , we can get that and . It follows that . By (i) and Proposition 3.11, . Thus by Lemma 3.14, we have that and so . According to Lemma 3.14, this implies that . It follows from Proposition 3.8 that and thus . Therefore, . On the other hand, the proof of is similar to that of . Therefore .(iii)We will show that . By the condition, we have that . By (ii), we have that . Thus . By Theorem 3.7, it is clear that . Thus . By Theorem 3.7, this implies that such that and so . By (i) and Proposition 3.9, we can conclude that . This implies that .
By (ii) and (iii), we have that . Thus . In the same way, we can prove that . It follows that . This completes the proof of the necessity.

For the covering of , since the lower approximation and the upper approximation are dual, they determine each other. That is to say, for two coverings and , if and only if . From the above analysis and Theorem 3.13, we can obtain the following two corollaries.

Corollary 3.15. Let and be two coverings of a nonempty set . Then , if and only if , .

Corollary 3.16. Let and be two coverings of a nonempty set . Then , if and only if , .

Theorem 3.13 is an important result for studying the covering reduction of the third type of neighborhood-based rough sets. In Section 4.1, we will present the concept of reduct based on this theorem for the third type of rough set model.

3.2. The Fourth Type of Neighborhood-Based Rough Sets and the Maximal Neighborhood Description

In this subsection, we will study the relationship between the fourth type of neighborhood-based rough sets and the notion of maximal neighborhood description. For this purpose, we first explore the properties of maximal neighborhood description.

Proposition 3.17. Let be a covering approximation space and . Then , .

Proof. Let . By Definition 3.2, we have that . Thus by Definition 2.3, .

Proposition 3.18. Let be a covering approximation space and . If , then there exists such that .

Proof. Since is a finite covering of , it follows from Definition 2.3 that the set has only finite elements. We will use this fact to prove the proposition.
Let . Assume that , . Then , hence by Definition 3.2, , . By the assumption, we have that . Clearly, , so again by Definition 3.2, , . Thus . By the assumption, . Clearly, , so again by Definition 3.2, , . Continue in this way, we have an infinite sequence in such that . But it is impossible since the set has only finite elements. This completes the proof.

By the above proposition, we can easily conclude the following result.

Corollary 3.19. Let be a covering approximation space and . If , then .

Remark 3.20. Since , , it follows from Proposition 3.18 that there exists such that . This implies that , .

Proposition 3.21. Let be a covering approximation space, , and . If then .

Proof. Suppose that . Then by and Definition 3.2, there exists such that , which contradicts with . Thus .

Now we use the concept of maximal neighborhood description to characterize the fourth type of neighborhood-based rough sets.

Theorem 3.22. Let be a covering approximation space. Then for ,

Proof. Let . We first show that . For all , we will prove that . For all, by Definition 3.2, . Since , it follows from the part (4) of Definition 2.4 that . Thus . This implies that . Thus . On the other hand, for all , we can get that . Further, and , by Corollary 3.19, we have that . Thus . It follows from the part (4) of Definition 2.4 that and so . In summary, .
Now we show that . For all , by Definition 2.4, there exists such that and . Taking , by Corollary 3.19, we have that and . Thus and so . Therefore, . On the other hand, for all , there exists such that . Thus , s.t. . By Definition 3.2, . Hence there exists such that and . It follows from Definition 2.4 that and so . In summary, .
This completes the proof of theorem.

The following example shows that two different coverings can induce the same the fourth type of neighborhood-based lower and upper approximation operations.

Example 3.23 (Two different coverings generate the same the fourth type of neighborhood-based rough sets). Let , , , , , and . Clearly, and are two different coverings of . By Definition 2.3, it clear that , , , and , . It is easy to check that , and , . Thus , and so , . It follows by Theorem 3.22 that and .

In the following, we study the conditions for two coverings generating the same fourth type of neighborhood-based lower and upper approximation operations. Firstly, we present two lemmas.

Lemma 3.24. Let be a covering of a nonempty set and . If and , then .

Proof. Suppose that . Then , s.t. . Thus by Definition 2.3, . Since , it follows from Definition 3.2 that . By and Definition 3.2, it is clear that . Thus , which contradicts the condition . Therefore, .

Lemma 3.25. Let and be two coverings of a nonempty set . And and satisfy the condition that , . If for , , , and , then .

Proof. Suppose that . Then, . Taking , that is, , . Thus by and Proposition 3.21, we have that and so . On the other hand, since and , it follows from Lemma 3.24 that . Thus , which contradicts the condition , . Therefore, .

Now we present the conditions under which the two different coverings generate the same fourth type of neighborhood-based upper approximation operation.

Theorem 3.26. Let and be two coverings of a nonempty set . Then the following assertions are equivalent: (1), ,(2), ,(3), ,(4).

Proof. (1)(2) , by Theorem 3.22, we have that and . Since , , it follows that . Thus .
(2)(1) It follows directly from Theorem 3.22.
(2)(3) Let . By assertion (2), . Thus, for all , there exists such that . Suppose that . Then by Lemma 3.25, we have that . Thus there exists such that . By Proposition 3.21, implies and so . In addition, by and Lemma 3.24, we have that . Thus , which is a contradiction with the assertion (2). Therefore, and so . Thus . In the same way, we can prove that . Thus .
(3)(2) It is obvious.
(3)(4) It is obvious.
(4)(3) Let . For all , clearly, . Since , there exists such that . By and Definition 3.2, we have that . Thus . Since , it follows that there exists such that . Thus again by Proposition 3.21, we can have that . It follows from that . Hence . In the same way, we can prove that . Thus .

For the covering of , since the lower approximation and the upper approximation are dual, they determine each other. That is, for two coverings and , if and only if . From the above analysis and Theorem 3.26, we can obtain the following results.

Corollary 3.27. Let and be two coverings of a nonempty set . Then the following assertions are equivalent: (1), ,(2), ,(3), ,(4).

Now we present the conditions under which the two different coverings generate the same the fourth type of neighborhood-based rough sets.

Theorem 3.28. Let and be two coverings of a nonempty set . Then the following assertions are equivalent: (1), and ,(2), ,(3), ,(4).

Proof. It follows directly from Theorem 3.26 and Corollary 3.27.

The above theorem is an important result for studying the covering reduction of the fourth type of neighborhood-based rough sets. In Section 4.2, we will present the concept of reduct based on this theorem for the fourth type of rough set model.

4. Reduction of the Third and the Fourth Types of Neighborhood-Based Rough Sets

Examples 3.12 and 3.23 show that for a covering, it could still be a covering by dropping some of its members. Furthermore, the resulting new covering might still produce the same neighborhood-based lower and upper approximations. Hence, a covering may have “redundant” members, and a procedure is needed to find its “smallest” covering that induces the same neighborhood-based lower and upper approximations. This technique can be used to reduce the redundant information in data mining.

In this section, we will investigate the reduction issues about the third and the fourth types of neighborhood-based generalized rough sets. Since for a covering it could not be a covering by dropping some of its members, we need to extend the concepts of neighborhood, minimal neighborhood description, and maximal neighborhood description to a general family of subsets of a universe case so as to reasonably explore the covering reduction of neighborhood-based rough sets.

Let be a nonempty set called the universe of discourse. The class of all subsets of will be denoted by . Naturally, we present the definitions of generalization of neighborhood, minimal neighborhood description, and maximal neighborhood description.

Definition 4.1 (Neighborhood). Let be a universe, and . is called the neighborhood of . Generally, we omit the subscript when there is no confusion.

Definition 4.2 (Minimal neighborhood description). Let be a universe, and . The family of sets is called the minimal neighborhood description of the element . When there is no confusion, we omit the subscript .

Definition 4.3 (Maximal neighborhood description). Let be a universe, and . The family of sets is called the maximal neighborhood description of the element . When there is no confusion, we omit the subscript .

It is easy to see that when is a covering of , Definitions 4.1, 4.2, and 4.3 are coincident with Definitions 2.3, 3.1, and 3.2, respectively.

4.1. Reduction of the Third Type of Neighborhood-Based Rough Sets

Throughout this subsection, we always assume that the rough set model which is discussed by us is the third type of neighborhood-based generalized rough sets. So the definitions of a reducible element, an irreducible covering, and a reduct are all based on the third type of neighborhood-based rough sets.

A reduct should be able to preserve the original classification power provided by the initial covering. In order to present a reasonable notion of reduct, we first give the definition of a reducible element of a covering.

Definition 4.4 (A reducible element about the third type of lower and upper approximation operations). Let be a covering of a universe and . If , , we say that is a reducible element of . Otherwise, is an irreducible element of .

Definition 4.5 (Irreducible covering about the third type of lower and upper approximation operations). Let be a covering of a universe . If every element of is an irreducible element, we say that is irreducible. Otherwise, is reducible.

Definition 4.6 (Reduct about the third type of lower and upper approximation operations). Let be a covering of a universe and . If is an irreducible covering and , , we say that is a reduct of . Let .

In the following, we will illustrate that, for a covering, the reduct always exists and is not unique. Further, we will show that every reduct and the initial covering induce the same lower and upper approximation operations.

Firstly, we give an important proposition.

Proposition 4.7. Let be a covering of a universe and . If and satisfy the condition that , , then is a covering of .

Proof. Suppose that is not a covering of . Then . Taking , by Definition 4.1, we have that . Thus by Definition 4.2, . On the other hand, since is a covering of , it follows from Remark 3.6 that . Thus , which contradicts with the conditions , . This completes the proof.

Corollary 4.8. Let be a covering of a universe and . If is a reducible element of , then is still a covering of .

Proof. It comes directly from Definition 4.4 and Proposition 4.7.

The following theorem shows that for a covering, there is at least one reduct.

Theorem 4.9. Let be a covering of a universe . Then there exists such that is a reduct of .

Proof. Suppose that for all , is not a reduct of . Then is not a reduct of . Thus by Definition 4.6, is a reducible covering. This implies that there exists such that is a reducible element of . We write . By Definition 4.4, we have that , . Further, by Corollary 4.8, is a covering of , and clearly, . Thus by the assumption, is not a reduct of . Since , , it follows from Definition 4.6 that is a reducible covering and so there exists such that is a reducible element of . We write . By Definition 4.4, we have that , . By , , this implies that , . Further, by Corollary 4.8, is a covering of , and clearly, , that is, . Thus by the assumption, is not a reduct of . Continue in this way, we have an infinite sequence such that . But it is impossible since the covering has only finite elements. This completes the proof.

Actually, the proving process of the above theorem provides a procedure to compute the reduct of a covering of a universe.

The following result shows that the definition of reduct is reasonable.

Theorem 4.10. Let be a covering of a universe . Then for all , and generate the same the third neighborhood-based lower and upper approximations.

Proof. It follows directly from Definition 4.6, Proposition 4.7, and Theorem 3.13.

Lemma 4.11. Let be a covering of a universe and . and satisfy the condition that , . Then , , we have that and .

Proof. Let . Then by the condition, . Thus , there exists such that . Hence and by Proposition 4.7, is a covering of . Thus, applying Proposition 3.9, we have that . It follows that and .

Proposition 4.12. Let be a covering of a universe and . and satisfy the condition that , . Then , is a reducible element of .

Proof. Let and . Then .(i)We will show that and are two coverings of . By the condition and Proposition 4.7, is a covering of . Clearly, , thus is also a covering of .By (i) and the condition, we know that , , and are all coverings of . Hence, in the following process of proof, we can use directly the concepts and conclusions obtained in Section 3.(ii)We will prove that , , . For all , by the condition and Lemma 4.11, we have that . Since , it follows by Definition 2.3 that . Thus .(iii)We will show that . For all , then by Definition 3.1, . By , we have that . Thus suppose that . Then by Definition 3.1, there exists such that . By (ii), we have that , thus, . Clearly, , thus , which contradicts the fact that . Thus . It follows by (ii) that . Thus .(iv)We will show that . For all . Since , it is clear that . By and Lemma 3.5, there exists such that . This implies that and so . Thus by Proposition 3.9, . Further, by and (ii), we have that . Thus . In addition, by Lemma 4.11, and . Thus and so . It follows by Proposition 3.9 that and so and . By Definition 3.1, it is clear that . Since , it follows that . Thus . It follows by Proposition 3.10 that and so . Thus, by the condition , we conclude that . It follows that .
By (iii) and (iv), we have that . We have proved that , . Thus by Definition 4.4, is a reducible element of .

Corollary 4.13. Let be a covering of a universe , a reducible element of and . If is an irreducible element of , then is an irreducible element of .

Proof. Suppose that is a reducible element of . Then by Definition 4.4, , . In addition, since is a reducible element of , it follows from Definition 4.4 that , . Thus , . Clearly, , thus by Proposition 4.12, is a reducible element of , which contradicts the condition that is an irreducible element of . This completes the proof.

The above proposition guarantees that omitting a reducible element in a covering will not make any current irreducible element reducible. Therefore, the set of all irreducible elements of is constant. We denote this set by , that is,

The following result establishes the relationship between and .

Theorem 4.14. Let be a covering of a universe . Then .

Proof. Let . Suppose that . Then there exists such that . Hence . By Definition 4.6 and Proposition 4.12, this implies that is a reducible element of , which contradicts the fact that . Hence and so . On the other hand, let . Suppose that . Then is a reducible element of . By Definition 4.4, we have that , and by Corollary 4.8, is a covering of . Thus by Theorem 4.9, there exists such that is a reduct of . By Definition 4.6, this implies that is an irreducible covering and , . Thus , and is an irreducible covering. It follows from Definition 4.6 that is a reduct of . Since , it follows that , which contradicts the fact that . Thus , and so . In summary, .

The above result states that an element will not be reduced in any reduction procedure if and only if it is irreducible. Hence the irreducible elements will be reserved in any reduction procedure, that is to say, is contained in any reduct of . So we can compute the reduct of based on .

Example 4.15. Let , , , , , and . Clearly, is a covering of .
By Definition 4.2, it easy to check that
By Definition 4.2, we can get that Hence and do not satisfy the condition , . It follows from Definition 4.4 that is an irreducible element of .
By Definition 4.2, we can get that Hence It follows from Definition 4.4 that is a reducible element of . In the same way, we can check that and are all reducible elements of .
Hence .
Since and are not coverings of , it follows from Corollaries 4.8 and 4.13 that every element of is an irreducible. Thus is a reduct of .
For , by Definition 4.2, it is easy to check that It follows from Definition 4.4 that is a reducible element of . Further, for , by Corollaries 4.8 and 4.13, it is clear that and are all irreducible elements of . Thus it is clear that is a reduct of . A similar analysis to , we can also get that is a reduct of .
To sum up, has two reducts that are and . It is easy to see that .

The above example also illustrates that for a covering, the reduct is not unique.

Remark 4.16. Let be a covering of a universe . For , , by Definitions 4.4, 4.5, and 4.6, it is easy to see that and do not satisfy the condition , . Thus by Theorem 3.13, we know that and cannot induce the same lower and upper approximation operations. This illustrates that , is a smallest covering that induces the same the third type of neighborhood-based rough sets.

4.2. Reduction of the Third Type of Neighborhood-Based Rough Sets

Throughout this subsection, we always assume that the rough set model which is discussed by us is the fourth type of neighborhood-based generalized rough sets. So the definitions of a reducible element, an irreducible covering and a reduct, are all based on the fourth type of neighborhood-based rough sets.

Notation 1. Let be a covering of . We write .

By Theorem 3.28, we know that if , then and generate the same the forth type of neighborhood-based lower and upper approximation operations. So we can give the following definition of a reducible element.

Definition 4.17 (A reducible element about the fourth type of lower and upper approximation operations). Let be a covering of a universe and . If , we say that is a reducible element of . Otherwise, is an irreducible element of .

Definition 4.18 (Irreducible covering about the fourth type of lower and upper approximation operations). Let be a covering of a universe . If every element of is an irreducible element, we say that is irreducible. Otherwise, is reducible.

Definition 4.19 (Reduct about the fourth type of lower and upper approximation operations). Let be a covering of a universe and . If is irreducible and , we say that is a reduct of . Let .

The following proposition is basic.

Proposition 4.20. Let be a covering of a universe and . If and satisfy the condition that , then is a covering of .

Proof. Suppose that is not a covering of . Then . Taking , by Definition 4.1, we have that , . Thus , . On the other hand, since is a covering of , it follows from Definition 3.2 that . Thus there exists such that . Hence and so , which is a contradiction with the condition . Thus is a covering of .

Corollary 4.21. Let be a covering of a universe and . If is a reducible element of , then is still a covering of .

Proof. It comes directly from Definition 4.17 and Proposition 4.20.

In the following, we will illustrate that for a covering, the reduct always exists.

Theorem 4.22. Let be a covering of a universe . Then there exists such that is a reduct of .

Proof. The proof is similar to that of Theorem 4.9.

Now we show that every reduct and the initial covering induce the same lower and upper approximation operations.

Theorem 4.23. Let be a covering of a universe . Then for all , and generate the same the fourth type of neighborhood-based lower and upper approximations.

Proof. By Definition 4.19 and Proposition 4.20, is a covering of . Thus by Definition 4.19 and Theorem 3.28, for all , and generate the same the fourth type of neighborhood-based lower and upper approximations.

Lemma 4.24. Let be a covering of a universe and . and satisfy the condition that . Then , we have that and .

Proof. By the conditions and Proposition 4.20, we know that and are all coverings of . Hence, in the following process of proof, we can use directly the concepts and conclusions obtained in Section 3.
Let . Then by , there exists such that and so . By Definition 2.3, it is clear that . In addition, since , it follows from Definition 2.3 that . By , we have that . By , Notation 1 and Definition 3.2, this implies . Thus and .

Proposition 4.25. Let be a covering of a universe and . and satisfy the condition that . Then , is a reducible element of .

Proof. Let , then .(i)We will show that and are two coverings of . By the condition and Proposition 4.20, is a covering of . Clearly, , thus is also a covering of .By (i) and the condition, we know that , and are all coverings of . Hence, in the following process of proof, we can use directly the concepts and conclusions obtained in Section 3.(ii)We will show that . For all , then by Lemma 4.24, and . Since , it follows from Definition 2.3 that . Thus by , we have that . Suppose that . Then . Thus by Definition 3.2, there exists such that and so , which contradicts the fact that . Thus . Combining and , we can get that and so . Thus, .(iii)We will show that , for all. By , it is clear that . By and Proposition 3.18, there exists such that . This implies . By Notation 1, it is clear that . Since , it follows that there exists such that . Thus . Clearly, , hence . In addition, since , it follows from Notation 1 and Definition 3.2 that . This implies that . It follows by that . Thus .
By (ii) and (iii), we have that . Thus by Definition 4.17, is a reducible element of .

Corollary 4.26. Let be a covering of a universe , a reducible element of and . If is an irreducible element of , then is an irreducible element of .

Proof. Suppose that is a reducible element of . Then by Definition 4.17, . Since is a reducible element of , it follows from Definition 4.17 that . Thus . Clearly, , thus by Proposition 4.25, is a reducible element of , which contradicts the condition that is an irreducible element of . This completes the proof.

The above proposition guarantees that omitting a reducible element in a covering will not make any current irreducible element reducible. Therefore, for a covering , the set of all irreducible elements is constant. We denote this set by , that is,

Theorem 4.27. Let be a covering of a universe . Then .

Proof. The proof is similar to that of Theorem 4.14.

The above result states that an element will not be reduced in any reduction procedure if and only if it is irreducible. Hence the irreducible elements will be reserved in any reduction procedure, that is to say, is contained in any reduct of . So we can compute the reduct of based on .

Example 4.28. Let , , , , , , and . Clearly, is a covering of .
By Definition 4.3 and Notation 1, it is easy to see that .
Since , it follows that . Thus is a reducible element of . In the same way, we can check that and are reducible elements of .
Since , it follows that . Thus is an irreducible element of .
Since , it follows that . Thus is an irreducible element of .
Hence .
It is easy to check that and . Thus and are irreducible elements of . By Corollary 4.26, and are also irreducible elements of . Hence is an irreducible covering of . It follows that is a reduct of .
It is easy to check that and . Thus is a reducible element of , and is an irreducible element of . Further, by Corollary 4.26, is an irreducible covering. Further, it is easy to verify that . Hence is a reduct of . A similar analysis to , we can also get that is a reduct of .
To sum up, has two reducts that are and . It is easy to see that .

Remark 4.29. Let be a covering of a universe . For , , by Definitions 4.17, 4.18, and 4.19, it is easy to see that and do not satisfy the condition , . Thus by Theorem 3.28, we know that and cannot induce the same lower and upper approximation operations. This illustrates that , is a smallest covering that induces the same the fourth type of neighborhood-based rough sets.

5. The Two Open Problems

In [28], Yun et al. proposed two open problems how to give sufficient and necessary conditions for to form a partition of by using only a single covering approximation operator . That is to say, the first one is how to characterize the conditions for to form a partition by applying the first type of generalized approximation operator, and the second one is how to characterize the conditions for to form a partition by applying the fourth type of generalized approximation operator. In this section, we present some conditions under which forms a partition of . As a result, the two open problems are solved (see Theorems 5.3 and 5.4).

Lemma 5.1. Let be a covering of a universe . If forms a partition of , then , .

Proof. Let . Suppose that . Then we choose , that is, and . Thus there exists such that . By Definition 2.3, this implies that . On the other hand, since forms a partition of , it follows from that and so . Thus . This is a contradiction with . Therefore, .

Lemma 5.2. Let be a covering of a universe and . If , then .

Proof. Let . Suppose that . Then . By Definition 2.3, this implies that . Since , it follows that and so , which contradicts the fact that . Thus . Hence .

Theorem 5.3. Let be a covering of a universe . Then forms a partition of if and only if for each , .

Proof. Let . By Lemma 5.1, we have that . This implies that . In addition, by the part (1) of Definition 2.4, we have that . Consequently, . On the other hand, , then . Thus , that is, . This implies that , . Thus , . This implies that and so , that is, . Since forms a partition of , it follows that and so . Thus . In summary, . This completes the proof of the necessity.
Conversely, suppose that is not a partition of . Then there exist such that and . Taking , then . Clearly, or . Without loss of generality, we may assume that , then . Thus, by Lemma 5.2, we have that . Thus and so . It follows from the part (1) of Definition 2.4 that . By the condition, we have that and . Thus , which contradicts with . Hence forms a partition of .

In fact, the above theorem establishes the relationship between the first type of generalized rough sets and the other types of neighborhood-based rough sets.

Theorem 5.4. Let be a covering of a universe . Then forms a partition of if and only if for each , .

Proof. The necessity is obvious.
Conversely, suppose that is not a partition of . Then there exist such that and . Taking , then . Clearly, or . Without loss of generality, we may assume that , then . Thus (Otherwise, by Definition 2.3, ). In addition, clearly, ; therefore, by the part (4) of Definition 2.4, we have that . Thus , which contradicts the condition , . Thus forms a partition of .

6. Conclusions

This paper defines the concepts of minimal neighborhood description and maximal neighborhood description in neighborhood-based rough set models. We give the new characterizations of the third and the fourth types of neighborhood-based rough sets. By means of these new characterizations, we explore the covering reduction of two types of neighborhood-based rough sets and have shown that the reduct of a covering is the minimal covering that generates the same lower and upper approximations. Clearly, the notions of minimal neighborhood description and maximal neighborhood description play essential roles in the studies of the reduction issues of the third and the fourth types of neighborhood-based rough sets. In fact, the two concepts are the essential characteristics related to the neighborhood-based rough sets. In particular, the notion of maximal neighborhood description is very useful. A similar notion was also discussed in [30]. In the future, we will further study neighborhood-based rough sets by means of these concepts.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions in improving this paper. This work is supported by The Foundation of National Nature Science of China (Grant no. 11071178).