Characterizations and applications of parametric covering-based rough sets

Abstract

As a technique for granular computing, rough sets deal with the vagueness and granularity in information systems. Covering-based rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings and have been applied in many fields. However, many vital issues in covering-based rough sets, including attribute reduction, are NP-hard and therefore the algorithms for addressing them are usually greedy. Hence, it is necessary and helpful to generalize the covering-based rough sets from different viewpoints. In this paper, a new type of covering-based rough sets, named parametric covering-based rough sets, are proposed and some properties and applications of the parametric covering-based rough sets are investigated. First, a concept of inclusion degree is introduced into covering-based rough set theory to explore some properties of the parametric covering approximation space. Second, the parametric covering-based rough sets are established on the basis of inclusion degree. Moreover, some properties of the parametric covering-based rough sets are introduced. Third, it is found that the calculations of corresponding parametric covering-based lower and upper approximations can be converted into the operations of matrices, which makes the calculations convenient. Finally, a simple application of the parametric covering-based rough sets to network security is introduced.

Keywords

Granular computing Covering Rough set Neighborhood Inclusion degree Parametric covering-based rough sets Network security

1 Introduction

Rough set theory (RST) was originally proposed by Pawlak [26, 27] in 1982 as a useful mathematical tool for dealing with the vagueness and granularity in information systems and data analysis. This theory can approximately characterize an arbitrary subset of a universe by using two definable subsets called lower and upper approximations [7]. Pawlak’s rough set is built on equivalence relations. However, an equivalence relation imposes restrictions and limitations on many applications [11 , 40]. To address this issue, many extensions have been made by replacing equivalence relations with other notions such as reflexive relations [20], similarity relations [32], tolerance relations [31], arbitrary relations [23], coverings [8 , 46–49], and fuzzy relations [12 , 22]. Meanwhile, different rough set models have not only combined with each other, but also connected with other theories [1 , 39]. For instance, Zakowski [43] established covering-based rough set theory by exploiting coverings of universal sets. The study on covering-based rough sets is very necessary and important. Particularly, in recent years, with the fast development of computer science and technology, how to use the effective mathematical tools to address practical problems has become more and more essential.

A further RST innovation has been the development by Ziarko [50] of a variable precision rough sets (VPRS) model, which incorporates probabilistic decision rules. This is an important extension, since as noted by Kattan et al. [19], “In real-world decision making, the patterns of classes often overlap, suggesting that predictor information may be incomplete... This lack of information results in probabilistic decision making, where perfect prediction accuracy is not expected”. VPRS deals with partial classification by introducing a precision parameter β (in the rough set the β value is zero). The β value represents a bound on the conditional probability of a proportion of objects in a condition class, which are classified to the same decision class. Ziarko [50] considered β as a classification error and it is defined to be in the domain [0.0, 0.5). Because the VPRS model has no formal historical background of having empirical evidence to support any particular method of β-reduct selection [2], VPRS-related research studies do not focus in detail on the choice of the precision parameter (β) value. Ziarko [50] proposed the β value to be specified by the decision maker. Beynon [3] proposed two methods of selecting a β-reduct without such a known β value. Beynon [4] proposed the allowable β value range to be an interval, where the quality of classification may be known prior to determining the β value range. The extended VPRS was introduced by Katzberg et al. [10], which allowed asymmetric bounds l and u to be used. The VPRS models the restriction l < 0.5 and u = 1 - l must hold. Beynon [5] introduced the (l, u)-quality graph, which elucidates the associated level of quality of classification, based on the selected l and u values. And another research works for VPRS can be found in [25, 33].

The covering-based rough set model based on variable precision was presented by Zhang and Wang [45] in order to expand rough set application space and to generalize the classical rough set. But there are two basic relational formulas of variable precision covering rough set inclusion ${\underline{C}}_{β} (X ⋂ Y) \subseteq {\underline{C}}_{β} (X) ⋂ {\underline{C}}_{β} (Y)$ and ${\bar{C}}_{β} (X ⋂ Y) \subseteq {\bar{C}}_{β} (X) ⋂ {\bar{C}}_{β} (Y)$ , which means that the information may be lost in the operations of intersection. In [44], Zhang and Mo solved the problems of information lost by introducing two operators in variable precision rough set model. Furthermore, Chen [9] studied the properties of information lost in the variable precision covering rough set. They solved this problem by introducing a new pair of covering upper and lower boundary regions operators imitate to the paper [44] in variable precision covering rough set. They also discussed the properties and made the β upper approximation and lower approximation operators keep intersection operation.

However, the above research works for the VPRS are all based on a parameter β. In this paper, by introducing the concept of inclusion degree, a new type of covering-based rough sets, named parametric covering-based rough sets are defined, which can not only generalize the theory of covering-based rough sets, but extend the applications of covering-based rough sets to diverse fields. Especially, this parameter covering-based rough set model is defined based on two parameters 0 ≤ β < α ≤ 1. On one hand, since there exist two parameters in this parameter covering-based rough set model, it is easy to see that this model is more complex than another VPRS models. On the other hand, the accuracy of this parameter covering-based rough set model is higher than another VPRS models in the practical problems. First, the definition of inclusion degree is proposed and some of its properties are explored. Moreover, through inclusion degree, some problems of covering-based rough sets can be addressed. Second, the parametric covering-based rough sets are proposed by means of inclusion degree and the properties of parametric covering-based rough sets are investigated. Third, it is found that the calculations of corresponding parametric covering-based lower and upper approximations can be represented by the operations of matrices. Finally, an application of parametric covering-based rough sets to network security is given.

The remainder of this paper is organized as follows: In Section 2, some basic concepts and properties of covering-based rough sets are introduced. In Section 3, the parametric covering-based rough sets are proposed by means of inclusion degree. Moreover, characterizations of parametric covering-based rough sets are investigated. In Section 4, the matrix representations of parametric covering-based lower and upper approximations are proposed. In Section 5, an application of the parametric covering-based rough sets is introduced. Section 6 concludes this paper.

2 Preliminaries

We introduce the definitions of covering and partition at first.

Definition 2.1. (Partition [30]) Let U be a universe of discourse and P be a family of subsets of U. P is called a partition of U if the following conditions hold: (1) ∅ ∉ P; (2) ∪P = U; (3) for any K, L ∈ P, K∩ L = ∅. Every element of P is called a partition block.

In the following discussion, unless stated to the contrary, the universe of discourse U is considered to be finite and nonempty.

Definition 2.2. (Covering [30]) Let U be a universe of discourse and C be a family of subsets of U. If ∅ ∉ C and ∪C = U, C is called a covering of U. Every element of C is called a covering block. We also call (U, C) a covering approximation space and denoted it by CAS, i.e., CAS = (U, C).

It is clear that a partition of U is certainly a covering of U. So the concept of covering is an extension of the concept of partition. Neighborhood [6 , 41] is a concept used widely in covering based rough sets. It is defined as follows.

Definition 2.3. (Neighborhood [6]) Let C be a covering of U. For any x ∈ U, ∩ {K ∈ C : x ∈ K} is denoted as N_C (x) and called the neighborhood of x.

It is clear that x ∈ N_C (x). The following proposition gives an important property of neighborhoods.

Theorem 2.4. ([34]) Let C be a covering of U. For any x, y ∈ U, if y ∈ N_C (x), then N_C (y) ⊆ N_C (x).

If y ∈ N_C (x) and x ∈ N_C (y), by the above proposition, we have N_C (x) = N_C (y). All the neighborhoods induced by a covering of a universe form a set family. This set family is still a covering of the universe. This type of set families has been studied by many scholars [39, 34]. However, both the term and the mark of it are not identical. In this paper, we call it covering of neighborhoods and cite the mark proposed by Wang et al. [34].

After the concept of neighborhood has been given, we can introduce the concept of neighborhoods.

Definition 2.5. (Covering of neighborhoods [34]) Let C be a covering of U. {N_C (x) : x ∈ U} is denoted as Cov (C) and called the covering of neighborhoods induced by C.

There is an important property of neighborhoods presented by the following proposition.

Theorem 2.6. ([34]) For any N_C (x) ∈ Cov (C), N_C (x) is not a union of other blocks in Cov (C).

By the definition of Cov (C), we see that Cov (C) is also a covering of universe U. In particular, if C is a partition, we have that Cov (C) = C.

3 Parametric covering-based rough sets

In this section, we introduce a new concept named inclusion degree into covering-based rough sets. Based on this, the parametric covering-based rough set model is established and characterizations of parametric covering-based rough sets are investigated.

3.1 The inclusion degree based on covering approximation space

Definition 3.1. Let CAS = (U, C) be a covering approximation space. For any x ∈ U and X ⊆ U, the inclusion degree of x to X with respect to C is defined as follows: $\begin{matrix} σ_{C}^{X} (x) & = \frac{| X ⋂ N_{C} (x) |}{| N_{C} (x) |} . \end{matrix}$

Example 3.2. Let C = {{a, b} , {a, c} , {b, d}} be a covering of U = {a, b, c, d}. Then N_C (a) = {a}, N_C (b) = {b}, N_C (c) = {a, c}, N_C (d) = {b, d}. Suppose X = {b, c} and Y = {a, b, d}. Then $\begin{matrix} σ_{C}^{X} (a) & = 0, σ_{C}^{X} (b) = 1, σ_{C}^{X} (c) = 0.5, σ_{C}^{X} (d) = 0; \\ σ_{C}^{Y} (a) & = 1, σ_{C}^{Y} (b) = 1, σ_{C}^{Y} (c) = 0.5, σ_{C}^{Y} (d) = 1; \\ σ_{C}^{U} (a) & = σ_{C}^{U} (b) = σ_{C}^{U} (c) = σ_{C}^{U} (d) = 1; \\ σ_{C}^{\emptyset} (a) & = σ_{C}^{\emptyset} (b) = σ_{C}^{\emptyset} (c) = σ_{C}^{\emptyset} (d) = 0 . \end{matrix}$

For any x ∈ U and X ⊆ U, the inclusion degree of x to X with respect to C in the CAS can be measured the inclusion degree of x to X from the perspective of quantity. Based on the definition of inclusion degree, some properties of inclusion degree are proposed as follows:

Proposition 3.3. Let C be a covering of U. Then

(1) $0 \leq σ_{C}^{X} (x) \leq 1$ ; (2) if X ⊆ Y, then $σ_{C}^{X} (x) \leq σ_{C}^{Y} (x)$ ; (3) if x ∈ N_C (y) and y ∈ N_C (x), then $σ_{C}^{X} (x) = σ_{C}^{X} (y)$ ; (4) $N_{C} (x) \subseteq X \Leftrightarrow σ_{C}^{X} (x) = 1$ ; (5) $N_{C} (x) ⋂ X \neq \emptyset \Leftrightarrow σ_{C}^{X} (x) \neq 0$ ; (6) $⋃_{z \in X} N_{C} (z) = X \Leftrightarrow σ_{C}^{X} (z) = 1$ for all z ∈ X; (7) $σ_{C}^{X} (x) + σ_{C}^{Y} (x) = σ_{C}^{(X ⋃ Y)} (x) + σ_{C}^{(X ⋂ Y)} (x)$ ; (8) $σ_{C}^{X} (x) = 1 - σ_{C}^{(\sim X)} (x)$ ;

are hold, where X, Y ⊆ U and x, y ∈ U.

Proof. It is easy to prove this proposition by the above definition of inclusion degree.□

Example 3.4. (Continued from Example 3.2) Suppose X₁ = {b} , X₂ = {b, c, d} , X₃ = {a, b, d}. Then $σ_{C}^{X_{1}} (a) = 0, σ_{C}^{X_{1}} (b) = 1, σ_{C}^{X_{1}} (c) = 0, σ_{C}^{X_{1}} (d) = 0.5$ ; $σ_{C}^{X_{2}} (a) = 0, σ_{C}^{X_{2}} (b) = 1, σ_{C}^{X_{2}} (c) = 0.5, σ_{C}^{X_{2}} (d) = 1$ ; $σ_{C}^{X_{3}} (a) = 1, σ_{C}^{X_{3}} (b) = 1, σ_{C}^{X_{3}} (c) = 0.5, σ_{C}^{X_{3}} (d) = 1$ . It is easy to find that $σ_{C}^{X_{1}} (a) = σ_{C}^{X_{2}} (a), σ_{C}^{X_{1}} (a) < σ_{C}^{X_{3}} (a);$ $σ_{C}^{X_{1}} (b) = σ_{C}^{X_{2}} (b), σ_{C}^{X_{1}} (b) = σ_{C}^{X_{3}} (b);$ $σ_{C}^{X_{1}} (c) < σ_{C}^{X_{2}} (c), σ_{C}^{X_{1}} (c) < σ_{C}^{X_{3}} (c);$ $σ_{C}^{X_{1}} (d) < σ_{C}^{X_{2}} (d), σ_{C}^{X_{1}} (d) < σ_{C}^{X_{3}} (d) .$ In addition, $X_{1} = ⋃_{x \in X_{1}} N_{C} (x)$ and $X_{3} = ⋃_{x \in X_{3}} N_{C} (x)$ .

According to the definition of inclusion degree, it is easy to see the inclusion degree of x to X with respect to C is determined by C. In fact, some properties of C can also be explored by inclusion degree.

Proposition 3.5. Let C = {K₁, K₂, …, K_s} be a covering of U. C is a partition of U if and only if, for any x ∈ U, there exists an 1 ≤ i ≤ s such that $σ_{C}^{K_{i}} (x) = 1$ and $σ_{C}^{K_{j}} (x) = 0$ , j = 1, 2, …, i - 1, i + 1, … s.

Proof. (⇒): If C is a partition of U, then there exists K_i (i ∈ {1, 2, …, s}) such that N_C (x) = K_i, and N_C (x) ⋂ K_j = ∅ (j = 1, 2, …, i - 1, i + 1, … s). Therefore $σ_{C}^{K_{i}} (x) = 1, σ_{C}^{K_{j}} (x) = 0$ .

(⇐): If there exist K_i, K_j ∈ C such that K_i⋂ K_j ≠ ∅, then there exists x₁ ∈ K_i ⋂ K_j such that $σ_{C}^{K_{i}} (x_{1}) \neq 0$ and $σ_{C}^{K_{j}} (x_{1}) \neq 0$ . (i ≠ j, i, j ∈ {1, 2, …, s}). It is a contradiction.□

The above proposition shows a necessary and sufficient condition for covering to be a partition from the viewpoint of inclusion degree.

Example 3.6. (Continued from Example 3.2) By Example 3.2, we have that $\begin{matrix} σ_{C}^{{a, b}} (a) & = 1, σ_{C}^{{a, b}} (b) = 1, σ_{C}^{{a, b}} (c) = 0.5, \\ σ_{C}^{{a, b}} (d) & = 0.5; σ_{C}^{{a, c}} (a) = 1, \\ σ_{C}^{{a, c}} (b) & = 0, σ_{C}^{{a, c}} (c) = 1, σ_{C}^{{a, c}} (d) = 0 . \end{matrix}$ Since $σ_{C}^{{a, b}} (c) = 0.5$ and $σ_{C}^{{a, c}} (c) = 1$ , we have that C is not a partition of U.

Neighborhood is an important concept in covering-based rough sets. That under what condition neighborhoods form a partition is a meaningful issue induced by this concept. Many scholars have paid attention to this issue and presented some necessary and sufficient conditions. In the following proposition, a new necessary and sufficient condition for neighborhoods to form a partition is presented from the viewpoint of inclusion degree.

Proposition 3.7. Let C be a covering of U. Cov (C) is a partition of U if and only if, for any x, y ∈ U, $σ_{C}^{N_{C} (x)} (y) = σ_{C}^{N_{C} (y)} (x) = 1$ or $σ_{C}^{N_{C} (x)} (y) = σ_{C}^{N_{C} (y)} (x) = 0$ .

Proof. (⇒): For all x, y ∈ U, if Cov (C) is a partition of U, then N_C (x) = N_C (y) or N_C (x)⋂ N_C (y) = ∅. If N_C (x) = N_C (y) for any x, y ∈ U, then $σ_{C}^{N_{C} (x)} (y) = \frac{| N_{C} (x) ⋂ N_{C} (y) |}{| N_{C} (y) |} = \frac{| N_{C} (x) ⋂ N_{C} (y) |}{| N_{C} (x) |} = σ_{C}^{N_{C} (y)} (x) = 1$ . If N_C (x)⋂ N_C (y) = ∅, then $σ_{C}^{N_{C} (x)} (y) = σ_{C}^{N_{C} (y)} (x) = 0$ .

(⇐): For any x, y ∈ U, if $σ_{C}^{N_{C} (x)} (y) = σ_{C}^{N_{C} (y)} (x) = 1$ , then N_C (x) = N_C (y); if $σ_{C}^{N_{C} (x)} (y) = σ_{C}^{N_{C} (y)} (x) = 0$ , then N_C (x)⋂ N_C (y) = ∅. Hence, Cov (C) is a partition of U.□

The above propositions and examples show some properties and applications of inclusion degree. In the following subsection, the parametric covering-based rough sets are established from the perspective of inclusion degree. Based on this work, characterizations of the parametric covering-based rough sets are investigated.

3.2 Characterizations of the parametric covering-based rough sets

We firstly define the parametric covering-based rough sets by using inclusion degree.

Definition 3.8. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X \in P (U)$ . Then the parametric covering-based lower and upper approximations of X about CAS with parameter α and β as follows, respectively. $\begin{matrix} {\underline{C}}^{α} (X) & = {x \in U : σ_{C}^{X} (x) \geq α}, \\ {\bar{C}}^{β} (X) & = {x \in U : σ_{C}^{X} (x) > β} . \end{matrix}$

Example 3.9. Let C = {{a, c} , {a, b, c} , {b, c, e} , {b, d, e, f}} be a covering of U = {a, b, c, d, e, f}. Then N_C (a) = {a, c}, N_C (b) = {b}, N_C (c) = {c}, N_C (d) = {b, d, e, f}, N_C (e) = {b, e}, N_C (f) = {b, d, e, f}. Suppose X = {a, b, e, f} and Y = {b, c, d}. Then $σ_{C}^{X} (a) = 0.5, σ_{C}^{X} (b) = 1, σ_{C}^{X} (c) = 0, σ_{C}^{X} (d) = 0.75,$ $σ_{C}^{X} (e) = 1, σ_{C}^{X} (f) = 0.75; σ_{C}^{Y} (a) = 0.5, σ_{C}^{Y} (b) = 1,$ $σ_{C}^{Y} (c) = 1, σ_{C}^{Y} (d) = 0.5, σ_{C}^{Y} (e) = 0.5, σ_{C}^{Y} (f) = 0.5 .$ Suppose β = 0, α = 0.5. Then $\begin{matrix} {\underline{C}}^{0.5} (X) & = {a, b, d, e, f}, \\ {\bar{C}}^{0} (X) & = {a, b, d, e, f}; \\ {\underline{C}}^{0.5} (Y) & = {a, b, c, d, e, f}, \\ {\bar{C}}^{0} (Y) & = {a, b, c, d, e, f} . \end{matrix}$

Suppose β = 0.4, α = 0.75. Then $\begin{matrix} {\underline{C}}^{0.75} (X) & = {b, d, e, f}, \\ {\bar{C}}^{0.4} (X) & = {a, b, d, e, f}; \\ {\underline{C}}^{0.75} (Y) & = {b, c}, {\bar{C}}^{0.4} (Y) = {a, b, c, d, e, f} . \end{matrix}$

For any 0 ≤ β < α ≤ 1, ${\underline{C}}^{α} (\emptyset) = \emptyset$ , ${\bar{C}}^{β} (\emptyset) = \emptyset$ ; ${\underline{C}}^{α} (U) = U$ , ${\bar{C}}^{β} (U) = U$ .

The above definition shows a new type covering-based rough sets with parameter α and β. In fact, this type of covering-based rough set model focuses on the approximation operations of X ⊆ U in CAS from the viewpoint of quantity. Specifically, this covering-based rough set model is decided by two parameters, i.e., 0 ≤ β < α ≤ 1. As we all known, rough set can approximately characterize an arbitrary subset of a universe by using two definable subsets called lower and upper approximation operators. Furthermore, we study some properties of this type of covering-based rough sets. In the first, the positive region, boundary region and negative region of $X \in P (U)$ in CAS = (U, C) with parameter α and β could be given as follows, respectively. $\begin{matrix} pos (X, α) & = {x \in U : σ_{C}^{X} (x) \geq α}, \\ bn (X, α, β) & = {x \in U : β < σ_{C}^{X} (x) < α}, \\ neg (X, β) & = {x \in U : σ_{C}^{X} (x) \leq β} \\ = U - {\bar{C}}^{β} (X) . \end{matrix}$

If ${\underline{C}}^{α} (X) = {\bar{C}}^{β} (X)$ , then X is called a definable set in covering approximation space CAS = (U, C). Otherwise, X is called a parametric covering-based rough set. Meanwhile, the relationships among covering lower and upper approximations, positive region, boundary region and negative region of X in CAS = (U, C) are obtained.

Proposition 3.10. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X \in P (U)$ , the following relationships hold. $\begin{matrix} {\bar{C}}^{β} (X) & = pos (X, α) ⋃ bn (X, α, β), or \\ bn (X, α, β) & = {\bar{C}}^{β} (X) - pos (X, α) . \end{matrix}$

Example 3.11. (Continued from Example 3.9) We have pos (X, 0.5) = {a, b, d, e, f}, bn (X, 0.5, 0) =∅, neg (X, 0) = {c}; pos (Y, 0.5) = {a, b, c, d, e, f}, bn (Y, 0.5, 0) =∅, neg (Y, 0) =∅; pos (X, 0.75) = {b, d, e, f}, bn (X, 0.75, 0.4) = {a}, neg (X, 0.4) = {c}; pos (Y, 0.75) = {b, c}, bn (Y, 0.75, 0.4) = {a, d, e, f}, neg (Y, 0.4) =∅.

Like many other types of covering-based rough sets on CAS = (U, C), we can also present the basic properties of covering approximation operators by the constructed method.

Proposition 3.12. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X, Y \in P (U)$ , the parametric covering-based lower approximation operator and upper approximation operator satisfies the following properties.

(1) ${\underline{C}}^{α} (\emptyset) = {\bar{C}}^{β} (\emptyset) = \emptyset$ , ${\underline{C}}^{α} (U) = {\bar{C}}^{β} (U) = U$ ,∥(2) ${\underline{C}}^{α} (X) = \sim {\bar{C}}^{(1 - α)} (\sim X)$ , ${\bar{C}}^{β} (X) = \sim {\underline{C}}^{(1 - β)} (\sim X)$ ,∥(3) ${\underline{C}}^{α} (X ⋂ Y) \subseteq {\underline{C}}^{α} (X) ⋂ {\underline{C}}^{α} (Y)$ , ${\bar{C}}^{β} (X ⋂ Y) \subseteq {\bar{C}}^{β} (X) ⋂ {\bar{C}}^{β} (Y)$ ,∥(4) ${\underline{C}}^{α} (X ⋃ Y) \supseteq {\underline{C}}^{α} (X) ⋃ {\underline{C}}^{α} (Y)$ , ${\bar{C}}^{β} (X ⋃ Y) \supseteq {\bar{C}}^{β} (X) ⋃ {\bar{C}}^{β} (Y)$ ,∥(5) If X ⊆ Y, then ${\underline{C}}^{α} (X) \subseteq {\underline{C}}^{α} (Y)$ , ${\bar{C}}^{β} (X) \subseteq {\bar{C}}^{β} (Y)$ ,∥(6) If α₁ ≤ α₂, β₁ ≤ β₂, then ${\underline{C}}^{α_{2}} (X) \subseteq {\underline{C}}^{α_{1}} (X)$ , ${\bar{C}}^{β_{2}} (X) \subseteq {\bar{C}}^{β_{1}} (X)$ ,∥(7) If $α \leq \min {\frac{1}{| N_{C} (x) |} : x \in U}$ , then $X \subseteq {\underline{C}}^{α} (X)$ , $X \subseteq {\bar{C}}^{α} (X)$ ,∥(8) $bn (X, α, β) = \emptyset \Leftrightarrow {\underline{C}}^{α} (X) = {\bar{C}}^{β} (X)$ .

Proof. It is easy to prove the results by the above definition of the covering lower and upper approximation operators.□

The above proposition shows some important properties of the parametric covering-based rough sets. In fact, it is clear that positive region will increase with parameter α decrease, negative region will increase with parameter α increase and that boundary region will dwindle for the parametric covering-based rough sets. In the following examples we study another properties of the parametric covering-based rough sets.

Example 3.13. (Continued from Example 3.9) We have ${\underline{C}}^{0.5} (X ⋂ Y) = {\underline{C}}^{0.5} ({b}) = {b} \subset {a, b, d, e,$ $f} = {\underline{C}}^{0.5} (X) ⋂ {\underline{C}}^{0.5} (Y)$ , ${\bar{C}}^{0} (X ⋂ Y) = {\bar{C}}^{0} ({b}) = {b, d, e, f} \subset {a, b, d, e, f} = {\bar{C}}^{0} (X) ⋂ {\bar{C}}^{0} (Y)$ . It is easy to see that ${\underline{C}}^{α} (X ⋂ Y) = {\underline{C}}^{α} (X) ⋂ {\underline{C}}^{α} (Y)$ and ${\bar{C}}^{β} (X ⋂ Y) = {\bar{C}}^{β} (X) ⋂ {\bar{C}}^{β} (Y)$ are not hold for any $0 \leq β < α \leq 1, X, Y \in P (U)$ .

Example 3.14. (Continued from Example 3.9) Suppose X = {b} and Y = {e}. Then ${\underline{C}}^{0.5} (X) = {b, e}$ , ${\bar{C}}^{0.25} (X) = {b, e}$ ; ${\underline{C}}^{0.5} (Y) = {e}$ , ${\bar{C}}^{0.25} (Y) = {e}$ ; ${\underline{C}}^{0.5} (X ⋃ Y) = {b, d, e, f}$ , ${\bar{C}}^{0.25} (X ⋃ Y) = {b, d, e, f}$ . It is easy to see ${\underline{C}}^{0.5} (X) ⋃ {\underline{C}}^{0.5} (Y) \subset {\underline{C}}^{0.5} (X ⋃ Y)$ , ${\bar{C}}^{0.25} (X) ⋃ {\bar{C}}^{0.25} (Y) \subset {\bar{C}}^{0.25} (X ⋃ Y)$ . Therefore ${\underline{C}}^{α} (X ⋃ Y) = {\underline{C}}^{α} (X) ⋃ {\underline{C}}^{α} (Y)$ and ${\bar{C}}^{β} (X ⋃ Y) = {\bar{C}}^{β} (X) ⋃ {\bar{C}}^{β} (Y)$ are not hold for any $0 \leq β < α \leq 1, X, Y \in P (U)$ .

In following, we give the roughness and precision of the parametric covering-based rough sets.

Definition 3.15. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X \in P (U)$ , we define the roughness ρ_C (X, α, β) and precision η_C (X, α, β) of X according to the parameter α, β about the covering approximation space CAS as follows: $η_{C} (X, α, β) = \frac{| {\underline{C}}^{α} (X) |}{| {\bar{C}}^{β} (X) |}$ , $ρ_{C} (X, α, β) = 1 - \frac{| {\underline{C}}^{α} (X) |}{| {\bar{C}}^{β} (X) |}$ .

Example 3.16. (Continued from Example 3.9) We have η_C (X, 0.5, 0) =1, ρ_C (X, 0.5, 0) =0; η_C (X, 0.75, 0.4) =0.8, ρ_C (X, 0.75, 0.4) =0.2; η_C (Y, 0.5, 0) =1, ρ_C (X, 0.5, 0) =0; $η_{C} (Y, 0.75, 0.4) = \frac{1}{3}, ρ_{C} (X, 0.75, 0.4) = \frac{2}{3}$ .

According to the roughness and precision of X, bn (X, α, β) =∅ if and only if η_C (X, α, β) =1, ρ_C (X, α, β) =0. Meanwhile, the following proposition is obviously from the above definition.

Proposition 3.17. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X \in P (U)$ , the roughness ρ_C (X, α, β) and precision η_C (X, α, β) are satisfied the following properties:

(1) 0 ≤ η_C (X, α, β) ≤1, 0 ≤ ρ_C (X, α, β) ≤1,∥(2) If 1 ≥ α₁ ≥ α₂ > β₁ ≥ β₂ ≥ 0, then ρ_C (X, α₁, β_i) ≥ ρ_C (X, α₂, β_i) and ρ_C (X, α_i, β₁) ≤ ρ_C (X, α_i, β₂) , i = 1, 2,∥(3) If 1 ≥ α₁ ≥ α₂ > β₁ ≥ β₂ ≥ 0, then η_C (X, α₁, β_i) ≤ η_C (X, α₂, β_i) and η_C (X, α_i, β₁) ≥ η_C (X, α_i, β₂) , i = 1, 2.

Proof. It is easy to prove this proposition by Definition 3.15 and Proposition 3.12.□

3.3 Interdependency of the lower and upper approximation operations

According to the above definitions and propositions, the parametric covering-based rough sets have been established from the viewpoint of inclusion degree. In this subsection, the conditions for two coverings to generate the same parametric covering-based lower and upper approximation operations are studied. For this purpose, we firstly introduce the key concept defined in [46], namely reduct of a covering.

Definition 3.18. (A reducible element of a covering [46]) Let C be a covering of U and K ∈ C. If K is the union of some sets in C - {K}, we say K is reducible in C, otherwise K is irreducible.

Definition 3.19. (A reducible covering [46]) Let C be a covering of U. If every element in C is irreducible, we say C is irreducible; otherwise C is reducible.

Proposition 3.20. ([46]) Let C be a covering of U. If K is reducible in C, C - {K} is still a covering of U.

Proposition 3.21. ([46]) Let C be a covering of U, K ∈ C, K be reducible in C, and K₁ ∈ C - {K}. K₁ is reducible in C, if and only if it is reducible in C - {K}.

Proposition 3.20 guarantees that after deleting a reducible subset in a covering, it is still a covering, while Proposition 3.21 shows that deleting a reducible subset in a covering will not generate any new reducible elements or make other previously reducible elements irreducible. Consequently, we can compute the reduct of a covering of a domain by deleting all reducible elements. The remainder still consists of a covering of the domain and is irreducible.

Definition 3.22. (Reduct of a covering [46]) Let C be a covering of U. The family of all irreducible elements of C is called the reduct of C, denoted as reduct(C).

By Proposition 3.21 and the definition of the reduct, it is clear that reduct(C) is unique for a covering C of U.

The following proposition on the condition under which two coverings generate the same parametric covering-based lower approximation operations.

Proposition 3.23. Let C₁ and C₂ be two coverings of U. For any $0 < α \leq 1, X \in P (U)$ , if reduct(C₁) = reduct(C₂), then ${\underline{C_{1}}}^{α} (X) = {\underline{C_{2}}}^{α} (X)$ .

Proof. If reduct(C₁) = reduct(C₂), then Cov (C₁) = Cov (C₂). In fact, if Cov (C₁) ≠ Cov (C₂), then there exists x ∈ U such that N_{C
₁} (x) ≠ N_{C
₂} (x), i.e., ⋂ {x ∈ K : K ∈ C₁} ≠ ⋂ {x ∈ K : K ∈ C₂}. But ⋂ {x ∈ K : K ∈ C₁} = ⋂ {x ∈ K : K ∈ reduct (C₁)} and ⋂ {x ∈ K : K ∈ C₂} = ⋂ {x ∈ K : K ∈ reduct (C₂)}. It is a contradiction. Therefore ${\underline{C_{1}}}^{α} (X) = {\underline{C_{2}}}^{α} (X)$ holds for any $0 < α \leq 1, X \in P (U)$ .□

The above proposition shows a sufficient but not necessary condition under which two coverings generate the same parametric covering-based lower approximation operations. Let us propose an example as follows to expound this proposition.

Example 3.24. Let C₁ = {{a, c} , {a, b, c} , {b, c, e} , {a, b, c, e} , {b, d, e, f}} and C₂ = {{a} , {b} , {c} , {a, b} , {b, c} , {a, e, f} , {b, d, e, f} , {e} , {f}} be two coverings of U = {a, b, c, d, e, f}. Suppose X = {a, b, e, f}. Then $σ_{C_{1}}^{X} (a) = 0.5, σ_{C_{1}}^{X} (b) = 1, σ_{C_{1}}^{X} (c) = 0, σ_{C_{1}}^{X} (d) = 0.75, σ_{C_{1}}^{X} (e) = 1, σ_{C_{1}}^{X} (f) = 0.75$ ; $σ_{C_{2}}^{X} (a) = 1, σ_{C_{2}}^{X} (b) = 1, σ_{C_{2}}^{X} (c) = 0, σ_{C_{2}}^{X} (d) = 0.75, σ_{C_{2}}^{X} (e) = 1, σ_{C_{2}}^{X} (f) = 1$ and ${\underline{C_{1}}}^{0.5} (X) = {\underline{C_{2}}}^{0.5} (X) = {a, b, d, e, f}$ . But reduct (C₁) = {{a, c} , {a, b, c} , {b, c, e} , {b, d, e, f}} ≠ {{a} , {b} , {c} , {b, d, e, f} , {e} , {f}} = reduct (C₂).

For parametric covering-based upper approximation operations in the parametric covering-based rough sets, we are still faced with the question of when two coverings will generate the same upper approximation operation. In the following discussions, the condition under which two coverings generate the same parametric covering-based lower approximation operations is explored.

Proposition 3.25. Let C₁ and C₂ be two coverings of U. For any $0 \leq β < 1, X \in P (U)$ , if reduct(C₁) = reduct(C₂), then ${\bar{C_{1}}}^{β} (X) = {\bar{C_{2}}}^{β} (X)$ .

Proof. It is easy to prove this proposition by Propositions 3.12 and 3.23.□

Same as Proposition 3.23, the above proposition shows a sufficient but not necessary condition under which two coverings generate the same parametric covering-based upper approximation operations.

Example 3.26. (Continued from Example 3.24) Let C₁, C₂ be two coverings in Example 3.24 and X = {a, b, e, f}. Then ${\bar{C_{1}}}^{0.4} (X) = {\bar{C_{2}}}^{0.4} (X) = {a, b, d, e, f}$ . But $\begin{matrix} reduct (C_{1}) = \\ {{a, c}, {a, b, c}, {b, c, e}, {b, d, e, f}} \neq \\ {{a}, {b}, {c}, {b, d, e, f}, {e}, {f}} = reduct (C_{2}) . \end{matrix}$

3.4 Topological properties of parametric covering-based rough sets

In this subsection, some topological characterizations of the parametric covering-based rough sets will be investigated. For this purpose, some fundamental definitions and results of topology should be introduced.

Definition 3.27. ([28]) Let U be a non-empty set. A topology on U is a family $T$ of subsets of U which satisfies the three conditions: $\emptyset, U \in T$ , the intersection of any finite members of $T$ is still a member of $T$ , and the union of the members of each subfamily of $T$ is also a member of $T$ . The pair $(U, T)$ , or briefly, U is called a topological space.

The members of the topology $T$ are called open relative to $T$ , and a subset of U is called closed if its complement is open. The closure of a subset A of a topological space $(U, T)$ is the intersection of the family of all closed sets containing A.

Definition 3.28. ([28, 47]) For an operator $cl : P (U) \to P (U)$ , if it satisfies the following rules, then we call it a closure operator on U. For all $X, Y \in P (U)$ , (1) cl (∅) = ∅; (2) X ⊆ cl (X); (3) cl (cl (X)) = cl (X); (4) cl (X ⋃ Y) = cl (X) ⋃ cl (Y).

For an operator $int : P (U) \to P (U)$ , if it satisfies the following rules, then we call it an interior operator on U. For all $X, Y \in P (U)$ ,(1) int (U) = U; (2) int (X) ⊆ X; (3) int (int (X)) = int (X); (4) int (X ⋂ Y) = int (X) ⋂ int (Y).

Based on the above basic definitions, the following propositions can be proposed.

Proposition 3.29. Let CAS = (U, C) be a covering approximation space. Then ${\underline{C}}^{1}$ is an interior operator on U.

Proof. It is easy to see ${\underline{C}}^{1} (U) = U$ . Then we need to prove ${\underline{C}}^{1}$ satisfies three rules of the interior operator. For all $X \in P (U)$ , if $x \in {\underline{C}}^{1} (X)$ , then $σ_{C}^{X} (x) = 1$ . Therefore x ∈ N_C (x) ⊆ X, i.e., ${\underline{C}}^{1} (X) \subseteq X$ . For all $X \in P (U)$ , ${\underline{C}}^{1} ({\underline{C}}^{1} (X)) \subseteq {\underline{C}}^{1} (X)$ is obviously. If there exists $x \in {\underline{C}}^{1} (X)$ such that $x \notin {\underline{C}}^{1} ({\underline{C}}^{1} (X))$ , then there exists y ∈ N_C (x) such that $y \notin {\underline{C}}^{1} (X)$ . It follows from y ∈ N_C (x) that N_C (y) ⊆ N_C (x). Since $y \notin {\underline{C}}^{1} (X)$ , then $\frac{| N_{C} (y) ⋂ X |}{| N_{C} (y) |} < 1$ . Therefore N_C (y) ⊈ X. It is a contradiction to $x \in {\underline{C}}^{1} (X)$ . Then ${\underline{C}}^{1} ({\underline{C}}^{1} (X)) \supseteq {\underline{C}}^{1} (X)$ , i.e., ${\underline{C}}^{1} ({\underline{C}}^{1} (X)) = {\underline{C}}^{1} (X)$ . Finally, for all $X, Y \in P (U)$ , if $x \in {\underline{C}}^{1} (X) ⋂ {\underline{C}}^{1} (Y)$ , then N_C (x) ⊆ X ⋂ Y, i.e., $x \in {\underline{C}}^{1} (X ⋂ Y)$ . Therefore ${\underline{C}}^{1} (X ⋂ Y) = {\underline{C}}^{1} (X) ⋂ {\underline{C}}^{1} (Y)$ holds. To sum up, ${\underline{C}}^{1}$ is an interior operator on U.□

The above proposition shows the parametric covering-based lower approximation operator is an interior operator on U when the parametric α = 1. In the following discussions, we prove that the parametric covering-based upper approximation operator is a closure operator on U under some conditions.

Proposition 3.30. Let CAS = (U, C) be a covering approximation space. Then ${\bar{C}}^{0}$ is a closure operator on U.

Proof. It is easy to prove this proposition by Propositions 3.12, 3.29 and Definition 3.28.□

Based on the above discussions, the following proposition is presented.

Proposition 3.31. Let CAS = (U, C) be a covering approximation space. Then $T_{1} = {X \in P (U) : {\underline{C}}^{1} (X) = X}$ and $T_{2} = {X \in P (U) : {\bar{C}}^{0} (X) = X}$ are two topologies on U.

Proof. It is easy to prove this proposition by Propositions 3.29, 3.30 and Definition 3.27.□

In this section, we mainly establish the parametric covering-based rough sets model and then investigate some properties of the parametric covering-based rough sets. Especially, the conditions under which two coverings generate the same parametric covering-based approximation operations are proposed. In the following section, the matrix representations of parametric covering-based lower and upper approximation operations will be explored.

4 Matrix representations of parametric covering-based lower and upper approximation operations

In this section, we will show that the calculations of the parametric covering-based lower and upper approximations of the parametric covering-based rough sets can be performed through operations of matrices, which makes the calculations much more convenient. For this purpose, some basic definitions and remarks are introduced as follows.

Definition 4.1. (Characteristic function [17]) Let U = {x₁, x₂, …, x_n} and X ⊆ U, we define χ_X (x) : U → {0, 1} as the characteristic function of X, where

$χ_{X} (x) = {\begin{matrix} 1, & x \in X, \\ 0, & x \notin X . \end{matrix}$ And the characteristic vector of X is denoted by χ_X = (χ_X (x₁) , …, χ_X (x_n)) ^T.

Definition 4.2. ([17]) Let A = (a_ik) _n×m and B = (b_kj) _m×l be two matrices. We define C = A · B = (c_ij) _n×l and ∼A = (e_ik) _n×m as follows:

$c_{ij} = \lor_{k = 1}^{m} (a_{ik} \land b_{kj}), i = 1, 2, \dots, n, j = 1, 2, \dots, l$ . e_ik = 1 - a_ik, i = 1, 2, …, n, k = 1, 2, …, m.

The definition of operations between Boolean matrix and Boolean vector is similar to above definition. Here ∧ denotes minimum and ∨ denotes maximum.

Definition 4.3. (Covering incidence matrix [17]) Let C = {K₁, K₂, …, K_s} be a covering of U = {x₁, x₂, …, x_n}. We define M_C = (m_ij) _n×s as follows:

$m_{ij} = {\begin{matrix} 1, & x_{i} \in K_{j}, \\ 0, & x_{i} \notin K_{j} . \end{matrix}$ M_C is called a covering incidence matrix of C. When there is no confusion, we omit C.

Let us illustrate the above definitions by a simple example.

Example 4.4. Let C = {K₁, K₂, K₃} be a covering of U = {a, b, c, d}, where K₁ = {a, b} , K₂ = {a, c} and K₃ = {b, d}. Then

$M_{C} = \begin{array}{l} a \\ b \\ c \\ d \end{array} \begin{array}{l} K_{1} K_{2} K_{3} \\ [\begin{array}{l} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] \end{array} .$

Neighborhood is an important concept in covering-based rough sets. In [17], authors proposed the matrix representation of neighborhood as the following theorem shown.

Theorem 4.5. Let C = {K₁, K₂, …, K_s} be a covering of U = {x₁, x₂, …, x_n}. Then $D = (d_{ij})_{n \times n} = \sim (M_{C} \cdot (\sim M_{C}^{T}))$ , where

$d_{ij} = {\begin{matrix} 1, & x_{j} \in N_{C} (x_{i}), \\ 0, & x_{j} \notin N_{C} (x_{i}) . \end{matrix}$

Definition 4.6. (Inclusion matrix) Let C = {K₁, K₂, …, K_s} be a covering of U. For any x ∈ U, we define M_C (x) = (x_i) _1×s as follows:

$x_{i} = {\begin{matrix} 1, & x \in K_{i}, \\ 0, & x \notin K_{i} . \end{matrix}$ M_C (x) is called a inclusion matrix of x. When there is no confusion, we omit C.

Example 4.7. (Continued from Example 4.4) It is easy to have

$\begin{array}{l} M_{C} (a) = \begin{array}{l} K_{1} K_{2} K_{3} \\ [\begin{array}{l} 1 & 1 & 0 \end{array}], \\ K_{1} K_{2} K_{3} \end{array} \\ M_{C} (b) = \begin{array}{l} K_{1} K_{2} K_{3} \\ [\begin{array}{l} 1 & 0 & 1 \end{array}], \\ K_{1} K_{2} K_{3} \end{array} \\ M_{C} (c) = \begin{array}{l} K_{1} K_{2} K_{3} \\ [\begin{array}{l} 0 & 1 & 0 \end{array}], \\ K_{1} K_{2} K_{3} \end{array} \\ M_{C} (d) = \begin{array}{l} K_{1} K_{2} K_{3} \\ [\begin{array}{l} 0 & 0 & 1 \end{array}] . \\ K_{1} K_{2} K_{3} \end{array} \end{array}$

It is easy to find that M_C = (M_C (a) , M_C (b) , M_C (c) , M_C (d)) by the above definitions. In fact, if C is a covering of U = {x₁, x₂, …, x_n}, then M_C = (M_C (x₁) , M_C (x₂) , …, M_C (x_n)).

Proposition 4.8. Let C = {K₁, K₂, …, K_s} be a covering of U = {x₁, x₂, …, x_n}. For all i ∈ {1, 2, …, n},

$χ_{N_{C} (x_{i})} = \sim (M_{C} (x_{i}) \cdot (\sim M_{C}^{T}))^{T}$ .

Proof. It is easy to prove this proposition by Theorem 4.5.□

Example 4.9. (Continued from Example 4.4) We have

$χ_{N_{C} (a)} = \sim (M_{C} (a) \cdot (\sim M_{C}^{T}))^{T} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ ,

$χ_{N_{C} (b)} = \sim (M_{C} (b) \cdot (\sim M_{C}^{T}))^{T} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}]$ ,

$χ_{N_{C} (c)} = \sim (M_{C} (c) \cdot (\sim M_{C}^{T}))^{T} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 0 \end{matrix}]$ , $χ_{N_{C} (d)} = \sim (M_{C} (d) \cdot (\sim M_{C}^{T}))^{T} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 1 \end{matrix}]$ . Hence N_C (a) = {a}, N_C (b) = {b} , N_C (c) = {a, c} , N_C (d) = {b, d}.

The above proposition shows a matrix representation of neighborhood of covering. Based on this representation, the inclusion degree can be represented by matrix approach.

Proposition 4.10. Let C = {K₁, K₂, …, K_s} be a covering of U = {x₁, x₂, …, x_n}. For all X ⊆ U and i ∈ {1, 2, …, n}, $σ_{C}^{X} (x_{i}) = \frac{\sum (\sim (M_{C} (x_{i}) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (x_{i}) \cdot (\sim M_{C}^{T})))}$ , where ∑ ((y₁, y₂, …, y_n) ^T) = y₁ + y₂ + … + y_n.

Proof. It is easy to prove this proposition by Definition 3.1 and Proposition 4.8.□

Example 4.11. (Continued from Example 4.4) Let X = {a, d}. Then $σ_{C}^{X} (a) = \frac{\sum (\sim (M_{C} (a) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (a) \cdot (\sim M_{C}^{T})))} = 1$ , $σ_{C}^{X} (b) = \frac{\sum (\sim (M_{C} (b) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (b) \cdot (\sim M_{C}^{T})))} = 0$ , $σ_{C}^{X} (c) = \frac{\sum (\sim (M_{C} (c) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (c) \cdot (\sim M_{C}^{T})))} = \frac{1}{2}$ , $σ_{C}^{X} (d) = \frac{\sum (\sim (M_{C} (d) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (d) \cdot (\sim M_{C}^{T})))} = \frac{1}{2}$ .

In the following discussions, a matrix representation of the parametric covering lower and upper approximations of X about CAS is presented.

Proposition 4.12. Let CAS = (U, C) be a covering approximation space. For any $0 \leq β < α \leq 1, X \in P (U)$ and x ∈ U. Then

$χ_{{\underline{C}}^{α} (X)} (x) = {\begin{matrix} 1, & \frac{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})))} - α \geq 0; \\ 0, & \frac{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})))} - α < 0 . \end{matrix}$

$χ_{{\bar{C}}^{β} (X)} (x) = {\begin{matrix} 1, & \frac{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})))} - β > 0; \\ 0, & \frac{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})) \land (χ_{X})^{T})}{\sum (\sim (M_{C} (x) \cdot (\sim M_{C}^{T})))} - β \leq 0 . \end{matrix}$

Proof. It is easy to prove this proposition by Definition 3.8 and Proposition 4.10.□

Example 4.13. (Continued from Examples 4.4 and 4.11) We have $χ_{{\bar{C}}^{0.4} (X)} = χ_{{\underline{C}}^{0.5} (X)} = [\begin{matrix} 1 \\ 0 \\ 1 \\ 1 \end{matrix}]$ , $χ_{{\bar{C}}^{0.5} (X)} = χ_{{\underline{C}}^{0.6} (X)} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ . Hence ${\bar{C}}^{0.4} (X) = {a, c, d}$ and ${\bar{C}}^{0.5} (X) = {a}$ .

In this section, we mainly present the matrix representations of parametric covering-based upper and lower approximations.

5 An application of the parametric covering-based rough sets to network

In [13], Ge investigated separations in covering approximation spaces and gave some characterizations of these separations and some relations among these separations. As an application of these results, investigations on network security were converted into investigations on separations in covering approximation spaces by taking covering approximation spaces as mathematical models of networks. In this section, we introduce the application of the new type of covering-based rough sets into network security based on [13]. First, some definitions and lemmas should be introduced.

Definition 5.1. (Network [13]) A network is a pair $(V, B)$ , where $B$ is a family of servers and V is a set of their users, such that each server in $B$ provides its service to some users in V and each user in V accepts some services from some servers in $B$ .

Definition 5.2. (S₁-security [13]) Let $(V, B)$ be a network. S₁-security: For each pair of distinct users x and y in V, there are servers E₁ and E₂ in $B$ providing their services to x and y but not providing their services to y and x, respectively.

Lemma 5.3. ([13]) Let $(V, B)$ be a network. For each server E in $B$ , let V_E be a set of some users in V such that x is a user in V_E if and only if E provides its service to x. Put U is the abstract set of V and K_E is the abstract set of V_E for each $E \in B$ . Then $C = {K_{E} : E \in B}$ is a covering of U, and so (U, C) is a covering approximation space.

Remark 1. Let $(V, B)$ and (U, C) be stated as Lemma 5.3. Then (U, C) is called a covering approximation space induced by $(V, B)$ .

Definition 5.4. (S₁-space [13]) Let (U, C) be a covering approximation space. Then (U, C) is an S₁ - space if and only if N_C (x) = {x} for all x ∈ U.

Lemma 5.2. ([13]) Let $(V, B)$ be a network and (U, C) be a covering approximation space induced by $(V, B)$ . Then $(V, B)$ has S₁-security if and only if (U, C) is an S₁-space.

Now, we give a simple application of the new type of covering-based rough sets into network security.

Proposition 5.6. Let $(V, B)$ be a network and (U, C) be a covering approximation space induced by $(V, B)$ . Then the following are equivalent. (1) $(V, B)$ has S₁-security. (2) $σ_{C}^{{x}} (x) = 1$ for any x ∈ U. (3) ${\underline{C}}^{1} (X) = X$ for any $X \in P (U)$ .

Proof. According to Definition 5.4 and Lemma 5.5, we only need to prove (2), (3) and (4) N_C (x) = {x} for any x ∈ U are equivalent.

(2)⇒ (4): If $σ_{C}^{{x}} (x) = 1$ for any x ∈ U, then N_C (x) ⊆ {x}. Hence N_C (x) = {x} for any x ∈ U. (4)⇒ (2): It is straightforward. Therefore, (2) and (4) are equivalent.

(3)⇔ (4): It is easy to prove by Definitions 3.1, 3.8 and Proposition 3.3.

This completes the proof of this proposition.□

The above proposition shows some sufficient and necessary conditions for a network has S₁-security from the viewpoint of the new type of covering-based rough sets. A successfully application of this new type of covering-based rough sets into network is a course examination network in Soochow University [13].

Example 5.7. A Course Examination Network $(V, B)$ .

Conditions of Network $(V, B)$ .

$(V, B)$ can be applied to a course examination for a group consisting of 30 students in Soochow University.

In order to prevent cheats in a course examination, it is necessary to guarantee that every student can only acquire his/her own exam questions in the network $(V, B)$ . Therefore, the designer of $(V, B)$ designs that every student x must accept services from all servers in $B_{x}$ , where $B_{x} = {E \in B : E$ provides its service to x}. Obviously, it is sufficient if $(V, B)$ has S₁-security.

Establishment of Network $(V, B)$ .

Put V is the set of 30 users: V = {x₁, x₂, …, x₃₀}.

Put $B$ is the set 10 servers: $B = {E_{1}, E_{2}, \dots,$ E₁₀}.

For each $E_{k} \in B (k = 1, 2, \dots, 10)$ , the designer of $(V, B)$ designs V_{E
_k} = {x ∈ V : E_k provides its service to x} as follows. V_{E
₁} = {x₁, x₂, x₃, x₄, x₅, x₆}, V_{E
₂} = {x₆, x₇, x₈, x₉, x₁₀, x₁₁}, V_{E
₃} = {x₁₁, x₁₂, x₁₃, x₁₄, x₁₅, x₁₆}, V_{E
₄} = {x₁₆, x₁₇, x₁₈, x₁₉, x₂₀, x₂₁}, V_{E
₅} = {x₂₁, x₂₂, x₂₃, x₂₄, x₂₅, x₂₆}, V_{E
₆} = {x₂₆, x₂₇, x₂₈, x₂₉, x₃₀, x₁}, V_{E
₇} = {x₂, x₇, x₁₂, x₁₇, x₂₂, x₂₇}, V_{E
₈} = {x₃, x₈, x₁₃, x₁₈, x₂₃, x₂₈}, V_{E
₉} = {x₄, x₉, x₁₄, x₁₉, x₂₄, x₂₉}, V_{E
₁₀} = {x₅, x₁₀, x₁₅, x₂₀, x₂₅, x₃₀}. Thus, a network $(V, B)$ is established.

Conversion of Network $(V, B)$ .

By Lemma 5.3, we convert the network $(V, B)$ to a covering approximation space (U, C) as follows, where (U, C) is induced by $(V, B)$ .

U is the abstract set of V: U = {x₁, x₂, …, x₃₀}.

C is a covering of U: C = {K₁, K₂, …, K₁₀}. K₁ = {x₁, x₂, x₃, x₄, x₅, x₆}, K₂ = {x₆, x₇, x₈, x₉, x₁₀, x₁₁}, K₃ = {x₁₁, x₁₂, x₁₃, x₁₄, x₁₅, x₁₆}, K₄ = {x₁₆, x₁₇, x₁₈, x₁₉, x₂₀, x₂₁}, K₅ = {x₂₁, x₂₂, x₂₃, x₂₄, x₂₅, x₂₆}, K₆ = {x₂₆, x₂₇, x₂₈, x₂₉, x₃₀, x₁}, K₇ = {x₂, x₇, x₁₂, x₁₇, x₂₂, x₂₇}, K₈ = {x₃, x₈, x₁₃, x₁₈, x₂₃, x₂₈}, K₉ = {x₄, x₉, x₁₄, x₁₉, x₂₄, x₂₉}, K₁₀ = {x₅, x₁₀, x₁₅, x₂₀, x₂₅, x₃₀}. Thus, a covering approximation space (U, C) is obtained, which is induced by $(V, B)$ .

Security of Network $(V, B)$ .

It is not difficult to check that $σ_{C}^{{x_{i}}} (x_{i}) = 1$ for each i = 1, 2, …, 30. By Proposition 5.6, (U, C) has S₁-security.

It is not difficult to compute that ${\underline{C}}^{1} (X) = X$ for each $X \in P (U)$ . By Proposition 5.6, (U, C) has S₁-security.

According to the analysis of the above example, it is easy to know that the conclusions are more scientific, reasonable and suitable to the reality when applying the new type of covering-based rough sets to the reality. Therefore, the decision of the reality will be more exact in the practice. Finally, we give a practical example of the parameter covering-based rough set model defined inDefinition 3.8.

Example 5.8. Let U = {x₁, x₂, x₃, x₄, x₅, x₆} be a universe of six associated symptoms, V = {y₁, y₂, y₃, y₄, y₅} be a set of sufferers and C = {C₁, C₂, C₃, C₄, C₅} be a family of subsets of U, where $σ_{C}^{C_{i}} (x_{j})$ denotes the critical value of the corresponding sufferer y_i ∈ V showing the symptom x_j ∈ U. Suppose C₁ = {x₁, x₃} , C₂ = {x₁, x₂, x₃} , C₃ = {x₂, x₃, x₅} , C₄ = {x₁, x₂, x₃, x₅} , C₅ = {x₂, x₄, x₅, x₆}. Then $σ_{C}^{C_{i}} (x_{j}) (i = 1, 2, 3, 4, 5; j = 1, 2, 3, 4, 5, 6)$ can be listed as follows table.

$σ_{C}^{C_{i}} (x_{j})$	C ₁	C ₂	C ₃	C ₄	C ₅
x ₁	1	1	0.5	1	0
x ₂	0	1	1	1	1
x ₃	1	1	1	1	0
x ₄	0	0.25	0.5	0.5	1
x ₅	0	0.5	1	1	1
x ₆	0	0.25	0.5	0.5	1

Suppose that the symptoms in X = {x₁, x₂, x₅, x₆} are all involved in a certain disease A, taking α = 1, β = 0.5 as the critical value, then we can calculate the lower approximation and upper approximation of X as follows. ${\underline{C}}^{1} (X) = {x_{2}, x_{5}}$ and ${\bar{C}}^{0.5} (X) = {x_{2}, x_{4}, x_{5}, x_{6}}$ .

Since $Pos (X, 1, 0.5) = {\underline{C}}^{1} (X) = {x_{2}, x_{5}}$ , and $σ_{C}^{C_{i}} (x_{2}) = σ_{C}^{C_{i}} (x_{5}) = 1 (i = 3, 4, 5)$ , then y₃, y₄ and y₅ must be suffered from the disease A, and should be given treatment immediately. Since Bn (X, 1, 0.5) = {x₄, x₆} and $σ_{C}^{C_{1}} (x_{4}) = σ_{C}^{C_{1}} (x_{6}) = 0$ , then y₁ has nothing to do with the disease A. Since $σ_{C}^{C_{2}} (x_{4}) = σ_{C}^{C_{2}} (x_{6}) = 0.25$ , $σ_{C}^{C_{2}} (x_{2}) = 1$ and $σ_{C}^{C_{2}} (x_{5}) = 0.5$ , then the sufferer y₂ are not necessarily ill with the disease A and should be further considered by the doctor.

6 Conclusions

In this paper, by introducing the concept of inclusion degree, a new type of covering-based rough sets, named parametric covering-based rough sets were proposed, which can not only generalize the theory of covering-based rough sets, but extend the applications of covering-based rough sets to diverse fields. First, the definition of inclusion degree was defined and some of its properties were investigated based on covering approximation space. Moreover, through inclusion degree, some problems of covering-based rough sets can be addressed. Second, the parametric covering-based rough sets were established by means of inclusion degree and the properties of parametric covering-based rough sets were explored. Third, it is found that the calculations of corresponding parametric covering-based lower and upper approximations can be represented by the operations of matrices. Finally, an application of parametric covering-based rough sets to network security was introduced. Though much research has been conducted in this paper, there are still many interesting issues worth studying.

Relationships between parametric covering-based rough sets and other types of covering-based rough sets.

Dependency of lower and upper approximation operations of the parametric covering-based rough sets.

Axiomatic system for approximation operations of the parametric covering-based rough sets.

Matroidal structures of the parametric covering-based rough sets.

In further work, we will conduct more specific research alone the lines of the above four issues, especially axiomatic systems and matroidal properties [21 , 39] for the parametric covering-based rough sets.

Compliance with ethical standards

Conflict of interest: Author Bin Yang declares that he has no conflict of interest.

Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

Footnotes

Acknowledgments

The authors are greatly thankful to anonymous reviewers for sharing their valuable comments that significantly improved the quality of the paper. This research was supported by the National Natural Science Foundation of China [grant No. 61562079] and the Initial Foundation for Scientific Research of Northwest A & F University [grant No. 2452018054].

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