Interpretations of belief functions in approximation operators by covering

Abstract

The rough set theory and the evidence theory are two important methods used to deal with uncertainty. The relationships between the rough set theory and the evidence theory have been discussed. In covering rough set theory, several pairs of covering approximation operators are characterized by belief and plausibility functions. The purpose of this paper is to review and examine interpretations of belief functions in covering approximation operators. Firstly, properties of the belief structures induced by two pairs of covering approximation operators are presented. Then, for a belief structure with the properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions induced by one of the two pairs of covering approximation operators. Moreover, two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators are presented.

Keywords

Belief and plausibility functions Covering approximation operators Covering rough set Evidence theory

1 Introduction

Rough set theory, proposed by Pawlak [13], is a kind of models of granular computing and is used to deal with the vagueness and granularity in information systems. In rough set theory, basic notions are the lower and upper approximation operators. By the lower and upper approximation operators, the uncertainty of knowledge can be characterized.

The Dempster-Shafer theory of evidence is used to model and manipulate uncertain information. The concept of lower and upper probabilities was presented by Dempster in 1967 [17], which was extended by Shafer as a theory in 1976, i.e. the Dempster-Shafer theory of evidence or the theory of belief function. In Dempster-Shafer theory of evidence, there exists a dual pair of uncertainty measures, i.e., the plausibility and belief functions. By the the plausibility and belief functions, the belief and plausibility degrees of a set can be measured.

The theory of rough sets and the Dempster-Shafer theory of evidence are two important methods used to deal with uncertainty. Then it is interesting to investigate connections between the rough set theory and the evidence theory. In fact, there exist many results about the relationships between the belief functions and rough sets, such as the Pawlak rough sets [18–20 , 31], relation-based generalized rough sets [10 , 42], covering-based generalized rough sets [4 , 23], random rough sets [25], multigranulation rough sets [9 , 22], and so on. Moreover, the Dempster-Shafer theory has been generalized to the fuzzy environment by Zadeh based on his work of information granularity and the theory of possibility [34, 35], and its relationships with random rough fuzzy sets, random fuzzy rough sets, random fuzzy t-rough sets and fuzzy rough sets [25 , 37] have been explored. Furthermore, the evidence theory was used to characterize knowledge reduction for rough sets in information systems [3 , 28]. One main result of these studies is that a belief structure can be derived from a rough set algebra. The converse, that every belief functions can be derived from a rough set algebra, is also an important issue. Harmanec et al. referred to the issue that every belief function can be derived from a Pawlak rough set algebra as the completeness of an interpretation [8]. Every belief function can be derived from a general relation rough set algebra was discussed by Yao [31], Wu [25], and Zhang [42].

Since Żakowski first extended the Pawlak rough set model from partition to covering in 1983 [36], the theory of covering rough sets become more and more attractive in the last years. In the covering rough set theory, there have been many results about definitions and properties of covering approximation operators [1 , 43], axiomization of covering approximation operators [33 , 43], relevance among different types of covering approximation operators [11 , 39], connections between the covering rough set theory and evidence theory [4, 5], and so on. On the relationships between covering rough set theory and evidence theory, Chen et al. presented that some pairs of covering approximation operators can be characterized by belief and plausibility functions and some kinds of covering approximation operators do not have this property [4, 5]. However, the converse, that a belief structure and its inducing dual pairs of belief and plausibility functions can be derived from a type of covering rough sets, was not considered in these studies.

The purpose of this paper is to present the interpretations of belief functions in the theory of covering rough sets. In Section 2, we introduce basic definitions of two types of covering rough sets and evidence theory. In Section 3, we discuss relationships between the plausibility and belief functions and the two pairs of covering lower and upper approximation operators. The probabilities of two pairs of covering lower and upper approximations construct two pairs of belief and plausibility functions and their evidence structures. Some same properties of the two belief structures induced by the two pairs of covering approximation operators are presented. Conversely, for a belief structure with some properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions by one of the two pairs of covering approximation operators. Then two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators are presented.

2 Preliminaries

In this section, we introduce some basic definitions of covering rough sets and evidence theory. Throughout this paper, we always assume that the universe of discourse U is a finite and nonempty set unless other statement. The class of all subsets of U will be denoted by $P (U)$ .

2.1 Basic definitions of covering rough sets

We present definitions of two pairs of covering approximation operators.

Definition 1. [24, 44] Let $C$ be a collection of nonempty subsets of the universe U. If $\cup C = U$ , $C$ is said to be a covering of U. For any x ∈ U, $(x)_{C} = \cap {K \in C | x \in K}$ . Define two pairs of covering approximation operators $({XL}_{C}, {XH}_{C})$ and $({IL}_{C}, {IH}_{C})$ as follows: ∀X ⊆ U,

${XL}_{C} (X) = {x \in U | (x)_{C} \subseteq X}$ ,

${XH}_{C} (X) = {x \in U | (x)_{C} \cap X \neq \emptyset}$ ,

${IL}_{C} (X) = {x \in U | \forall u \in U, x \in (u)_{C} \Rightarrow u \in X}$ ,

${IH}_{C} (X) = \cup {(x)_{C} | x \in X}$ .

When the covering is clear, we omit the lowercase $C$ for the approximation operators hereinafter.

The pair of approximation operators (XL, XH) was first defined in [24]. Xu and Wang also introduced the pair of covering approximation operators (XL, XH) and explored the relationship between (XL, XH) and a pair of relation approximation operators [30]. Xu and Zhang introduced (XL, XH) in another form and defined a measure of roughness with (XL, XH) [29]. The pair of approximation operators (XL, XH) is used for the feature selection of covering information systems [6]. The covering upper approximation operator IH was defined by Zhu, who also explored topological properties of IH and gave an axiomatic characterization for it [43]. Chen et al. presented that (XL, XH) and (IL, IH) can be characterized by belief and plausibility functions [4, 5]. (XL, XH) and (IL, IH) are also discussed by many authors [15 , 41].

It is easy to see that for any x, y ∈ U,

(1) $x \in (x)_{C}$ , (2) if $x \in (y)_{C}$ , then $(x)_{C} \subseteq (y)_{C}$ .

2.2 Basic concepts related to evidence theory

This subsection will recall some basic definitions about evidence theory.

Definition 2. [17] Let U be a non-empty finite set. A set function $m : P (U) \to [0, 1]$ is referred to as a basic probability assignment or mass distribution if it satisfies

(M1) m (∅) =0, (M2) ∑_X⊆Um (X) =1 .

A set X ⊆ U with m (X) >0 is referred to as a focal element. Denote $M = {X \subseteq U | m (X) > 0}$ . $(M, m)$ is called a belief structure on U.

A pair of belief function and plausibility function can be defined in the following

Definition 3. [7, 17] A set function $Bel : P (U) \to [0, 1]$ is referred to as a belief function if it satisfies the following axioms:

(1) Bel (∅) =0, Bel (U) =1,

(2) for any X₁, X₂, ⋯ , X_n ⊆ U (n ≥ 1),

$Bel (\cup_{i = 1}^{n} X_{i}) \geq \sum_{\emptyset \neq I \subseteq {1, 2, \dots, n}} (- 1)^{| I | + 1} Bel (\cap_{i \in I} X_{i}) .$

A set function $Pl : P (U) \to [0, 1]$ is referred to as a plausibility function, if

(1) Pl (∅) =0, Pl (U) =1;

(2) for any X₁, X₂, ⋯ , X_n ⊆ U (n ≥ 1),

$Pl (\cap_{i = 1}^{n} X_{i}) \leq \sum_{\emptyset \neq I \subseteq {1, 2, \dots, n}} (- 1)^{| I | + 1} Pl (\cup_{i \in I} X_{i}) .$

Belief function and plausibility function based on the same belief structure are connected by the dual property Pl (X) =1 - Bel (- X). Furthermore, Bel (X) ≤ Pl (X) for all X ⊆ U. The belief and plausibility functions can also be derived based on the belief structure by

Bel (X) = ∑_A⊆Xm (A),

Pl (X) = ∑_A∩X≠∅m (A) , ∀ X ⊆ U .

Conversely, a basic probability assignment m can be represented by its belief function Bel by

m (X) = ∑_A⊆X (-1) ^|X\A|Bel (A), ∀X ⊆ U.

Definition 4. [5] Let Ω be a sample space. $F$ is a σ-algebra on Ω. A real-valued function $P : F \to [0, 1]$ is referred to as a probability on $(Ω, F)$ if it satisfies

(1) for any $X \in F$ , 0 ≤ P (X) ≤1,

(2) P (Ω) =1,

(3) for any $X_{i} \in F$ (i = 1, 2, ⋯), if X_i ∩ X_j = ∅ (i ≠ j), then $P (\cup_{i = 1}^{\infty} X_{i}) = \sum_{i = 1}^{\infty} P (X_{i})$ .

Moreover, $(Ω, F, P)$ is a probability space.

3 Interpretations of belief structures in two types of covering rough sets

In this section, we interpret belief structures with two types of covering rough sets.

3.1 Relationships between the first type of covering rough sets and evidence theory

In [4], the pair of covering approximation operators (XL, XH) is measured by belief and plausibility functions.

Theorem 1. [4] Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. For any X ⊆ U, define

${Bel}_{C} (X) = P (XL (X))$ , ${Pl}_{C} (X) = P (XH (X))$ .

Then ${Bel}_{C}$ and ${Pl}_{C}$ are the belief and plausibility functions, respectively.

In [4], the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ is not be presented. We give the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ in the following

Proposition 1. Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. For any X ⊆ U, define

$j (X) = {x \in U | (x)_{C} = X}$ , $m_{C} (X) = P (j (X))$ .

Then $m_{C} : P (U) \to [0, 1]$ is the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ .

Proof. (1) XL (X) = ∪ _A⊆Xj (A), XH (X) = ∪ _A∩X≠∅j (A).

For any x ∈ XL (X), we get $(x)_{C} \subseteq X$ . Then $x \in j ((x)_{C}) \subseteq \cup_{A \subseteq X} j (A)$ . Hence XL (X) ⊆ ∪ _A⊆Xj (A). Conversely, for any x ∈ ∪ _A⊆Xj (A), there exists an A ⊆ X such that x ∈ j (A). It follows that $(x)_{C} = A$ , which implies that x ∈ XL (X). Thus, ∪_A⊆Xj (A) ⊆ XL (X). We can conclude that XL (X) = ∪ _A⊆Xj (A).

For any x ∈ XH (X), we have that $(x)_{C} \cap X \neq \emptyset$ . Thus, $x \in j ((x)_{C}) \subseteq \cup_{A \cap X \neq \emptyset} j (A)$ . Then XH (X) ⊆ ∪ _A∩X≠∅j (A). Conversely, for any x ∈ ∪ _A∩X≠∅j (A), there exists an A ⊆ U such that A∩ X ≠ ∅ and x ∈ j (A). It implies that $(x)_{C} = A$ . Hence $(x)_{C} \cap X \neq \emptyset$ , which follows that x ∈ XH (X). So ∪_A∩X≠∅j (A) ⊆ XH (X). Therefore, XH (X) = ∪ _A∩X≠∅j (A).

(2) {j (X) ≠ ∅ |X ⊆ U} is a partition of U.

For any x ∈ U, $x \in j ((x)_{C}) \subseteq \cup {j (X) \neq \emptyset | X \subseteq U}$ . Then ∪ {j (X) ≠ ∅ |X ⊆ U} = U.

For any j (X₁) , j (X₂) ∈ {j (X) ≠ ∅ |X ⊆ U} with j (X₁)∩ j (X₂) ≠ ∅, there exists an x ∈ U such that x ∈ j (X₁) ∩ j (X₂). Then $X_{1} = (x)_{C} = X_{2}$ . Hence j (X₁) = j (X₂).

(3) $m_{C} (\emptyset) = 0$ , $Σ_{X \subseteq U} m_{C} (X) = 1$ .

$m_{C} (\emptyset) = P (j (\emptyset)) = P (\emptyset) = 0$ .

$Σ_{X \subseteq U} m_{C} (X) = Σ_{X \subseteq U} P (j (X)) = P (\cup_{X \subseteq U} j (X)) = P (U) = 1$ .

(4) ${Bel}_{C} (X) = \sum_{A \subseteq X} m_{C} (A)$ ,

$Pl C(X) =∑A∩ X≠ ∅ m C(A) . Bel C(X)= P(XL(X))=P(∪A⊆ Xj(A))$ $= \sum_{A \subseteq X} P (j (A)) = \sum_{A \subseteq X} m_{C} (A)$ ,

${Pl}_{C} (X) = P (XH (X)) = P (\cup_{A \cap X \neq \emptyset} j (A))$

$= \sum_{A \cap X \neq \emptyset} P (j (A)) = \sum_{A \cap X \neq \emptyset} m_{C} (A)$ .

As a consequence, $m_{C}$ is the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ . □

In order to present properties of the belief structure of the belief function ${Bel}_{C}$ and plausibility function ${Pl}_{C}$ in Theorem 1, we introduce the following definition.

Definition 5. [1, 43] Let $C$ be a covering of the universe U and $K \in C$ .

(1) If K is a union of some sets in $C - {K}$ , we say K is reducible in $C$ ; otherwise, K is irreducible. If every element in $C$ is irreducible, we say $C$ is irreducible; otherwise, $C$ is reducible.

(2) For any x ∈ K satisfying the condition $\forall G \in C (x \in G \Rightarrow K \subseteq G)$ , x is called a representative element of the set K.

Theorem 2. Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. If P (A) >0 for all A≠ ∅, then the belief structure $(M, m_{C})$ of the belief function ${Bel}_{C} (X) = P (XL (X))$ and plausibility function ${Pl}_{C} (X) = P (XH (X))$ has properties as follows:

(1) $\cup M = U$ ,

(2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B,

(3) $M$ is irreducible,

(4) there exists a surjection $φ : U \to M$ such that for any x ∈ U, x is a representative element of φ (x), that is,

(4a) x ∈ φ (x), (4b) for any $G \in M$ , if x ∈ G, then φ (x) ⊆ G.

Proof. $M = {(x)_{C} | x \in U}$ . In fact, for any x ∈ U, $x \in j ((x)_{C}) \neq \emptyset$ . Then $m_{C} ((x)_{C}) = P (j ((x)_{C})) > 0$ . Therefore, $(x)_{C} \in M$ , which implies that ${(x)_{C} | x \in U} \subseteq M$ . Conversely, for any $X \in M$ , $m_{C} (X) = P (j (X)) > 0$ . It follows that $j (X) = {x | (x)_{C} = X} \neq \emptyset$ . Then, there exists an x₀ ∈ U such that $(x_{0})_{C} = X$ , which implies that $X \in {(x)_{C} | x \in U}$ . Therefore, $M \subseteq {(x)_{C} | x \in U}$ .

(1) For any x ∈ U, $x \in (x)_{C}$ . Then $U \subseteq \cup M = \cup {(x)_{C} | x \in U} \subseteq U$ . It implies that $\cup M = U$ .

(2) For any $A, B \in M$ , there exist y_A,y_B ∈ U such that $A = (y_{A})_{C}$ and $B = (y_{B})_{C}$ . For any $x \in A \cap B = (y_{A})_{C} \cap (y_{B})_{C}$ , we have that $(x)_{C} \subseteq (y_{A})_{C} \cap (y_{B})_{C}$ . Then there exists a $G = (x)_{C} \in M$ such that x ∈ G ⊆ A ∩ B.

(3) If not, there exist x₀, x₁, x₂, ⋯ , x_n ∈ U such that $\cup_{i = 1}^{n} (x_{i})_{C} = (x_{0})_{C}$ and $(x_{i})_{C} \neq (x_{0})_{C} (i = 1, 2, \dots, n)$ . Since $x_{0} \in (x_{0})_{C} = \cup_{i = 1}^{n} (x_{i})_{C}$ . There exists an i₀ ∈ {1, 2, ⋯ , n} such that $x_{0} \in (x_{i_{0}})_{C}$ . Then $(x_{0})_{C} \subseteq (x_{i_{0}})_{C}$ . Due to $(x_{i_{0}})_{C} \subseteq (x_{0})_{C}$ , $(x_{i_{0}})_{C} = (x_{0})_{C}$ , which contradicts the fact $(x_{i_{0}})_{C} \neq (x_{0})_{C}$ .

(4) Define a mapping $φ : U \to M$ as follows: for any x ∈ U, $φ (x) = (x)_{C} \in M$ . Then φ is a surjection. We also have that $x \in (x)_{C} = φ (x)$ for all x ∈ U. Moreover, for any $G \in M$ , there exists a y ∈ U such that $G = (y)_{C}$ . If x ∈ G, then $φ (x) = (x)_{C} \subseteq (y)_{C} = G$ . □

The properties of a belief structure $(M, m)$ have relationships as follows:

Lemma 1. [42] Let $M$ be a family of subsets of U. Then $M$ satisfies the conditions:

(M1) $\cup M = U$ ,

(M2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B,

(M3) $M$ is irreducible,

if and only if there exists a surjection $φ : U \to M$ such that for any x ∈ U, x is a representative element of φ (x), that is,

(a1) x ∈ φ (x),

(a2) for any $G \in M$ , if x ∈ G, then φ (x) ⊆ G.

Proof. Sufficiency. (1) By (a1), for any x ∈ U, we get that $x \in φ (x) \in M$ . Since φ is a surjection, we obtain that $U \subseteq \cup {φ (x) | x \in U} = \cup M$ . Hence, $\cup M = U$ .

(2) For any $A, B \in M$ and x ∈ A ∩ B, by (a2), φ (x) ⊆ A and φ (x) ⊆ B. It follows that φ (x) ⊆ A ∩ B. Then there exists a $φ (x) \in M$ such that x ∈ φ (x) ⊆ A ∩ B.

(3) Suppose that $M$ is reducible. Then there exist $K, K_{1}, K_{2}, \dots, K_{r} \in M$ such that K_i ≠ K (i = 1, 2, ⋯ , r) and $K = \cup_{i = 1}^{r} K_{i}$ . Let x_K be a representative element of K. Hence, there exists an i₀ ∈ {1, 2, ⋯ , r} such that x_K ∈ K_{i
₀}. According to (a1), K ⊆ K_{i
₀}. Then K = K_{i
₀}, which contradicts the fact K ≠ K_{i
₀}.

Necessity. For any x ∈ U, by (M1), there exist sets in $M$ containing x. Denote the family of all the sets in $M$ containing x be $C_{x}$ . Suppose that $C_{x} = {K_{1}, K_{2}, \dots, K_{r}}$ . According to (M2), there exists a $K_{i_{1}} \in C_{x}$ such that x ∈ K_{i
₁} ⊆ K₁ ∩ K₂. Similarly, there exists a $K_{i_{2}} \in C_{x}$ such that x ∈ K_{i
₂} ⊆ K_{i
₁} ∩ K₃. And, there exists a $K_{i_{3}} \in C_{x}$ such that x ∈ K_{i
₃} ⊆ K_{i
₂} ∩ K₄. Likewise, there exists a $K_{i_{r - 1}} \in C_{x}$ such that x ∈ K_{i
_r-1} ⊆ K_{i
_r-2} ∩ K_r. Then we have that K_{i
_r-1} ⊆ K_i for all $K_{i} \in C_{x}$ . Thus, $K_{i_{r - 1}} \subseteq \cap C_{x} \subseteq K_{i_{r - 1}}$ . It implies that x is a representative element of $K_{i_{r - 1}} = \cap C_{x} \in M$ .

Define $φ : U \to M$ as follows: for any x ∈ U, $φ (x) = \cap C_{x}$ . Then, for any x ∈ U, x is a representative element of φ (x).

For any $K \in M$ , there exists an x ∈ K such that $K = \cap C_{x}$ . If not, for any x ∈ K, we have that $K \neq \cap C_{x}$ . Since $\cap C_{x} \in M$ and $\cap C_{x} \subseteq K$ , we obtain that $K = \cup_{x \in K} (\cap C_{x})$ . Then K is reducible, which contradicts the fact that $M$ is irreducible. Consequently, we get that φ a surjection. □

By Proposition 1, we can see that in a probability space $(U, P (U), P)$ , the belief structure $(M, m_{C})$ induced by the covering approximation operators ${XL}_{C}$ , ${XH}_{C}$ satisfies (M1), (M2) and (M3). This conclusion can be showed as follows:

$U \to C {XL}_{C}, {XH}_{C} \to P {Bel}_{C}, {Pl}_{C}$ with $(M, m_{C})$

satisfying (M1), (M2) and (M3).

Naturally, there is a question: for a pair of belief and plausibility functions (Bel, Pl) with belief structure $(M, m)$ satisfying (M1), (M2) and (M3), do there exist a probability P and a covering $C$ such that the belief structure $(M, m_{C})$ of ${Bel}_{C}, {Pl}_{C}$ with $m_{C} = m$ ? The question can be expressed in the following:

Bel, Pl with $(M, m)$ satisfying (M1), (M2) and (M3)

$\to P, C ?$

${Bel}_{C}, {Pl}_{C}$ and $(M, m_{C})$ with ${Bel}_{C} = Bel, {Pl}_{C} = Pl$ and $m_{C} = m$ .

Now we present a necessary and sufficient condition for a belief structure to be the belief structure induced by the covering approximation operators (XL, XH).

Theorem 3. Let m be a mass distribution on U, its belief structure be $(M, m)$ , and belief function and plausibility function be Bel and Pl, respectively. Then the following are equivalent.

(1) $M$ satisfies that (M1) $\cup M = U$ , (M2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B, (M3) $M$ is irreducible.

(2) There exists a surjection $φ : U \to M$ such that x is a representative element of φ (x) for all x ∈ U.

(3) There exist a covering $C$ and a probability $P : P (U) \to [0, 1]$ with P (A) >0 for all A≠ ∅ such that ${Bel}_{C} (X) = P (XL (X)) = Bel (X)$ and ${Pl}_{C} (X) = P (XH (X)) = Pl (X)$ for all X ⊆ U.

Proof. (1) ⇔ (2). It follows from Lemma 1.

(3) ⇒ (2). Since ${Bel}_{C} = Bel$ and ${Pl}_{C} = Pl$ , we have that $M$ is the family of all the focal elements of ${Bel}_{C}$ . Then, according to Theorem 2, we can obtain (2).

(2) ⇒ (3). 1) Suppose that $M = {B_{1}, B_{2}, \dots, B_{n}}$ . Let $C = M$ . Then $C$ is a covering on U. In fact, for any x ∈ U, we have $x \in φ (x) \in M$ . It follows that $U \subseteq \cup_{x \in U} φ (x) = \cup M = \cup C$ . It implies that $\cup C = U$ , that is, $C$ is a covering on U.

2) For any A ⊆ U, define $j (A) = {x \in X | (x)_{C} = A}$ . Then

(a1) j (∅) = ∅,

(a2) $\cup_{i = 1}^{n} j (B_{i}) = U$ ,

(a3) B_i≠ B_j ⇒ j (B_i) ∩ j (B_j) = ∅,

(a4) for any B ⊆ U, if $B \notin M$ , then j (B) =∅,

(a5) for any B ⊆ U, if $B \in M$ , then j (B)≠ ∅.

Now, we present the proof of (a1)-(a5).

(a1) For any x ∈ U, $(x)_{C} \neq \emptyset$ . Then x ∉ j (∅), which follows that j (∅) = ∅.

(a2) For any x ∈ U, $φ (x) \in M = C$ . Assume that $φ (x) = B_{i} \in C$ . Since x ∈ B_i, we obtain that $(x)_{C} \subseteq B_{i}$ . Conversely, since x is a representative element of φ (x) = B_i, if $x \in G \in C$ , then B_i ⊆ G. It follows that $B_{i} \subseteq (x)_{C}$ . Then $(x)_{C} = B_{i}$ . Hence x ∈ j (B_i). Therefore, $U \subseteq \cup_{i = 1}^{n} j (B_{i})$ , which implies that $\cup_{i = 1}^{n} j (B_{i}) = U$ .

(a3) For any x ∈ U, if x ∈ j (B_i), then $(x)_{C} = B_{i} \neq B_{j}$ . Hence x ∉ j (B_j). Similarly, for any x ∈ U, if x ∈ j (B_j), then x ∉ j (B_i). Therefore, j (B_i)∩ j (B_j) = ∅.

(a4) For any $B \notin M$ , suppose that j (B)≠ ∅. Then there exists an x ∈ U such that x ∈ j (B). By (a2), there exists a $B_{i} \in M$ such that x ∈ j (B_i). It follows that $B_{i} = (x)_{C} = B$ , which contradicts the fact $B \notin M$ .

(a5) For any $B \in M$ , suppose that B = B_i (i ∈ {1, 2, ⋯ , n}). Since φ is a surjection, there exists an x ∈ U such that φ (x) = B_i. According to the proof of (a2), $(x)_{C} = B_{i}$ . It means that x∈ j (B_i) ≠ ∅.

3) By (a2) and (a3), {j (B_i) |i = 1, 2, ⋯ , n} is a partition of U. For any x ∈ U, x belongs to one and only one set in {j (B_i) |i = 1, 2, ⋯ , n}, which is denoted by j (B_x). Define a real-value function $P : P (U) \to [0, 1]$ by:

P (∅) =0;

$P ({x}) = \frac{m (B_{x})}{| j (B_{x}) |}$ , for any x ∈ U;

P (A) = ∑_x∈AP ({x}), for any A ⊆ U.

Then P is a probability on U. Initially, for any A ⊆ U, P (A) ≥0. Next, due to (a2), (a3) and (a5), we have

$P (U) = P (\cup_{i = 1}^{n} j (B_{i})) = \sum_{i = 1}^{n} P (j (B_{i})))$

$=∑i=1n∑x∈ j(Bi)P({ x})$ $= \sum_{i = 1}^{n} \sum_{x \in j (B_{i})} \frac{m (B_{i})}{| j (B_{i}) |}$

$=∑i=1nm(Bi)=1 . Finally, for any A,B⊆ U, if A∩ B=∅, then P(A∪ B)=∑x∈ A∪ BP({ x})$ = ∑_x∈AP ({x}) + ∑_x∈BP ({x})

$=P(A)+P(B) . 4) For any B∉ M, according to (a 4), m C(B)=P(j(B))=P(∅)=0=m(B) . For any B∈ M, by (a 5), m C(B)=P(j(B))=∑x∈ j(B)P({ x})$ $= \sum_{x \in j (B)} \frac{m (B)}{| j (B) |} = m (B)$ .

Then, according to Theorem 2, we have that for any X ⊆ U,

${Bel}_{C} (X) = P (XL (X)) = P (\cup_{B \subseteq X} j (B))$

$=∑B⊆ XP(j(B))=∑B⊆ Xm C(B)$ = ∑_B⊆Xm (B) = Bel (X),

${Pl}_{C} (X) = P (XH (X)) = P (\cup_{B \cap X \neq \emptyset} j (B))$

$=∑B∩ X≠ ∅ P(j(B))=∑B∩ X≠ ∅ m C(B)$ = ∑_B∩X≠∅m (B) = Pl (X). □

By Theorem 3, we can know that for a belief structure (M, m) satisfying (M1),(M2) and (M3), there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions by the pair of the covering approximation operators (XL, XH). We employ Example 3 next to state the idea.

Example 1. Let U = {a, b, c, d, e, f, g}. A mass distribution m on U is defined as follows:

$m ({a, c}) = \frac{2 - \sqrt{3}}{15}$ , $m ({b}) = \frac{1}{15}$ , $m ({c}) = \frac{1 + \sqrt{3}}{15}$ ,

$m ({c, d, g}) = \frac{3 - \sqrt{2}}{15}$ , $m ({b, e, f, g}) = \frac{1 + \sqrt{2}}{15}$ ,

$m ({g}) = \frac{7}{15}$ , m (A) =0 for all other A ⊆ U.

Then the set of all the focal elements of m is $M = {{a, c}, {b}, {c}, {c, d, g}, {b, e, f, g}, {g}}$ . Clearly, $M$ satisfies the conditions (1a)-(1c) in Theorem 3.

Define a mapping $φ : U \to M$ as follows:

φ (a) = {a, c} , φ (b) = {b} , φ (c) = {c},

φ (d) = {c, d, g} , φ (e) = φ (f) = {b, e, f, g} , φ (g) = {g}.

Then φ is a surjection and x is a representative element of φ (x) for all x ∈ U.

Let $C = M$ . We can obtain that $(a)_{C} = {a, c}$ , $(b)_{C} = {b}$ , $(c)_{C} = {c}$ , $(d)_{C} = {c, d, g}$ , $(e)_{C} = (f)_{C} = {b, e, f, g}$ , $(g)_{C} = {g}$ . Therefore, j ({a, c}) = {a}, j ({b}) = {b}, j ({c}) = {c}, j ({c, d, g}) = {d}, j ({b, e, f, g}) = {e, f}, j ({g}) = {g}, j (A) =∅ for all other A ⊆ U.

Thus a probability on U is defined as following:

P (∅) = ∅,

$P ({a}) = \frac{m ({a, c})}{| j ({a, c}) |} = \frac{2 - \sqrt{3}}{15}$ ,

$P ({b}) = \frac{m ({b})}{| j ({b}) |} = \frac{1}{15}$ ,

$P ({c}) = \frac{m ({c})}{| j ({c}) |} = \frac{1 + \sqrt{3}}{15}$ ,

$P ({d}) = \frac{m ({c, d, g})}{| j ({c, d, g}) |} = \frac{3 - \sqrt{2}}{15}$ ,

$P ({e}) = P ({f}) = \frac{m ({b, e, f, g})}{| j ({b, e, f, g}) |} = \frac{1 + \sqrt{2}}{30}$ ,

$P ({g}) = \frac{m ({g})}{| j ({g}) |} = \frac{7}{15}$ ,

P (A) = ∑_x∈AP ({x}) for all other A ⊆ U.

Therefore,

$m_{C} ({a, c}) = P (j ({a, c})) = P ({a})$

$= 2- 315=m({ a,c}), m C({ b})=P(j({ b}))=P({ b})= 115=m({ b}), m C({ c})=P(j({ c}))=P({ c})= 1+ 315=m({ c}), m C({ c,d,g})=P(j({ c,d,g}))=P({ d})$ $= \frac{3 - \sqrt{2}}{15} = m ({c, d, g})$ ,

$m_{C} ({a, b}) = P (j ({a, b})) = P ({a})$

$= 1+ 220=m({ a,b}), m C({ b,e,f,g})=P(j({ b,e,f,g}))=P({ e,f})$ $= P ({e}) + P ({f}) = \frac{1 + \sqrt{2}}{30} + \frac{1 + \sqrt{2}}{30}$

$= \frac{1 + \sqrt{2}}{15} = m ({b, e, f, g})$ , $m_{C} ({g}) = P (j ({g})) = P ({g}) = \frac{7}{15} = m ({g})$ , $m_{C} (B) = P (j (B)) = P (\emptyset) = 0 = m (B)$ for all other B ⊆ U.

Then we get that for any X ⊆ U,

${Bel}_{C} (X) = P (XL (X)) = \sum_{B \subseteq X} m_{C} (B)$

$= ∑B⊆ Xm(B)= Bel(X), Pl C(X)=P(XH(X))= ∑B∩ X≠ ∅ m C(B)$ = ∑_B∩X≠∅m (B) = Pl (X).

3.2 Relationships between the second type of covering rough sets and evidence theory

The pair of covering approximation operators (IL, IH) can be measured by belief and plausibility functions.

Theorem 4. [4] Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. For any X ⊆ U, define

${Bel}_{C} (X) = P (IL (X))$ , ${Pl}_{C} (X) = P (IH (X))$ .

Then ${Bel}_{C}$ and ${Pl}_{C}$ are the belief and plausibility functions, respectively.

We present the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ in the following Proposition 2.

Proposition 2. Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. For any x ∈ U, define $n (x) = {u \in U | x \in (u)_{C}}$ . For any X ⊆ U, define

j (X) = {x ∈ U|n (x) = X}, $m_{C} (X) = P (j (X))$ .

Then $m_{C} : P (U) \to [0, 1]$ is the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ .

Proof. (1) IL (X) = ∪ _A⊆Xj (A), IH (X) = ∪ _A∩X≠∅j (A).

For any x ∈ IL (X), any u ∈ U with $x \in (u)_{C}$ , we have u ∈ X. Then $n (x) = {u \in U | x \in (u)_{C}} \subseteq X$ . It follows that x ∈ j (n (x)) ⊆ ∪ _A⊆Xj (A). Hence IL (X) ⊆ ∪ _A⊆Xj (A). Conversely, for any x ∈ ∪ _A⊆Xj (A), there exists an A ⊆ X such that x ∈ j (A). Hence n (x) = A ⊆ X, which implies that x ∈ IL (X). Therefore, ∪_A⊆Xj (A) ⊆ IL (X). We can conclude that IL (X) = ∪ _A⊆Xj (A).

For any x ∈ IH (X), there exists a y ∈ X such that $x \in (y)_{C}$ . Then y ∈ n (x) ∩ X, which means that n (x)∩ X ≠ ∅. It follows that x ∈ j (n (x)) ⊆ ∪ _A∩X≠∅j (A). Thus, IH (X) ⊆ ∪ _A∩X≠∅j (A). Conversely, for any x ∈ ∪ _A∩X≠∅j (A), there exists an A ⊆ U such that A∩ X ≠ ∅ and x ∈ j (A). Hence n (x) = A. Let y ∈ A ∩ X. Since y ∈ A = n (x), we have that $x \in (y)_{C} \subseteq IH (X)$ . It implies that ∪_A∩X≠∅j (A) ⊆ IH (X). Therefore, IH (X) = ∪ _A∩X≠∅j (A).

(2) {j (X) ≠ ∅ |X ⊆ U} is a partition of U.

For any x ∈ U, we obtain that $x \in (x)_{C}$ . It follows that x ∈ j (n (x)) ⊆ ∪ {j (X) ≠ ∅ |X ⊆ U}. Then U ⊆ ∪ {j (X) ≠ ∅ |X ⊆ U}, which follows that ∪ {j (X) ≠ ∅ |X ⊆ U} = U.

Suppose that there exist j (X₁) , j (X₂) ∈ {j (X) ≠ ∅ |X ⊆ U} such that j (X₁)∩ j (X₂) ≠ ∅. Let x ∈ j (X₁) ∩ j (X₂). Then X₁ = n (x) = X₂. Hence, j (X₁) = j (X₂).

(3) $m_{C} (\emptyset) = 0$ , $Σ_{X \subseteq U} m_{C} (X) = 1$ .

$m_{C} (\emptyset) = P (j (\emptyset)) = P (\emptyset) = 0$ .

$Σ_{X \subseteq U} m_{C} (X) = Σ_{X \subseteq U} P (j (X)) = P (\cup_{X \subseteq U} j (X)) = P (U) = 1$ .

(4) ${Bel}_{C} (X) = \sum_{A \subseteq X} m_{C} (A)$ ,

$Pl C(X) =∑A∩ X≠ ∅ m C(A) . Bel C(X)= P(IL(X))=P(∪A⊆ Xj(A))$ $= \sum_{A \subseteq X} P (j (A)) = \sum_{A \subseteq X} m_{C} (A)$ ,

${Pl}_{C} (X) = P (IH (X)) = P (\cup_{A \cap X \neq \emptyset} j (A))$

$= \sum_{A \cap X \neq \emptyset} P (j (A)) = \sum_{A \cap X \neq \emptyset} m_{C} (A)$ .

As a consequence, $m_{C}$ is the mass distribution of ${Bel}_{C}$ and ${Pl}_{C}$ . □

Corollary 1. Let $(U, C)$ be a covering approximation space. For any X ⊆ U, define

${Bel}_{C} (X) = \frac{| IL (X) |}{| U |}$ , ${Pl}_{C} (X) = \frac{| IH (X) |}{| U |}$ .

Then ${Bel}_{C}$ and ${Pl}_{C}$ are the belief and plausibility functions, respectively. The corresponding mass distribution is $m_{C} (X) = \frac{| j (X) |}{| U |}$ .

Proof. Define a set function $P : P (X) \to [0, 1]$ as following: for any X ⊆ U,

$P (X) = \frac{| X |}{| U |}$ .

Then it is clear that P is a probability. By Theorem 4 and Proposition 2, we can get the proposition. □

To explore properties of the belief structure $(M, m_{C})$ of the belief function ${Bel}_{C} (X) = P (IL (X))$ and plausibility function ${Pl}_{C} (X) = P (IH (X))$ , we present properties of the family of sets {n (x) |x ∈ U}.

Proposition 3. Let $(U, C)$ be a covering approximation space. Then

(1) for any x ∈ U, x ∈ n (x),

(2) for any x, y ∈ U, if y ∈ n (x), then n (y) ⊆ n (x),

(3) for any x, y, z ∈ U, if z ∈ n (x) ∩ n (y), then n (z) ⊆ n (x) ∩ n (y),

(4) {n (x) |x ∈ U} is irreducible.

Proof. (1) For any x ∈ U, we deduce that $x \in (x)_{C}$ . Then x ∈ n (x).

(2) For any x, y ∈ U, if y ∈ n (x), we get that $x \in (y)_{C}$ . For any z ∈ n (y), we have that $y \in (z)_{C}$ . Then $x \in (y)_{C} \subseteq (z)_{C}$ . It follows that z ∈ n (x). Hence, n (y) ⊆ n (x).

(3) For any x, y, z ∈ U, if z ∈ n (x) ∩ n (y), by (2), n (z) ⊆ n (x) and n (z) ⊆ n (y). Then n (z) ⊆ n (x) ∩ n (y).

(4) Suppose that there exist x, x₁, x₂, ⋯ , x_n such that n (x_i) ⊊ n (x) (i = 1, ⋯ , n) and $n (x) = \cup_{i = 1}^{n} n (x_{i})$ . It means that $x \in n (x) = \cup_{i = 1}^{n} n (x_{i})$ . Hence, there exists an i₀ ∈ {1, 2, ⋯ , n} such that x ∈ n (x_{i
₀}). By (2), we obtain that n (x) ⊆ n (x_{i
₀}). Since n (x_{i
₀}) ⊆ n (x), we get that n (x_{i
₀}) = n (x), which contradicts the fact n (x_{i
₀}) ≠ n (x). Then we conclude that {n (x) |x ∈ U} is irreducible. □

Corollary 2. Let $(U, P (U), P)$ be a probability space, and $C$ a covering on U. If P (A) >0 for all A≠ ∅, then the belief structure $(M, m_{C})$ of the belief function ${Bel}_{C} (X) = P (IL (X))$ and plausibility function ${Pl}_{C} (X) = P (IH (X))$ has properties as follows:

(1) $\cup M = U$ ,

(2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B,

(3) $M$ is irreducible,

(4) there exists a surjection $φ : U \to M$ such that

(4a) x ∈ φ (x), (4b) for any $G \in M$ , if x ∈ G, then φ (x) ⊆ G.}

Proof. $M = {n (x) | x \in U}$ . In fact, for any x ∈ U, we get that x∈ j (n (x)) ≠ ∅. Then $m_{C} (n (x)) = P (j (n (x))) > 0$ . Hence, $n (x) \in M$ . It follows that ${n (x) | x \in U} \subseteq M$ . Conversely, for any $X \in M$ , $m_{C} (X) = P (j (X)) > 0$ . It follows that j (X) = {x∈ U|n (x) = X} ≠ ∅. Then, there exists an x₀ ∈ U such that n (x₀) = X, which implies that X ∈ {n (x) |x ∈ U}. Therefore, $M \subseteq {(x)_{C} | x \in U}$ .

(1) By Proposition 3(1), $U \subseteq \cup M = \cup {n (x) | x \in U} \subseteq U$ . It implies that $\cup M = U$ .

(2) For any $A, B \in M$ , there exist y_A,y_B ∈ U such that A = n (y_A) and B = n (y_B). For any x ∈ A ∩ B = n (y_A) ∩ n (y_B), by Proposition 3(3), we deduce that n (x) ⊆ n (y_A) ∩ n (y_B). Then there exists a $G = n (x) \in M$ such that x ∈ G ⊆ A ∩ B.

(3) By Proposition 3(4), we have that $M$ is irreducible.

(4) Define a mapping $φ : U \to M$ as following: for any x ∈ U, $φ (x) = n (x) \in M$ . Then φ is a surjection. For any x ∈ U, by Proposition 3(1), x ∈ n (x) = φ (x). For any $G \in M$ , there exists a y ∈ U such that G = n (y). If x ∈ G, then according to Proposition 3(2), we get that φ (x) = n (x) ⊆ n (y) = G. □

Remark 1. By Theorem 2 and Corollary 2, the belief structure $(M, m_{C})$ from (XL, XH) has some same properties with the belief structure $(M, m_{C})$ by (IL, IH).

We present a necessary and sufficient condition for a belief structure to be the belief structure induced by the covering approximation operators (IL, IH) as follows:

Theorem 5. Let m be a mass distribution on U, its belief structure be $(M, m)$ , and belief function and plausibility function be Bel and Pl. Then the following are equivalent.

(1) $M$ satisfies that (M1) $\cup M = U$ , (M2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B, (M3) $M$ is irreducible.

(2) There exists a surjection $φ : U \to M$ such that x is a representative element of φ (x) for all x ∈ U.

(3) There exist a covering $C$ and a probability $P : P (U) \to [0, 1]$ with P (A) >0 for all A≠ ∅ such that ${Bel}_{C} (X) = P (IL (X)) = Bel (X)$ and ${Pl}_{C} (X) = P (IH (X)) = Pl (X)$ for all X ⊆ U.

Proof. (1) ⇔ (2). It follows from Lemma 1.

(3) ⇒ (2). Since ${Bel}_{C} = Bel$ and ${Pl}_{C} = Pl$ , we deduce that $M$ is the family of all the focal elements of ${Bel}_{C}$ . Then, according to Corollary 2, we obtain (2).

(2) ⇒ (3). Suppose that $M = {B_{1}, B_{2}, \dots, B_{n}}$ . For any x ∈ U, define K_x = {u ∈ U|x ∈ φ (u)}. Let $C = {K_{x} | x \in U}$ . Then, $C$ is a covering on U. In fact, for any x ∈ U, we have that $x \in φ (x) \in M$ . Thus, x ∈ K_x. Therefore, $U = \cup_{x \in U} {x} \subseteq \cup_{x \in U} K_{x} = \cup C$ . It implies that $\cup C = U$ , i.e., $C$ is a covering on U.

For any x ∈ U, define $n (x) = {u \in U | x \in (u)_{C}}$ . Hence, n (x) = φ (x) for all x ∈ U. Indeed, for any y ∈ φ (x), by (2), φ (y) ⊆ φ (x). For any $K_{z} \in C$ , if y ∈ K_z, then z ∈ φ (y) ⊆ φ (x). It follows that x ∈ K_z. So $x \in \cap {K_{z} | z \in U, y \in K_{z}} = (y)_{C}$ . It implies that y ∈ n (x). Thus, φ (x) ⊆ n (x). Conversely, for any y ∈ n (x), we have that $x \in (y)_{C}$ . According to (2), y ∈ φ (y). Then $y \in K_{y} \in C$ . It follows that $(y)_{C} \subseteq K_{y}$ . Then, x ∈ K_y, which means that y ∈ φ (x). Therefore, n (x) ⊆ φ (x).

2) For any A ⊆ U, define j (A) = {x ∈ X|n (x) = A}. Then we have that

(a1) j (∅) = ∅,

(a2) $\cup_{i = 1}^{n} j (B_{i}) = U$ ,

(a3) B_i≠ B_j ⇒ j (B_i) ∩ j (B_j) = ∅,

(a4) for any $B \notin M$ , j (B) =∅,

(a5) for any $B \in M$ , j (B)≠ ∅.

Now, we present the proof of (a1)-(a5).

(a1) For any x ∈ U, we get that x∈ φ (x) = n (x) ≠ ∅, then x ∉ j (∅). It follows that j (∅) = ∅.

(a2) For any x ∈ U, assume that $φ (x) = B_{i} \in M$ . Then n (x) = φ (x) = B_i. It implies that x ∈ j (B_i). Then $U \subseteq \cup_{i = 1}^{n} j (B_{i})$ . Therefore, $\cup_{i = 1}^{n} j (B_{i}) = U$ .

(a3) For any x ∈ U, if x ∈ j (B_i), then n (x) = B_i ≠ B_j. Hence x ∉ j (B_j). Similarly, for any x ∈ U, if x ∈ j (B_j), then x ∉ j (B_i). Therefore, j (B_i)∩ j (B_j) = ∅.

(a4) For any $B \notin M$ , suppose that j (B)≠ ∅. Then there exists an x ∈ U such that x ∈ j (B). By (a2), there exists a $B_{i} \in M$ such that x ∈ j (B_i). It follows that B_i = n (x) = B, which contradicts the fact $B \notin M$ .

(a5) For any $B \in M$ , suppose that B = B_{i
₀} (i₀ ∈ {1, 2, ⋯ , n}). By (2), there exists an x ∈ B_{i
₀} such that φ (x) = B_{i
₀}. Then n (x) = B_{i
₀}. Thus, x∈ j (B_{i
₀}) ≠ ∅.

3) By (a2) and (a3), {j (B_i) |i = 1, 2, ⋯ , n} is a partition of U. For any x ∈ U, denote the one and only one set containing x in {j (B_i) |i = 1, 2, ⋯ , n} by j (B_x). Define a real-value function $P : P (U) \to [0, 1]$ by:

P (∅) =0;

$P ({x}) = \frac{m (B_{x})}{| j (B_{x}) |}$ , for any x ∈ U;

P (A) = ∑_x∈AP ({x}), for any A ⊆ U.

Similar to the proof of Theorem 3, P is a probability on U.

4) For any B ⊆ U with $B \notin M$ , according to (a4),

$m_{C} (B) = P (j (B)) = P (\emptyset) = 0 = m (B)$ .

For any B ⊆ U with $B \in M$ , by (a5),

$m_{C} (B) = P (j (B)) = \sum_{x \in j (B)} P ({x})$

$=∑x∈ j(B) m(B)|j(B)|=m(B) . Then, due to Proposition 2, we can conclude that for any X⊆ U, Bel C(X)= P(IL(X))=P(∪B⊆ Xj(B))$ $= \sum_{B \subseteq X} P (j (B)) = \sum_{B \subseteq X} m_{C} (B)$

$=∑B⊆ Xm(B)=Bel(X), Pl C(X)= P(IH(X))=P(∪B∩ X≠ ∅ j(B))$ $= \sum_{B \cap X \neq \emptyset} P (j (B)) = \sum_{B \cap X \neq \emptyset} m_{C} (B)$

= ∑_B∩X≠∅m (B) = Pl (X). □

From Theorem 5, we can see that for a belief structure with some properties, there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions induced by the covering approximations operators IL and IH.

Example 2. Let U = {a, b, c, d, e, f, g, h}. A mass distribution m on U is defined as follows:

$m ({a, g, h}) = \frac{4 - \sqrt{5}}{17}$ , $m ({b, e}) = \frac{1}{17}$ ,

$m ({c, d, f}) = \frac{1 + \sqrt{5}}{17}$ , $m ({d, f}) = \frac{3 - \sqrt{3}}{17}$ ,

$m ({e}) = \frac{1 + \sqrt{3}}{17}$ , $m ({g, h}) = \frac{7}{17}$ ,

m (A) =0 for all other A ⊆ U.

Then the set of all the focal elements of m is $M = {{a, g, h}, {b, e}, {c, d, f}, {d, f}, {e}, {g, h}}$ . Clearly, $M$ satisfies (1) in Theorem 5.

Define a mapping $φ : U \to M$ as:

φ (a) = {a, g, h}, φ (b) = {b, e}, φ (c) = {c, d, f},

φ (d) = φ (f) = {d, f}, φ (e) = {e},

φ (g) = φ (h) = {g, h}.

Then φ is a surjection and x is a representative element of φ (x) for all x ∈ U. We also have K_a = {a}, K_b = {b}, K_c = {c}, K_d = K_f = {c, d, f}, K_e = {b, e}, K_g = K_h = {a, g, h}.

Let $C = {{a}, {b}, {c}, {c, d, f}, {b, e}, {a, g, h}}$ . We can deduce that $(a)_{C} = {a}$ , $(b)_{C} = {b}$ , $(c)_{C} = {c}$ , $(d)_{C} = (f)_{C} = {c, d, f}$ , $(e)_{C} = {b, e}$ , $(g)_{C} = (h)_{C} = {a, g, h}$ . Then n (a) = {a, g, h} = φ (a), n (b) = {b, e} = φ (b), n (c) = {c, d, f} = φ (c), n (d) = n (f) = {d, f} = φ (d) = φ (f), n (e) = {e} = φ (e), n (g) = n (h) = {g, h} = φ (g) = φ (h). Therefore, j ({a, g, h}) = {a}, j ({b, e}) = {b}, j ({c, d, f}) = {c}, j ({d, f}) = {d, f}, j ({e}) = {e}, j ({g, h}) = {g, h}, j (A) =∅ for all other A ⊆ U.

Thus a probability on U is defined as:

P (∅) = ∅,

$P ({a}) = \frac{m ({a, g, h})}{| j ({a, g, h}) |} = \frac{4 - \sqrt{5}}{17}$ ,

$P ({b}) = \frac{m ({b, e})}{| j ({b, e}) |} = \frac{1}{17}$ ,

$P ({c}) = \frac{m ({c, d, f})}{| j ({c, d, f}) |} = \frac{1 + \sqrt{5}}{17}$ ,

$P ({d}) = P ({f}) = \frac{m ({d, f})}{| j ({d, f}) |} = \frac{3 - \sqrt{3}}{34}$ ,

$P ({e}) = \frac{m ({e})}{| j ({e}) |} = \frac{1 + \sqrt{3}}{17}$ ,

$P ({g}) = P ({h}) = \frac{m ({g, h})}{| j ({g, h}) |} = \frac{7}{34}$ ,

P (A) = ∑_x∈AP ({x}) for all other A ⊆ U.

Therefore,

$m_{C} ({a, g, h}) = P (j ({a, g, h})) = P ({a})$

$= 4- 517=m({ a,g,h}), m C({ b,e})=P(j({ b,e}))=P({ b}) = 117=m({ b,e}), m C({ c,d,f})=P(j({ c}))=P({ c})$ $= \frac{1 + \sqrt{5}}{17} = m ({c, d, f})$ ,

$m_{C} ({d, f}) = P (j ({d, f})) = P ({d, f})$

$= 3- 317=m({ d,f}), m C({ e})=P(j({ e}))=P({ e})= 1+ 317=m({ e}), m C({ g,h})=P(j({ g,h}))=P({ g,h})$ $= \frac{1 + \sqrt{7}}{17} = m ({g, h})$ ,

$m_{C} (A) = P (j (A)) = P (\emptyset) = 0 = m (A)$ for all other A ⊆ U.

Then we obtain that for any X ⊆ U,

${Bel}_{C} (X) = \sum_{B \subseteq X} m_{C} (B) = \sum_{B \subseteq X} m (B) = Bel (X)$ ,

${Pl}_{C} (X) = \sum_{B \cap X \neq \emptyset} m_{C} (B) = \sum_{B \cap X \neq \emptyset} m (B) = Pl (X)$ .

By Theorem 3 and Theorem 5, we get

Corollary 3. Let m be a mass distribution on U, its belief structure be $(M, m)$ , and belief function and plausibility function be Bel and Pl. Then the following are equivalence:

(1) $M$ satisfies that (M1) $\cup M = U$ , (M2) for any $A, B \in M$ , any x ∈ A ∩ B, there exists a $G \in M$ such that x ∈ G ⊆ A ∩ B, (M3) $M$ is irreducible.

(2) There exists a surjection $φ : U \to M$ such that x is a representative element of φ (x) for all x ∈ U.

(4) There exist a covering $C$ and a probability $P : P (U) \to [0, 1]$ with P (A) >0 for all A≠ ∅ such that ${Bel}_{C} (X) = P (IL (X)) = Bel (X)$ and ${Pl}_{C} (X) = P (IH (X)) = Pl (X)$ for all X ⊆ U.

4 Conclusion

In this paper, we have explored and established relationships between the plausibility and belief functions in evidence theory and the two pairs of covering lower and upper approximation operators in covering rough set theory. Properties of two belief structures induced by the two pairs of covering approximation operators have been presented. We have obtained that the two belief structures have several same properties. We have also proved that a belief structures with some properties, if and only if there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are the belief and plausibility functions by the covering approximation operators (XL, XH) respectively, if and only if there exists a probability space with a covering such that the belief and plausibility functions defined by the given belief structure are the belief and plausibility functions by the covering approximation operators (IL, IH) respectively. Then two necessary and sufficient conditions for a belief structure to be the belief structure induced by one of the two pairs of covering approximation operators have been presented.

5 Compliance with ethical standards

Funding

This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11701258, 11871259), Natural Science Foundation of Fujian (Nos. 2019J01749, 2020J01801, 2020J02043).

Conflict of interest

The authors declare that there is no conflict of interests.

Ethical approval

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

References

Bonikowski

and Bryniarski

, Wybraniec-Skardowska U Extensions and intentions in the rough set theory, Inf Sci 107 (1998), 149–167.

Bryniarski

, A calculus of rough sets of the first order, Bull Pol Acad Sci 36(16) (1989), 71–77.

Che

and Mi

, Attributes set reduction in multigranulationapproximation space of a multi-source decision information system, Int J Mach Learn Cyber 10 (2019), 2297–2311.

Chen

, Li

, Zhang

and Kwong

, Evidencetheory-basednumerical algorithms of attribute reduction withneighborhood-covering rough sets, Int J Approximate Reason 55 (2014), 908–923.

Chen

, Zhang

and Li

, On measurements of covering rough setsbased on granules and evidence theory, Inf Sci 317(2015), 329–348.

Chen

, Wang

and Hu

, A new approach to attribute reduction ofconsistent and inconsistent covering decision systems with coveringrough sets, Inf Sci 177 (2007), 3500–3518.

Dempster

A.P.

, Upper and lower probabilities induced by amultivalued mapping, Ann Math Sta 38(2) (1967), 325–339.

Harmanec

, Klir

and Resconi

, On modal logic interpretationof Dempster Shafer theory of evidence, Int J lntell Syst 9 (1994), 941–951.

Lin

, Liang

and Qian

, An information fusion approach bycombining multigranulation rough sets and evidence theory, InfSci 314 (2015), 184–199.

10.

Lin

T.Y.

, Heidelberg 18 (1998), 122–140 Granular computing on binary relations II: rough setrepresentations and belief functions, L.olkowski and A.Skowron eds, Rough Sets in Knowledge Discovery: 1. Methodolodgy andApplications Study in Fuzziness and Soft computing, Physica-Verlag.

11.

Liu

, The relationship among different covering approximations, Inf Sci 250 (2013), 178–183.

12.

, Li

, Zhai

and Bai

, Belief and plausibility functionsof type-2 fuzzy rough sets, Int J Approximate Reason 105(2019), 194–216.

13.

Pawlak

, Rough sets, Int J Comput Inf Sci 11 (1982), 341–356.

14.

Pomykala

J.A.

, Approximation operations in approximation space, Bull Pol Acad Sci 35(9-10) (1987), 653–662.

15.

Qin

, Gao

and Pei

, On covering rough sets. In: The Second International Conference on Rough Sets and Knowledge Technology (RSKT 2007), Lecture Notes in Computer Science 4481(2007) (2007), 34–41.

16.

Restrepo

, Cornelis

and Gómez

, Partial order relationfor approximation operators in covering based rough sets, InfSci 284 (2014), 44–59.

17.

Shafer

, A mathematical theory of evidence, Princeton University Press, Princeton.

18.

Skowron

, The relationship between rough settheory and evidence theory, Bull Pol Acad Sci Math 37(1989), 87–90.

19.

Skowron

, The rough sets theory and evidence theory, FundamInform 13 (1990), 245–262.

20.

Skowron

and Grzymala-Busse

, From rough set theory to evidence theory. In: R.R. Yager, M. Fedrizzi, J. Kacprzyk(Eds.), Advance in the Dempster-Shafer Theory of Evidence, Wiley, NewYork, 193–236.

21.

Tan

, Wu

, Li

and Lin

, Evidence-theory-based numericalcharacterization of multigranulation rough sets in incompleteinformation systems, Fuzzy Sets Syst 294 (2016), 18–35.

22.

Tan

, Wu

and Tao

, On the belief structures and reductions ofmultigranulation spaces with decisions, Int J ApproximateReason 88 (2017), 39–52.

23.

Tan

, Wu

and Tao

, A unified framework for characterizingrough sets with evidence theory in various approximation spaces, Inf Sci 454–455 (2018), 144–160.

24.

Wang

, Dai

and Zhou

, Fuzzy covering generalized rough sets, J Zhoukou Teachers College 21(2) (2004), 20–22(in Chinese).

25.

, Leung

and Zhang

, Connections between rough set theoryand Dempster-Shafer theory of evidence, Int J Gen Syst 31 (2002), 405–430.

26.

, Mi

and Li

, Rough approximation spaces and beliefstructures in infinite universes of discourse, J ComputResearch Develop 49(2) (2012), 327–336 (in Chinese).

27.

, Zhang

, Li

and Mi

, Knowledge reduction in randominformation systems via Dempster-Shafer theory of evidence, Inf Sci 174 (2005), 143–164.

28.

, Knowledge reduction in random incomplete decision tables viaevidence theory, Fundam Inform 115 (2012), 203–218.

29.

and Zhang

, Measuring roughness of generalized rough setsinduced by a covering, Fuzzy Sets Syst 158 (2007), 2443–2455.

30.

and Wang

, On the properties of covering rough sets model, J Henan Normal University (Natural Sciences) 33(1) (2005), 130–132 (in Chinese).

31.

Yao

and Lingras

, Interpretations of belief functions in thetheory of rough sets, Inf Sci 104 (1998), 81–106.

32.

Yao

and Yao

, Covering based rough set approximations, InfSci 200 (2012), 91–107.

33.

Yun

, Ge

and Bai

, Axiomatization and conditions forneighborhoods in a covering to form a partition, Inf Sci 181 (2011), 1735–1740.

34.

Zadeh

L.A.

, Fuzzy sets and information granularity. In: M. Gupta, R. Ragade, R. Yager (Eds.), Advances in Fuzzy Set Theory and Application, North Holland, Amsterdam, (1979), 3–18.

35.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Syst 1 (1978), 3–28.

36.

Żakowski

, Approximations in the space (∏), Demonstratio Mathematica 16 (1983), 761–769.

37.

Zhang

, Sun

, Mi

and Feng

, Belief function of Pythagoreanfuzzy rough approximation space and its applications, Int JApproximate Reason 119 (2020), 58–80.

38.

Zhang

, Li

and Li

, On characterizations of a pair ofcovering-based approximation operators, J Soft Comput 23(12) (2019), 3965–3972.

39.

Zhang

, Li

, Lin

and Lin

, Relationships betweengeneralized rough sets based on covering and reflexive neighborhoodsystem, Inf Sci 319 (2015), 56–67.

40.

Zhang

, Li

and Wu

, On axiomatic characterizations of threepairs of covering-based approximation operators, Inf Sci 180 (2010), 174–187.

41.

Zhang

and Luo

, Relationships between coveringbased rough setsand relation-based rough sets, Inf Sci 225 (2013), 55–71.

42.

Zhang

and Li

, Relationships between relationbased rough setsand belief structures, Int J Approximate Reason 127(2020), 83–98.

43.

Zhu

, Topological approaches to covering rough sets, Inf Sci 177 (2007), 1499–1508.

44.

Zhu

and Wang

, Reduction and axiomization of coveringgeneralized rough sets, Inf Sci 152 (2003), 217–230.