Numerical Characterizations of Topological Reductions of Covering Information Systems in Evidence Theory

1e reductions of covering information systems in terms of covering approximation operators are one of the most important applications of covering rough set theory. Based on the connections between the theory of topology and the covering rough set theory, two kinds of topological reductions of covering information systems are discussed in this paper, which are characterized by the belief and plausibility functions from the evidence theory. 1e topological spaces by two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. 1en, using plausibility function values of the sets in the partitions, the definitions of significance and relative significance of coverings are presented. Hence, topological reduction algorithms based on the evidence theory are proposed in covering information systems, and an example is adopted to illustrate the validity of the algorithms.

e theory of topology has many applications in almost all branches of mathematics and many real life applications. e topological interior and closure operators are two basic definitions in the topological theory. e theory of topology has a close contact with the theory of rough set based on the connections between the topological interior and closure operators and the lower and upper approximation operators, and there exists much result on relationships between topology and covering rough sets [20,[33][34][35][36][37][38][39]. Moreover, topological reductions for three types of covering rough sets in covering information systems have been discussed [40]. e Dempster-Shafer theory of evidence or the theory of belief function [41,42] is an important method to deal with uncertainty in information systems. e plausibility and belief functions construct a dual pair of uncertainty measures in Dempster-Shafer theory of evidence. ere exist strong connections between the Dempster-Shafer theory of evidence and the rough set theory. For example, the relationships between the belief functions and covering rough sets are discussed [23,[43][44][45]. Furthermore, the evidence theory was used to characterize knowledge reductions for covering rough sets in covering information systems [23,[46][47][48]. e purpose of this paper is to characterize two types of topological reductions of covering information systems by evidence theory. In Section 2, we review basic definitions of covering rough sets, topology, and evidence theory. Properties of two pairs of covering approximation operators and the topologies induced by the two pairs of covering approximation operators are also presented. In Section 3, topological reductions in covering information systems are characterized by the belief and plausibility functions from the evidence theory. Using plausibility function values of the sets in the partitions, the definitions of significance and relative significance of coverings are also developed. en, topological reduction algorithms based on the evidence theory are proposed in covering information systems, and an example is adopted to illustrate the validity of the algorithms. In Section 4, we compare a type of topological reduction with a kind of reduction proposed in [23].

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

Basic Concepts in Topology.
In this section, some basic concepts of topological spaces are reviewed. For the other basic topological concepts, we refer to [49].
Definition 1 (see [49]). Let U be a nonempty set. A topology on U is a collection τ of subsets of U having the following properties: (1) ∅ and U are in τ (2) e union of the elements of any subcollection of τ is in τ (3) e intersection of the elements of any finite subcollection of τ is in τ en, (U, τ) is called a topological space, each element in τ is called an open set, and the complement of an open set is called a closed set. In a topological space (U, τ), if A ⊆ U is open in U and if and only if A is closed in U, then (U, τ) is called a pseudo-discrete space.
Definition 2 (see [49]). Let (U, τ) be a topological space and X ∈ P(U). en, the topological interior and closure of X are, respectively, defined by int τ (X) � ∪ G| G is an open set and G ⊆ X , cl τ (X) � ∩ K| K is a closed set and X ⊆ K { }. int τ and cl τ are, respectively, called the topological interior operator and the topological closure operator of τ.
It can be shown that cl τ (X) is a closed set and int τ (X) is an open set in (U, τ). X is an open set in (U, τ) if and only if int τ (X) � X, and X is a closed set in (U, τ) if and only if cl τ (X) � X. e topological interior and closure operators can be also defined by Kuratowski interior and closure axioms.
In an interior space (U, int), it is easy to prove that

Basic Definitions of Covering Rough Sets.
We present definitions of two pairs of covering approximation operators.
Definition 4 (see [7,8,18,24]). Let C be a covering of the universe U. For any x ∈ U, (x) C � ∩ K ∈ C|x ∈ K { } is called a neighborhood of x by C. Define two pairs of covering approximation operators (FL C , FH C ) and (LL C , LH C ) as follows: ∀X ⊆ U, [8,24]. e pair of covering approximation operators (FL C , FH C ) was discussed in [7,18,21,51,52], and the pair of covering approximation operators (LL C , LH C ) was explored in [8,24,53]. For It is easy to obtain that for any x, y ∈ U, x ∈ st(x, C), and (4) x ∈ st(y, C)⇔y ∈ st(x, C).
Some basic properties of the pairs of approximation operators (FL C , FH C ) and (LL C , LH C ) are presented in Proposition 1.
Proposition 1 (see [7,18,53,54]). Let C be a covering of the universe U. en, for any X, Y ⊆ U, we get Mathematical Problems in Engineering According to Definition 3 and Proposition 1, it is easy to get the following.
By Corollary 1(4) and Lemma 1, it is easy to get the following.
□ Corollary 2 (see [40]). Let C be a covering of the universe U.

Basic Notions Related to Evidence eory.
is section will recall some basic definitions about evidence theory.
Definition 5 (see [41,42]). A set function Bel: P(U) ⟶ [0, 1] is referred to as a belief function if (2) For every collection of subsets X 1 , X 2 , . . . , X n ⊆ U, n ≥ 1, where |I| is the cardinality of the set I. A set function Pl: Belief and plausibility functions based on the same belief structure are connected by the dual property Pl(X)

Definition 6.
Let Ω be a sample space and F be a σ-algebra on Ω. en, a real-valued function P: e probability of covering lower approximate operator and covering upper approximate operator are, respectively, belief and plausibility functions.
Proposition 2 (see [44,47]). Let (U, P(U), P) be a probability space and C be a covering on U. For any A ⊆ U, define where P(X) � (|X|/|U|). en, Bel F C and Bel L C are belief functions and Pl F C and Pl L C are plausibility functions.

Evidence-Theory-Based Numerical Characterization of Topological Reductions in Covering Information Systems
A covering on U can be induced from a family of coverings C on U as follows.
Definition 7 (see [55]). Let C be a family of coverings on U.
It is easy to obtain that ∧B is a covering of U. For simplicity, It is clear that for any B � C 1 , C 2 , . . . , C n ⊆ C, X ⊆ U, N). Let C be a family of coverings on U; then, (U, C) is called a covering information system in [23]. Two kinds of reductions of covering information systems are defined as follows.
, then B is called as an L topological consistent set of C. If B is an L topological consistent set of C and no proper subset of B is an L topological consistent set of C, then B is referred to as an L topological reduct of C. e intersection of all L topological reduct of C is called L topological core of C, which is denoted by Core L (C).
We employ Example 1 below to state Definition 8, which is a modified example in [23,24]. Example 1. Consider the problem of evaluating credit card applicants. Let U � x 1 , x 2 , . . . , x 5 be a set of five applicants and E � education; salary; assets be a set of three attributes, where the values of "education" are higher; secondary; primary , the values of "salary" are high; middle; low , and the values of "assets" are high; middle; low . Suppose we have three specialists A, B, C { } to evaluate the attribute values for these applicants, and their evaluation results of the same attribute may not be the same. e results are listed below.
For the attribute "education": For the attribute "salary": For the attribute "assets": Suppose that the weights of the specialists A, B, C { } are equal. To combine the evaluations without losing information, the evaluations provided by each specialist for every attribute value should be union. en, we obtain three coverings from the attribute set E: Let C � C 1 , C 2 , C 3 , and hence (U, C) is a covering information system.
(1) By Definition 7, we obtain We can see that x 1 , x 2 in C 1 ∧C 2 is the set of applicants, whose education is higher and salary is higher. e other sets in C 1 ∧C 2 , C 1 ∧C 3 , C 2 ∧C 3 , ∧C have the same meanings. By Definition 4 and Corollary 1, According to Definition 8, the F topological reduct of C is C 2 , C 3 .
(2) e neighborhoods of elements are presented in Table 1. (x 1 ) C 1 ,C 2 { } contains the elements whose values of "education" and "salary" are the same with x 1 . e other neighborhoods of elements have the same meanings. 5 . According to Definition 8, the L topological reducts of C are C 1 , C 2 and C 2 , C 3 .
(2) It is similar to the proof of (1). □ Theorem 3. Let (U, C) be a covering information system, B ⊆ C.

, and for any nonempty subset
Proof. It is immediately obtained from eorem 2 and Definition 8.

Algorithms for Computing Topological Reducts of a
Covering Information System. To give algorithms for finding topological reducts, we define the significance of a covering in a covering information system. Definition 9. Let (U, C) be a covering information system. Define the F significance of the covering C ∈ C by Define the L significance of the covering C ∈ C by By the definition of F significance of a covering (or L significance of a covering), the Core F (C) (or Core L (C)) can be characterized. Proposition 3. Let (U, C) be a covering information system. en, Proof. For any C ∈ Core F (C), C∖ C { } is not an F topological consistent set. Otherwise, there exists an F topological reduct B ⊆ C∖ C { }. en, C ∉ B, which contradicts the fact C ∈ Core F (C). By eorem 2, m i�1 Pl F By eorem 2, C∖ C { } is not an F topological consistent set, which implies that C∖ C { } is not an F topological reduct. Hence, C belongs to each F topological reduct of C. It follows that C ∈ Core F (C). en, . Similarly, we can get Core L (C) � C ∈ C|Sig L C (C) > 0 . From Proposition 3, the significance of each covering in the core of C is larger than zero. Now we present a concept of the significance of a covering C ∈ C∖B relative to the family of coverings B. □ Definition 10. Let (U, C) be a covering information system, B ⊆ C. Define the F significance of C ∈ C∖B relative to B by Define the L significance of C ∈ C∖B relative to B by Core L (C) � ∅, then go to step (6), else go to step (5); (5) If n j�1 pl L Core L (C) (G j ) � 1, return Core L (C) as a reduct, else go to step (6); , stop and output B as a reduct, else go to step (7).
ALGORITHM 2: Computing the L topological core and L topological reduct of (U, C).
Core F (C) � ∅, then go to step (6), else go to step (5); (5) If m i�1 pl F Core F (C) (K i ) � 1, return Core F (C) as a reduct, else go to step (6); , stop and output B as a reduct, else go to step (7).
ALGORITHM 1: Computing the F topological core and F topological reduct of (U, C).

Mathematical Problems in Engineering
Let m i�1 Pl F ∅ (K i ) � m and n j�1 Pl L ∅ (G j ) � n. e relative significance Sig F B (C) (or Sig L B (C)) can measure importance degree of the covering C relative to B. Now we design algorithms to find an F topological reduct or an L topological reduct of the covering information system (U, C). e time complexity of Algorithms 1 and 2 is O(|C| 3 × |U| 2 ). In the following, an example is given to illustrate the mechanism of Algorithms 1 and 2.
Similarly, we can obtain that C 2 , C 3 is an L topological reduct. erefore, Core L (C) � C 2 , and C 1 , C 2 and C 2 , C 3 are L topological reducts.

Comparing L Topological Reduction with a Kind of Reduction in Covering Information Systems
Chen et al. presented a definition of reduction in a covering information system [23].
Proof. Since B is a reduct, by Definition 11, (x) C � (x) B for all x ∈ U. It follows that for any X ⊆ U, LH C (X) � LH B (X). us, τ(LH C ) � τ(LH B ), which implies that B is an L topological consistent set. (1) F topological reducts of (U, C) are not necessary correlated with the reducts of (U, C). From Example 1 and Example 3, we can know that the F topological reduct of (U, C) is C 2 , C 3 , and the reduct of (U, C) is C 1 , C 2 .
(2) L topological reduct of (U, C) could not be a reduct of (U, C).
For example, C 2 , C 3 is an L topological reduct of the covering information system (U, C) in Example 1. However, C 2 , C 3 is not a reduct of (U, C).

Conclusion
In this paper, the L topological reduction and the F topological reduction of covering information systems have been characterized by the belief and plausibility functions from the evidence theory. e topological spaces by the two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. en, using plausibility function values of sets in the partitions, the definitions of significance and relative significance of coverings in covering information systems have been also developed. Hence, topological reduction algorithms based on the evidence theory have been proposed in covering information systems, and an example has been adopted to illustrate the validity of the algorithms. We have also compared the L topological reduction with a kind of reduction proposed in [23].
Data Availability e underlying data supporting the results of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.