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

Authors: Yan-Lan Zhang; Chang-Qing Li
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The 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. The topological spaces by two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. Then, 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.


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## Body

## 1. Introduction

The theory of rough set is a new mathematical method to deal with imprecision, vagueness, and uncertainty in data analysis [1]. The basic notions in the rough set theory are the lower and upper approximation operators. The attribute reduction of datasets in terms of the lower and upper approximation operators is one of most important applications of the rough set theory. The covering rough set is one of the most important extensions of the classical Pawlak rough sets, which has received increasing concern in recent years [2–21]. By using the covering approximation operators, attribute reductions of covering information systems may be unraveled [22–32].The theory of topology has many applications in almost all branches of mathematics and many real life applications. The topological interior and closure operators are two basic definitions in the topological theory. The 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, 33–39]. Moreover, topological reductions for three types of covering rough sets in covering information systems have been discussed [40].The Dempster–Shafer theory of evidence or the theory of belief function [41, 42] is an important method to deal with uncertainty in information systems. The plausibility and belief functions construct a dual pair of uncertainty measures in Dempster–Shafer theory of evidence. There 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, 43–45]. Furthermore, the evidence theory was used to characterize knowledge reductions for covering rough sets in covering information systems [23, 46–48].The purpose of this paper is to characterize two types of topological reductions of covering information systems by evidence theory. InSection 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. Then, 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].


## 2. Preliminaries

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


### 2.1. 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)
The union of the elements of any subcollection ofτ is in τ(3)
The intersection of the elements of any finite subcollection ofτ is in τ
Then,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∈PU. Then, the topological interior and closure of X are, respectively, defined by intτX=∪G|Gis an open set andG⊆X, clτX=∩K|Kis a closed set andX⊆K. intτ and clτ are, respectively, called the topological interior operator and the topological closure operator of τ.
It can be shown thatclτX is a closed set and intτX is an open set in U,τ. X is an open set in U,τ if and only if intτX=X, and X is a closed set in U,τ if and only if clτX=X. The topological interior and closure operators can be also defined by Kuratowski interior and closure axioms.Definition 3.
(see [49, 50]). Let U be a nonempty set, int:PU⟶PU, and cl:PU⟶PU. For any X,Y⊆U,i1intU=U.i2intX⊆X.i3intX∩Y=intX∪intY.i4intintX=intX.c1cl∅=∅.c2X⊆clX.c3clX∪Y=clX∪clY.c4clclX=clX.
Ifint satisfies i1–i3, then int is called an interior operator, and U,int is called an interior space [50]. If int satisfies i1–i4, that is, int satisfies Kuratowski interior axiom, then int is called a topological interior operator [49]. If cl satisfies c1–c3, then cl is called a closure operator, and U,cl is called a closure space [50]. If cl satisfies c1–c4, that is, cl satisfies Kuratowski closure axiom, then cl is called a topological closure operator [49].
In an interior spaceU,int, it is easy to prove that τint=X|intX=X is a topology. In a closure space U,cl, it is easy to verify that τcl=U−X|clX=X is a topology.


### 2.2. 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, xC=∩K∈C|x∈K is called a neighborhood of x by C. Define two pairs of covering approximation operators FLC,FHC and LLC,LHC as follows: ∀X⊆U,FLCX=x|∀K∈Cx∈K⟺K⊆X [7].FHCX=∪K|K∈CK∩X≠∅ [18].LLCX=x|∀y∈U,x∈yC⟺yC⊆X [8].LHCX=∪xC|xC∩X≠∅ [8, 24].
The pair of covering approximation operatorsFLC,FHC was discussed in [7, 18, 21, 51, 52], and the pair of covering approximation operators LLC,LHC was explored in [8, 24, 53].
For anyx∈U, define stx,C=∪K∈C|x∈K. Then, FLCX=x∈U|stx,C⊆X. It is easy to obtain that for any x,y∈U, (1) x∈xC, (2) if x∈yC, then xC⊆yC, (3) x∈stx,C, and (4) x∈sty,C⇔y∈stx,C.
Some basic properties of the pairs of approximation operatorsFLC,FHC and LLC,LHC are presented in Proposition 1.Proposition 1.
(see [7, 18, 53, 54]). Let C be a covering of the universe U. Then, for any X,Y⊆U, we get(1)
FLC∅=FHC∅=∅, FLCU=FHCU=U.(2)
FLCX∩Y=FLCX∩FLCY, FHCX∪Y=FHCX∪FHCY.(3)
FLCX⊆X⊆FHCX.(4)
FLCX=X⇔FHCX=X.(5)
FHCU−X=U−FLCX.(6)
LLC∅=LHC∅=∅, LLCU=LHCU=U.(7)
LLCX∩Y=LLCX∩LLCY, LHCX∪Y=LHCX∪LHCY.(8)
LLCX⊆X⊆LHCX.(9)
LLCX=X⇔LHCX=X.(10)
LHCU−X=U−LLCX.
According to Definition3 and Proposition 1, it is easy to get the following.Corollary 1.
(see [40, 54]). Let C be a covering of the universe U. Then,(1)
FLC and LLC are interior operators.(2)
FHC and LHC are closure operators.(3)
τFHC=X|FHCU−X=U−X=τFLC, τLHC=X|LHCU−X=U−X=τLLC.(4)
U,τFHC and U,τLHC are pseudo-discrete spaces.
By Corollary1, U,τFHC and U,τLHC are pseudo-discrete spaces; then, τFHC=X|FHCX=X and τLHC=X|LHCX=X. There also exists a conclusion about a pseudo-discrete space.Lemma 1.
(see [40]). Let U,τ be a pseudo-discrete space; then, xτ|x∈U is a partition of U.Proof.
Suppose that there existx,y∈U such that xτ∩yτ≠∅ and xτ≠yτ. Then, x∉yτ or y∉xτ. Otherwise, xτ⊆yτ and yτ⊆xτ, and it follows that xτ⊆yτ, which contradicts that xτ≠yτ. Without loss of generality, we assume that x∉yτ. Since U,τ is pseudo-discrete, xτ∩yτ is open and closed. It implies that x∈xτ−xτ∩yτ and xτ−xτ∩yτ is an open and closed set. Since xτ∩yτ≠∅, xτ−xτ∩yτ⊊xτ, which contradicts the definition of xτ. Consequently, xτ|x∈U is a partition of U.
By Corollary1(4) and Lemma 1, it is easy to get the following.Corollary 2.
(see [40]). Let C be a covering of the universe U. Then, xτFHC|x∈U and xτLHC|x∈U are two partitions of U.


### 2.3. Basic Notions Related to Evidence Theory

This section will recall some basic definitions about evidence theory.Definition 5.
(see [41, 42]). A set function Bel:PU⟶0,1 is referred to as a belief function if(1)
Bel∅=0, BelU=1.(2)
For every collection of subsetsX1,X2,…,Xn⊆U, n≥1,(1)Bel∪i=1nXi≥∑∅≠I⊆1,2,…,n−1I+1Bel∩i∈IXi,whereI is the cardinality of the set I. A set function Pl:PU⟶0,1 is referred to as a plausibility function if(1)
Pl∅=0, PlU=1.(2)
For every collection of subsetsX1,X2,…,Xn⊆U, n≥1,(2)Pl∩i=1nXi≤∑∅≠I⊆1,2,…,n−1I+1Pl∪i∈IXi.
Belief and plausibility functions based on the same belief structure are connected by the dual propertyPlX=1−Bel−X. Furthermore, BelX≤PlX for all X⊆U.Definition 6.
LetΩ be a sample space and ℱ be a σ-algebra on Ω. Then, a real-valued function P:ℱ⟶0,1 is referred to as a probability on Ω,ℱ if it satisfies(1)
For anyX∈ℱ, 0≤PX≤1.(2)
PΩ=1.(3)
For anyXi∈ℱi=1,2,…, if Xi∩Xj=∅i≠j, then P∪i=1∞Xi=∑i=1∞PXi.
Moreover,Ω,ℱ,P is a probability space.
The probability of covering lower approximate operator and covering upper approximate operator are, respectively, belief and plausibility functions.Proposition 2 (see [44, 47]).
LetU,PU,P be a probability space and C be a covering on U. For any A⊆U, define(3)BelCFX=PFLCX,PlCFX=PFHCX,BelCLX=PLLCX,PlCLX=PLHCX,where PX=X/U. Then, BelCF and BelCL are belief functions and PlCF and PlCL are plausibility functions.


## 2.1. 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)
The union of the elements of any subcollection ofτ is in τ(3)
The intersection of the elements of any finite subcollection ofτ is in τ
Then,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∈PU. Then, the topological interior and closure of X are, respectively, defined by intτX=∪G|Gis an open set andG⊆X, clτX=∩K|Kis a closed set andX⊆K. intτ and clτ are, respectively, called the topological interior operator and the topological closure operator of τ.
It can be shown thatclτX is a closed set and intτX is an open set in U,τ. X is an open set in U,τ if and only if intτX=X, and X is a closed set in U,τ if and only if clτX=X. The topological interior and closure operators can be also defined by Kuratowski interior and closure axioms.Definition 3.
(see [49, 50]). Let U be a nonempty set, int:PU⟶PU, and cl:PU⟶PU. For any X,Y⊆U,i1intU=U.i2intX⊆X.i3intX∩Y=intX∪intY.i4intintX=intX.c1cl∅=∅.c2X⊆clX.c3clX∪Y=clX∪clY.c4clclX=clX.
Ifint satisfies i1–i3, then int is called an interior operator, and U,int is called an interior space [50]. If int satisfies i1–i4, that is, int satisfies Kuratowski interior axiom, then int is called a topological interior operator [49]. If cl satisfies c1–c3, then cl is called a closure operator, and U,cl is called a closure space [50]. If cl satisfies c1–c4, that is, cl satisfies Kuratowski closure axiom, then cl is called a topological closure operator [49].
In an interior spaceU,int, it is easy to prove that τint=X|intX=X is a topology. In a closure space U,cl, it is easy to verify that τcl=U−X|clX=X is a topology.


## 2.2. 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, xC=∩K∈C|x∈K is called a neighborhood of x by C. Define two pairs of covering approximation operators FLC,FHC and LLC,LHC as follows: ∀X⊆U,FLCX=x|∀K∈Cx∈K⟺K⊆X [7].FHCX=∪K|K∈CK∩X≠∅ [18].LLCX=x|∀y∈U,x∈yC⟺yC⊆X [8].LHCX=∪xC|xC∩X≠∅ [8, 24].
The pair of covering approximation operatorsFLC,FHC was discussed in [7, 18, 21, 51, 52], and the pair of covering approximation operators LLC,LHC was explored in [8, 24, 53].
For anyx∈U, define stx,C=∪K∈C|x∈K. Then, FLCX=x∈U|stx,C⊆X. It is easy to obtain that for any x,y∈U, (1) x∈xC, (2) if x∈yC, then xC⊆yC, (3) x∈stx,C, and (4) x∈sty,C⇔y∈stx,C.
Some basic properties of the pairs of approximation operatorsFLC,FHC and LLC,LHC are presented in Proposition 1.Proposition 1.
(see [7, 18, 53, 54]). Let C be a covering of the universe U. Then, for any X,Y⊆U, we get(1)
FLC∅=FHC∅=∅, FLCU=FHCU=U.(2)
FLCX∩Y=FLCX∩FLCY, FHCX∪Y=FHCX∪FHCY.(3)
FLCX⊆X⊆FHCX.(4)
FLCX=X⇔FHCX=X.(5)
FHCU−X=U−FLCX.(6)
LLC∅=LHC∅=∅, LLCU=LHCU=U.(7)
LLCX∩Y=LLCX∩LLCY, LHCX∪Y=LHCX∪LHCY.(8)
LLCX⊆X⊆LHCX.(9)
LLCX=X⇔LHCX=X.(10)
LHCU−X=U−LLCX.
According to Definition3 and Proposition 1, it is easy to get the following.Corollary 1.
(see [40, 54]). Let C be a covering of the universe U. Then,(1)
FLC and LLC are interior operators.(2)
FHC and LHC are closure operators.(3)
τFHC=X|FHCU−X=U−X=τFLC, τLHC=X|LHCU−X=U−X=τLLC.(4)
U,τFHC and U,τLHC are pseudo-discrete spaces.
By Corollary1, U,τFHC and U,τLHC are pseudo-discrete spaces; then, τFHC=X|FHCX=X and τLHC=X|LHCX=X. There also exists a conclusion about a pseudo-discrete space.Lemma 1.
(see [40]). Let U,τ be a pseudo-discrete space; then, xτ|x∈U is a partition of U.Proof.
Suppose that there existx,y∈U such that xτ∩yτ≠∅ and xτ≠yτ. Then, x∉yτ or y∉xτ. Otherwise, xτ⊆yτ and yτ⊆xτ, and it follows that xτ⊆yτ, which contradicts that xτ≠yτ. Without loss of generality, we assume that x∉yτ. Since U,τ is pseudo-discrete, xτ∩yτ is open and closed. It implies that x∈xτ−xτ∩yτ and xτ−xτ∩yτ is an open and closed set. Since xτ∩yτ≠∅, xτ−xτ∩yτ⊊xτ, which contradicts the definition of xτ. Consequently, xτ|x∈U is a partition of U.
By Corollary1(4) and Lemma 1, it is easy to get the following.Corollary 2.
(see [40]). Let C be a covering of the universe U. Then, xτFHC|x∈U and xτLHC|x∈U are two partitions of U.


## 2.3. Basic Notions Related to Evidence Theory

This section will recall some basic definitions about evidence theory.Definition 5.
(see [41, 42]). A set function Bel:PU⟶0,1 is referred to as a belief function if(1)
Bel∅=0, BelU=1.(2)
For every collection of subsetsX1,X2,…,Xn⊆U, n≥1,(1)Bel∪i=1nXi≥∑∅≠I⊆1,2,…,n−1I+1Bel∩i∈IXi,whereI is the cardinality of the set I. A set function Pl:PU⟶0,1 is referred to as a plausibility function if(1)
Pl∅=0, PlU=1.(2)
For every collection of subsetsX1,X2,…,Xn⊆U, n≥1,(2)Pl∩i=1nXi≤∑∅≠I⊆1,2,…,n−1I+1Pl∪i∈IXi.
Belief and plausibility functions based on the same belief structure are connected by the dual propertyPlX=1−Bel−X. Furthermore, BelX≤PlX for all X⊆U.Definition 6.
LetΩ be a sample space and ℱ be a σ-algebra on Ω. Then, a real-valued function P:ℱ⟶0,1 is referred to as a probability on Ω,ℱ if it satisfies(1)
For anyX∈ℱ, 0≤PX≤1.(2)
PΩ=1.(3)
For anyXi∈ℱi=1,2,…, if Xi∩Xj=∅i≠j, then P∪i=1∞Xi=∑i=1∞PXi.
Moreover,Ω,ℱ,P is a probability space.
The probability of covering lower approximate operator and covering upper approximate operator are, respectively, belief and plausibility functions.Proposition 2 (see [44, 47]).
LetU,PU,P be a probability space and C be a covering on U. For any A⊆U, define(3)BelCFX=PFLCX,PlCFX=PFHCX,BelCLX=PLLCX,PlCLX=PLHCX,where PX=X/U. Then, BelCF and BelCL are belief functions and PlCF and PlCL are plausibility functions.


## 3. Evidence-Theory-Based Numerical Characterization of Topological Reductions in Covering Information Systems

A covering onU 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. For any B=C1,C2,…,Cn⊆C, define ∧B≜C1∧C2∧⋯∧Cn=K1∩K2∩⋯∩Kn≠∅|Ki∈Ci,i=1,2,…,n.
It is easy to obtain that∧B is a covering of U. For simplicity, FL∧B, FH∧B, LL∧B, LH∧B, x∧B, Bel∧BF, Pl∧BF, Bel∧BL, and Pl∧BL are written as FLB, FHB, LLB, LHB, xB, BelBF, PlBF, BelBL, and PlBL, respectively.
It is clear that for anyB=C1,C2,…,Cn⊆C, X⊆U,(1)
stx,∧B=stx,C1∩stx,C2∩⋯∩stx,Cn.(2)
xB=∩C∈BxC.(3)
FLBX⊆FLCX⊆X⊆FHCX⊆FHBX.(4)
LLBX⊆LLCX⊆X⊆LHCX⊆LHBX.Remark 1.
From Corollary2, we can know that xτFHC|x∈U and xτLHC|x∈U are two partitions of U. We denote xτFHC|x∈U=K1,K2,…,Kmm∈N and xτFHC|x∈U=G1,G2,…,Gnn∈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.Definition 8.
(see [40]). Let U,C be a covering information system.(1)
For anyB⊆C, if τFHB=τFHC (or τFLB=τFLC), then B is called as an F topological consistent set of C. If B is an F topological consistent set of C and no proper subset of B is an F topological consistent set of C, then B is referred to as an F topological reduct of C. The intersection of all F topological reduct of C is called F topological core of C, which is denoted by CoreFC.(2)
For anyB⊆C, if τLHB=τLHC (or τLLB=τLLC), 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. The intersection of all L topological reduct of C is called L topological core of C, which is denoted by CoreLC.
We employ Example1 below to state Definition 8, which is a modified example in [23, 24].Example 1.
Consider the problem of evaluating credit card applicants. LetU=x1,x2,…,x5 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. The results are listed below.
For the attribute “education”:A: higher =x1,x2, secondary = x3,x4, low = x5.B: higher =x1, secondary = x2,x3, low = x4,x5.C: higher =x1,x2, secondary = x3, low = x4,x5.
For the attribute “salary”:A: higher =x1,x2, middle = x3,x4, low = x5.B: higher =x1,x2,x3, middle = x4, low = x5.C: higher =x1,x2, middle = x3,x4, low = x5.
For the attribute “assets”:A: higher =x1,x2, middle = x3, low = x4,x5.B: higher =x1,x2, middle = x3,x4, low = x5.C: higher =x1,x2, middle = x3,x4, low = x5.
Suppose that the weights of the specialistsA,B,C are equal. To combine the evaluations without losing information, the evaluations provided by each specialist for every attribute value should be union. Then, we obtain three coverings from the attribute set E:C1=x1,x2,x2,x3,x4,x4,x5.C2=x1,x2,x3,x3,x4,x5.C3=x1,x2,x3,x4,x4,x5.
LetC=C1,C2,C3, and hence U,C is a covering information system.(1)
By Definition7, we obtainC1∧C2=x1,x2,x2,x3,x3,x4,x4,x5.C1∧C3=x1,x2,x2,x3,x4,x4,x4,x5.C2∧C3=x1,x2,x3,x3,x4,x4,x5.∧C=C1∧C2∧C3=x1,x2,x2,x3,x3,x4,x4,x5.We can see thatx1,x2 in C1∧C2 is the set of applicants, whose education is higher and salary is higher. The other sets in C1∧C2, C1∧C3, C2∧C3, ∧C have the same meanings.By Definition4 and Corollary 1, τFHC=∅x1,x2x5,x1,x2,x5,x3,x4,x5,x1,x2,x3,x4U. Thus, x1τFHC=x2τFHC=x1,x2, x3τFHC=x4τFHC=x3,x4, x5τFHC=x5, that is, the partition induced by τFHC is K1=x1,x2,K2=x3,x4,K3=x5. According to Definition 8, the F topological reduct of C is C2,C3.(2)
The neighborhoods of elements are presented in Table1. x1C1,C2 contains the elements whose values of “education” and “salary” are the same with x1. The other neighborhoods of elements have the same meanings.
By Definition4 and Corollary 1, τLHC=∅,x1,x2, x3, x4,x5, x1,x2,x3,x1,x2,x4, x1,x2,x5, x3,x4, x3,x5,x4,x5, x1,x2,x3,x4, x1,x2,x3,x5, x1,x2,x4,x5,x3,x4,x5,U. It follows that x1τLHC=x2τLHC=x1,x2, x3τLHC=x3, x4τLHC=x4, x5τLHC=x5. Thus, the partition induced for τLHC is G1=x1,x2,G2=x3,G3=x4,G4=x5. According to Definition 8, the L topological reducts of C are C1,C2 and C2,C3.Table 1
The neighborhoods of elements in Example1.
xx1x2x3x4x5xC1x1,x2x2x2,x3,x4x4x4,x5xC2x1,x2,x3x1,x2,x3x3x3,x4x5xC3x1,x2x1,x2x3,x4x4x4,x5xC1,C2x1,x2x2x3x4x5xC1,C3x1,x2x2x3,x4x4x4,x5xC2,C3x1,x2x1,x2x3x4x5xCx1,x2x2x3x4x5Theorem 1.
(see [40]). Let U,C be a covering information system, B⊆C.(1)
The following are equivalent:(1a)
B is an F topological consistent set.(1b)
FHBKi=Ki for all i∈1,2,…,m.(1c)
FLBKi=Ki for all i∈1,2,…,m.(2)
The following are equivalent:(2a)
B is an L topological consistent set.(2b)
LHBGj=Gj for all j∈1,2,…,n.(2c)
LLBGj=Gj for all j∈1,2,…,n.Proof.
(1). (1a)⟺ (1b). B is an F topological consistent set; then, τFHB=τFHC. It follows that for any x∈U, xτFHB=xτFHC. By Corollary 1, U,τFHB is pseudo-discrete. Then, xτFHB is a closed set in U,τFHC, which implies that FHBxτFHC=xτFHC. That is, FHBKi=Ki for all i∈1,2,…,m.(1b) ⇔ (1c). By Proposition 1 (4), FHBKi=Ki⇔FLBKi=Ki for all i∈1,2,…,m.(1b) ⟺ (1a). For any A∈τFHB, FHBA=A. By Proposition 1, A⊆FHCA⊆FHBA. It follows that FHCA=A. Then, A∈τFHC. It implies that τFHB⊆τFHC. Conversely, for any A∈τFHC, A=∪x∈AxτFHC. Then, FHBA=FHB∪x∈AxτFHC=∪x∈AFHBxτFHC=∪x∈AxτFHC=A. Thus, A∈τFHB. Consequently, τFHC⊆τFHB.Theorem 2.
LetU,C be a covering information system, B⊆C.(1)
The following are equivalent:(1a)
B is an F topological consistent set.(1b)
∑i=1mPlBFKi=1.(1c)
∑i=1mBelBFKi=1.(2)
The following are equivalent:(2a)
B is an L topological consistent set.(2b)
∑j=1nPlBLGj=1.(2c)
∑j=1NBelBLGj=1.Proof.
(1)
(1a)⇒ (1b). Since B is an F topological consistent set, by Theorem 1, FHBKi=Ki for all i∈1,2,…,m. Then,(4)∑i=1mPlBFKi=∑i=1mPFHBKi=∑i=1mPKi=P∪i=1mKi=PU=1.(1b)⇒ (1c). For any i∈1,2,…,m, Ki⊆FHBKi. According to Definition 6 and Proposition 2,(5)1=∑i=1mPlBFKi=∑i=1mPFHBKi≥∑i=1mPKi=P∪i=1mKi=PU=1.SincePKi≤PFHBKi for all i∈1,2,…,m, we get PKi=PFHBKi, that is, PKi=Ki/U=FHBKi/U=PFHBKi. Then, for any i∈1,2,…,m, Ki=FHBKi. Since Ki⊆FHBKi, Ki=FHBKi. By Proposition 1 (4), Ki=FLBKi. Hence,(6)∑i=1mBelBFKi=∑i=1mPFLBKi=∑i=1mPKi=P∪i=1mKi=PU=1.(1c)⇒ (1a). For any i∈1,2,…,m, FLBKi⊆Ki. By Definition 2 and Proposition 2,(7)1=∑i=1mBelBFKi=∑i=1mPFLBKi≤∑i=1mPKi=P∪i=1mKi=PU=1.SincePFLBKi≤PKi for all i∈1,2,…,m, we get PKi=PFLBKi, that is, PKi=Ki/U=FLBKi/U=PFLBKi. Then, for any i∈1,2,…,m, Ki=FLBKi. It follows from FLBKi⊆Ki that FLBKi=Ki. Therefore, according to Theorem 1, B is an F topological consistent set.(2)
It is similar to the proof of (1).Theorem 3.
LetU,C be a covering information system, B⊆C.(1)
The following are equivalent:(1a)
B is an F topological reduct.(1b)
∑i=1mPlBFKi=1, and for any nonempty subset B′⊂B, ∑i=1mPlB′FKi>1.(1c)
∑i=1mBelBFKi=1, and for any nonempty subset B′⊂B, ∑i=1mBelB′FKi<1.(2)
The following are equivalent:(2a)
B is an L topological reduct.(2b)
∑j=1nPlBLGj=1, and for any nonempty proper subset B′⊂A, ∑j=1nPlB′LGj>1.(2c)
∑j=1nBelBLGj=1, and for any nonempty proper subset B′⊂A, ∑j=1nBelB′LGj<1.Proof.
It is immediately obtained from Theorem2 and Definition 8.


### 3.1. 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.
LetU,C be a covering information system. Define the F significance of the covering C∈C by(8)SigCFC=∑i=1mPlC∖CFKi−∑i=1mPlCFKi.
Define theL significance of the covering C∈C by(9)SigCLC=∑j=1nPlC∖CLGj−∑j=1nPlCLGj.
By the definition ofF significance of a covering (or L significance of a covering), the CoreFC (or CoreLC) can be characterized.Proposition 3.
LetU,C be a covering information system. Then,(10)CoreFC=C∈C|SigCFC>0,CoreLC=C∈C|SigCLC>0.Proof.
For anyC∈CoreFC, C∖C is not an F topological consistent set. Otherwise, there exists an F topological reduct B⊆C∖C. Then, C∉B, which contradicts the fact C∈CoreFC. By Theorem 2, ∑i=1mPlC∖CFKi>1. It follows that SigCFC>0. Therefore, CoreFC⊆C∈C|SigCFC>0. Conversely, for any C∈C∈C|SigCFC>0, SigCFC>0. Hence, SigCFC=∑i=1mPlC∖CFKi−∑i=1mPlCFKi>0. Since ∑i=1mPlCFKi=1, ∑i=1mPlC∖CFKi>1. By Theorem 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∈CoreFC. Then, C∈C|SigCFC>0⊆CoreFC.
Similarly, we can getCoreLC=C∈C|SigCLC>0.
From Proposition3, 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.
LetU,C be a covering information system, B⊆C. Define the F significance of C∈C∖B relative to B by(11)SigBFC=∑i=1mPlBFKi−∑i=1mPlB∪CFKi.
Define theL significance of C∈C∖B relative to B by(12)SigBLC=∑j=1nPlBLGj−∑j=1mPlB∪CLGj.
Let∑i=1mPl∅FKi=m and ∑j=1nPl∅LGj=n. The relative significance SigBFC (or SigBLC) can measure importance degree of the covering C relative to B.
Now we design algorithms to find anF topological reduct or an L topological reduct of the covering information system U,C.
The time complexity of Algorithms1 and 2 is OC3×U2. In the following, an example is given to illustrate the mechanism of Algorithms 1 and 2.Example 2.
Continued from Example1.(1)
ComputingF topological reducts: first, FHC\C1K1=x1,x2, FHC\C1K2=x3,x4, and FHCC1K3=x5. Thus, we get(13)SigCFC1=∑i=13PlC\C1FKi−∑i=13PlCFKi=25+25+15−1=0.ByFHC\C2K1=x1,x2, FHC\C2K2=x3,x4,x5, and FHC\C2K3=x4,x5, we have(14)SigCFC2=∑i=13PlC\C2FKi−∑i=13PlCFKi=25+35+25−1=25>0.SinceFHC\C3K1=x1,x2,x3, FHC\C3K2=x2,x3,x4, and FHC\C3K3=x5, we obtain(15)SigC1C3=∑i=13PlC\C3FKi−∑i=13PlCFKi=35+35+15−1=25>0.Hence,CoreFC=C2,C3.Second, since∑i=13PlCoreFCFKi=1, CoreFC=C2,C3 is an F topological reduct.(2)
ComputingL topological reducts: first, LHC\C1G1=x1,x2, LHC\C1G2=x3, LHC\C1G3=x4, and LHC\C1G4=x5; then,(16)SigCLC1=∑j=14PlC\C1LGj−∑j=14PlCLGj=25+15+15+15−1=0.ByLHC\C2G1=x1,x2, LHC\C2G2=x3,x4, LHC\C2G3=x3,x4,x5, and LHC\C2G4=x4,x5, we obtain(17)SigCLC2=∑j=14PlC\C1LGj−∑i=14PlCLGj=25+25+35+25−1=45>0.SinceLHC\C3G1=x1,x2, LHC\C3G2=x3, LHC\C3G3=x4, and LHC\C3G4=x5, we have(18)SigCLC3=∑j=14PlC\C1LGj−∑i=14PlCLGj=25+15+15+15−1=0.Hence,CoreLC=C2.Second, it follows from∑j=14PlCoreLCLGj=10/5≠1 that CoreLC=C2 is not an L topological reduct.Since(19)SigCoreLCLC1=∑j=14PlC2LGj−∑i=14PlC1,C2LGj=105−1=1,SigCoreLCLC3=∑j=14PlC2LGj−∑i=14PlC2,C3LGj=105−1=1,letB=CoreLC∪C1=C1,C2. We get ∑j=14PlBLGj=1; then, B=C1,C2 is an L topological reduct.
Similarly, we can obtain thatC2,C3 is an L topological reduct.
Therefore,CoreLC=C2, and C1,C2 and C2,C3 are L topological reducts.Algorithm 1: Computing the F topological core and F topological reduct of U,C.
(1)
LetCoreFC=∅, ∑i=1mPl∅FKi=m;(2)
For eachC∈C, calculate SigCFC=∑i=1mplC\CFKi−∑i=1mplCFKi;(3)
IfSigCFC>0, then CoreFC=CoreFC∪C;(4)
IfCoreFC=∅, then go to step (6), else go to step (5);(5)
If∑i=1mplCoreFCFKi=1, return CoreFC as a reduct, else go to step (6);(6)
LetB=CoreFC;(7)
For eachC∈C/B, calculate SigBFC=∑i=1mplBFKi−∑i=1mplB∪CFKi;(8)
IfSigBFC0=maxC∈C\BSigBFC, then B=B∪C0;(9)
If∑i=1mplBFKi=1, 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.
(1)
LetCoreLC=∅, ∑j=1npl∅LGj=n;(2)
For eachC∈C, calculate SigCLC=∑j=1nplCCLGj−∑j=1nplCLGi;(3)
IfSigCLC>0, then CoreLC=CoreLC∪C;(4)
IfCoreLC=∅, then go to step (6), else go to step (5);(5)
If∑j=1nplCoreLCLGj=1, return CoreLC as a reduct, else go to step (6);(6)
LetB=CoreLC;(7)
For eachC∈C\B, calculate SigBLC=∑j=1nplBLGj−∑j=1nplB∪CLGj;(8)
IfSigBLC0=maxC∈C∖BSigBLC, then B=B∪C0;(9)
If∑j=1nPlBLGj=1, stop and output B as a reduct, else go to step (7).


## 3.1. 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.
LetU,C be a covering information system. Define the F significance of the covering C∈C by(8)SigCFC=∑i=1mPlC∖CFKi−∑i=1mPlCFKi.
Define theL significance of the covering C∈C by(9)SigCLC=∑j=1nPlC∖CLGj−∑j=1nPlCLGj.
By the definition ofF significance of a covering (or L significance of a covering), the CoreFC (or CoreLC) can be characterized.Proposition 3.
LetU,C be a covering information system. Then,(10)CoreFC=C∈C|SigCFC>0,CoreLC=C∈C|SigCLC>0.Proof.
For anyC∈CoreFC, C∖C is not an F topological consistent set. Otherwise, there exists an F topological reduct B⊆C∖C. Then, C∉B, which contradicts the fact C∈CoreFC. By Theorem 2, ∑i=1mPlC∖CFKi>1. It follows that SigCFC>0. Therefore, CoreFC⊆C∈C|SigCFC>0. Conversely, for any C∈C∈C|SigCFC>0, SigCFC>0. Hence, SigCFC=∑i=1mPlC∖CFKi−∑i=1mPlCFKi>0. Since ∑i=1mPlCFKi=1, ∑i=1mPlC∖CFKi>1. By Theorem 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∈CoreFC. Then, C∈C|SigCFC>0⊆CoreFC.
Similarly, we can getCoreLC=C∈C|SigCLC>0.
From Proposition3, 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.
LetU,C be a covering information system, B⊆C. Define the F significance of C∈C∖B relative to B by(11)SigBFC=∑i=1mPlBFKi−∑i=1mPlB∪CFKi.
Define theL significance of C∈C∖B relative to B by(12)SigBLC=∑j=1nPlBLGj−∑j=1mPlB∪CLGj.
Let∑i=1mPl∅FKi=m and ∑j=1nPl∅LGj=n. The relative significance SigBFC (or SigBLC) can measure importance degree of the covering C relative to B.
Now we design algorithms to find anF topological reduct or an L topological reduct of the covering information system U,C.
The time complexity of Algorithms1 and 2 is OC3×U2. In the following, an example is given to illustrate the mechanism of Algorithms 1 and 2.Example 2.
Continued from Example1.(1)
ComputingF topological reducts: first, FHC\C1K1=x1,x2, FHC\C1K2=x3,x4, and FHCC1K3=x5. Thus, we get(13)SigCFC1=∑i=13PlC\C1FKi−∑i=13PlCFKi=25+25+15−1=0.ByFHC\C2K1=x1,x2, FHC\C2K2=x3,x4,x5, and FHC\C2K3=x4,x5, we have(14)SigCFC2=∑i=13PlC\C2FKi−∑i=13PlCFKi=25+35+25−1=25>0.SinceFHC\C3K1=x1,x2,x3, FHC\C3K2=x2,x3,x4, and FHC\C3K3=x5, we obtain(15)SigC1C3=∑i=13PlC\C3FKi−∑i=13PlCFKi=35+35+15−1=25>0.Hence,CoreFC=C2,C3.Second, since∑i=13PlCoreFCFKi=1, CoreFC=C2,C3 is an F topological reduct.(2)
ComputingL topological reducts: first, LHC\C1G1=x1,x2, LHC\C1G2=x3, LHC\C1G3=x4, and LHC\C1G4=x5; then,(16)SigCLC1=∑j=14PlC\C1LGj−∑j=14PlCLGj=25+15+15+15−1=0.ByLHC\C2G1=x1,x2, LHC\C2G2=x3,x4, LHC\C2G3=x3,x4,x5, and LHC\C2G4=x4,x5, we obtain(17)SigCLC2=∑j=14PlC\C1LGj−∑i=14PlCLGj=25+25+35+25−1=45>0.SinceLHC\C3G1=x1,x2, LHC\C3G2=x3, LHC\C3G3=x4, and LHC\C3G4=x5, we have(18)SigCLC3=∑j=14PlC\C1LGj−∑i=14PlCLGj=25+15+15+15−1=0.Hence,CoreLC=C2.Second, it follows from∑j=14PlCoreLCLGj=10/5≠1 that CoreLC=C2 is not an L topological reduct.Since(19)SigCoreLCLC1=∑j=14PlC2LGj−∑i=14PlC1,C2LGj=105−1=1,SigCoreLCLC3=∑j=14PlC2LGj−∑i=14PlC2,C3LGj=105−1=1,letB=CoreLC∪C1=C1,C2. We get ∑j=14PlBLGj=1; then, B=C1,C2 is an L topological reduct.
Similarly, we can obtain thatC2,C3 is an L topological reduct.
Therefore,CoreLC=C2, and C1,C2 and C2,C3 are L topological reducts.Algorithm 1: Computing the F topological core and F topological reduct of U,C.
(1)
LetCoreFC=∅, ∑i=1mPl∅FKi=m;(2)
For eachC∈C, calculate SigCFC=∑i=1mplC\CFKi−∑i=1mplCFKi;(3)
IfSigCFC>0, then CoreFC=CoreFC∪C;(4)
IfCoreFC=∅, then go to step (6), else go to step (5);(5)
If∑i=1mplCoreFCFKi=1, return CoreFC as a reduct, else go to step (6);(6)
LetB=CoreFC;(7)
For eachC∈C/B, calculate SigBFC=∑i=1mplBFKi−∑i=1mplB∪CFKi;(8)
IfSigBFC0=maxC∈C\BSigBFC, then B=B∪C0;(9)
If∑i=1mplBFKi=1, 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.
(1)
LetCoreLC=∅, ∑j=1npl∅LGj=n;(2)
For eachC∈C, calculate SigCLC=∑j=1nplCCLGj−∑j=1nplCLGi;(3)
IfSigCLC>0, then CoreLC=CoreLC∪C;(4)
IfCoreLC=∅, then go to step (6), else go to step (5);(5)
If∑j=1nplCoreLCLGj=1, return CoreLC as a reduct, else go to step (6);(6)
LetB=CoreLC;(7)
For eachC∈C\B, calculate SigBLC=∑j=1nplBLGj−∑j=1nplB∪CLGj;(8)
IfSigBLC0=maxC∈C∖BSigBLC, then B=B∪C0;(9)
If∑j=1nPlBLGj=1, stop and output B as a reduct, else go to step (7).


## 4. ComparingL 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].Definition 11.
(see [23]). Let U,C be a covering information system. For any B⊆C, denote CovB=xB|x∈U. If CovB=CovC and for any C∈B, CovB≠CovB−C, then B is called a reduct of U,C.Example 3.
LetU and C be the same as those in Example 1. By Definition 11 and Table 1, CovC=x1,x2,x2,x3,x4,x5, CovC1,C2=CovC, CovC1,C2≠CovC1, and CovC1,C2≠CovC2. Then, C1,C2 is a reduct of U,C. Furthermore, C1,C2 is the only reduct of U,C.Proposition 2.
LetU,C be a covering information system, B⊆C. If B is a reduct, then B is an L topological consistent set.Proof.
SinceB is a reduct, by Definition 11, xC=xB for all x∈U. It follows that for any X⊆U, LHCX=LHBX. Thus, τLHC=τLHB, which implies that B is an L topological consistent set.Remark 2.
(1)
F topological reducts of U,C are not necessary correlated with the reducts of U,C.From Example1 and Example 3, we can know that the F topological reduct of U,C is C2,C3, and the reduct of U,C is C1,C2.(2)
L topological reduct of U,C could not be a reduct of U,C.
For example,C2,C3 is an L topological reduct of the covering information system U,C in Example 1. However, C2,C3 is not a reduct of U,C.


## 5. Conclusion

In this paper, theL topological reduction and the F topological reduction of covering information systems have been characterized by the belief and plausibility functions from the evidence theory. The topological spaces by the two pairs of covering approximation operators in covering information systems are pseudo-discrete, which deduce partitions. Then, 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].



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