Xiaohe Zhang, Jusheng Mi ?, Meizheng Li, Meishe Liang,3

1College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, People’s Republic of China

2College of Computer and Cyber Security, Hebei Normal University, Shijiazhuang, People’s Republic of China

3Shijiazhuang Vocational Technology Institute, Shijiazhuang, People’s Republic of China

Abstract: Attribute reduction of formal decision context mainly uses the relationship between two concept lattices generated by the condition and decision attributes to remove redundant condition attributes. By using decision attributes to observe the covering of objects, this study defines two types of consistent sets and reducts in a consistent formal decision context based on neighbourhood systems. Four types of reductions in inconsistent formal decision contexts are also studied. The methods to calculate all types of reductions are formulated by discernibility matrix.Finally, an approach to obtain the decision rules in consistent formal decision context is proposed.

1 Introduction

Formal concept analysis (FCA) [1] was introduced by Wille in the 1980s, and then Ganter further developed the theoretical framework in [2]. The main concerns in FCA include the following topics: the construction of concept lattices, the acquisition of rules, the generalisation of concept lattices and attribute reduction of concept lattice.

With the development of FCA, much attention has been paid to attribute reduction in formal decision contexts. Zhang proposed an attribute reduction such that the new concept lattice is isomorphism with the original concept lattice [3]. Wu studied the granular structure of concept lattices with application in knowledge reduction in FCA [4]. Li formulated a new framework of knowledge reduction from the perspective of rule acquisition and developed the method of corresponding reduction by using the discernibility matrix and Boolean function [5, 6], investigated the issue of reducing the object set of a formal decision context without losing the decision rule information provided by the entire set of objects[7]. Li and Mi proposed two new kinds of attribute reduction in the decision formal context based on maximal rules [8]. By combing minimal vertex covers in graph theory, Chen proposed a fast attribute reduction method for large formal decision contexts [9,10]. Li and Wang investigated the inconsistent formal decision context based on congruence relations[11].

With the increase of the size of data, the number of nodes in the induced concept lattice grows rapidly and the lattice becomes too complex to research. Various methods are proposed to control the size of concept lattices by eliminating some nodes. Mi further studied the attribute reduction of the formal contexts based on axialities [12] and constructed the multi-scaled concept lattices based on inclusion degree [13]. Three-way concept lattice theories[14] are obtained by combining the three-way decision theory with the concept lattice [15]. Ma combined inclusion degree and a special covering neighbourhood systems to reduce the number of concepts by introducing a Galois connection between the two power sets of objects and attributes and then constructed multiscaled concept lattices from the induced context [16]. Attribute reduction in formal decision contexts is an important issue in FCA, which can help us to discover the knowledge hidden in formal decision contexts. The research on this issue has also made some achievements, such as [17–24]. The rule acquisition has also been studied in [25, 26]. However, there is no in-depth study on attribute reduction and rule acquisition in formal decision contexts based on neighbourhood systems. This paper studies the attribute reduction in both consistent and inconsistent formal decision contexts based on neighbourhood systems also constructed the rule acquisition method in consistent formal decision context.

The structure of this paper is as follows. Section 2 reviews some notions about formal concept analysis and neighbourhood systems.Section 3 introduces the method of attribute reduction and rule acquisition in consistent formal decision contexts based on neighbourhood systems. In Section 4, four types of attribute reductions in inconsistent formal decision context have been proposed. Section 5 concludes the proposed method in the paper with a summary.

2 Preliminaries

In this section,some basic notions in the formal contexts are briefly introduced.

Definition 1: [2]: A formal context is a triplet F =(U,A,I),where U is a non-empty set of objects, A is a non-empty set of attributes.I is a binary relation from U to A, (x,a)∈I means that object x has attribute a. For X ?U and B ?A, two operations were defined as

To simplify notation,we write x?instead of{x}?for all x ∈U and a?instead of {a}?for all a ∈A. In this study, we assume that I is regular, i.e. ?x ∈U, x?≠?, x?≠A hold and ?a ∈A, a?≠?,a?≠U hold.

Definition 2:[2]:Let F =(U,A,I)be a formal context.For X ?U and B ?A,a pair(X,B)is called a formal concept of F,if X =B?and B=X?. X and B are called the extension and intension of(X,B), respectively.

The set of all the formal concepts of F is denoted by L(U,A,I).L(U,A,I) can form a complete lattice called concept lattice and the corresponding partial order relation ‘≤’ in L(U,A,I) is defined as: (X1,B1)≤(X2,B2)?X1?X2(or B1?B2).

The meet and join of L(U,A,I) is defined as

Definition 3: Let U be a non-empty universe and R be a family of subsets of U. If U ={G:G ∈R}, then R is a covering of U.Therefore, we call (U,R) a covering approximation space of U.Let F =(U,A,I,D,G) be a formal decision context, where AD=?, I ?U ×A. For any B ?A, we denoteis a formal context. For X ?U,X?B={a ∈B:?x ∈X,(x,a)∈I}. ?x ∈U, ?a ∈A, when X ={x}, B={a}, we have

Definition 4: [16]: Let (U,D,G) be an information system, where D={d1,d2, ···,dt}, G is a binary relation between U and D

We assume that for all x ∈U,x?≠?.Then Ω=is a covering of U. If x ∈, we callis a neighbourhood of x,denoted by Nk(x)=If x has more than one neighbourhood, we call it a multi-neighbourhood.

The sets

are called the strong neighbourhood and weakly neighbourhood of x,respectively.

Through those properties,we know that ?x ∈U,can, respectively, constitute a covering of U.

Definition 5: Let Vl(l ≤m) be a non-empty and finite set, denote

Then P is called a set vector space. ?E =(E1, ...,Em) and S =(S1, ...,Sm), if El≤Sl(l ≤m), then we denote E ≤S.

3 Attribute reduction and rule acquisition in consistent formal decision contexts

In this section, we first investigate attribute reduction and then discuss the induction and fusion method of rules in consistent formal decision contexts.

3.1 Attribute reduction in consistent formal decision contexts

Definition 6: Let F =(U,A,I,D,G) be a formal decision context,where AD=?. For any xi∈U, if?F is strong (weak) neighbourhood consistent. For B ?A, ?xi∈U,?Ns(xi)then we call B as a strong (weak)neighbourhood consistent set. Furthermore, if there no proper subset of B is a strong (weak) neighbourhood consistent set of F,then B is a strong (weak) neighbourhood reduction of F.

Theorem 1: Let F =(U,A,I,D,G) be a formal decision context,for B ?A, the following statements hold:

(1) If F is strong neighbourhood consistent, then F must be a weak neighbourhood consistent.

(2)If B is a strong neighbourhood consistent set of F,then B must be a weak neighbourhood consistent set of F.

(3)If B is a strong neighbourhood reduction of F,then B must be a weak neighbourhood reduction of F.

Proof:

(1) If F is a strong neighbourhood consistent, then for any xi∈U,we have x?i??Ns(xi). Also, we knowThus,F is a weak neighbourhood consistent.

(2)If B is a strong neighbourhood consistent set of F, it means that for all xi∈U, we have x?iB??Also, thenThus,?F is a weak neighbourhood consistent set of F.

(3) By (2), B must be a weak neighbourhood consistent set of F.Therefore, we need only to prove that for any b ∈B,In fact, B is a strong neighbourhood reduction of F, so for any b ∈B, we knowandThus,B is a weak neighbourhood reduction of F. □

Definition 7: Let F =(U,A,I,D,G) be a formal decision context,we define

D1d(xi,xj) and D2d(xi,xj) are called the strong neighbourhood discernibility sets and the weak neighbourhood discernibility sets betweenand, respectively.

Dq={Dqd(xi,xj)|xi,xj∈U}(q=1,2) are referred to as strong and weak neighbourhood discernibility matrix of F, respectively.

Property 2:For the discernibility matrix Dqof F,?xi,xj,xk∈U,the following properties hold:

The proof process is available by Definition 5.

Let F =(U,A,I,D,G) be a formal strong (weak) neighbourhood consistent decision context, for B ?A, we denote that

Then we have

(1) x?B?i ={x ∈U:x?{a}i?x?{a},?a ∈B};

(2) RB?R1D?x?B?i?Ns(xi);

(3) RB?R2D?x?B?i?Ns(xi).

Thus, B is a strong (weak) neighbourhood consistent set of F ?RB?RqD,q=1,2.

Theorem 2: Let F =(U,A,I,D,G) be a formal strong (weak)neighbourhood consistent decision context and B ?A. B is a strong (weak) neighbourhood consistent set of F iff≠?for all Dqd(xi,xj)≠?.

Proof: For simplicity, we only prove the case of a strong neighbourhood consistent set.(Necessity): For all D1d(xi,xj)≠?, by Definition 6, we haveand then (xi,xj)?R1D. As B is a strong neighbourhood consistent set of F, we have RB?R1D. Hence,(xi,xj)?RB. There must be an attribute b ∈B such that(xi,xj)?RB. SoThus, b ∈D1d(xi,xj), and then b ∈BConsequently, B≠?.

Theorem 3: Let F =(U,A,I,D,G) be a formal strong (weak)neighbourhood consistent decision context. a ∈A is a strong(weak) core attribute of F iff there exists xi,xj∈U such that Dqd(xi,xj)={a}.

Proof:(Necessity):If a ∈A is a core attribute of F,then A-{a}is not a consistent set of F, i.e. RA-{a}■R1D. There exist xi,xj∈U such thatand ?b ∈A-a, x?ib?xbjhold. It is easy to proveso by definition we have a ∈D1d(xi,xj). Therefore, D1(xi,xj)={a}. (If x?{a}i?x?{a}j, then from(?b ∈A-a), we have x?ic?x?

jc(?c ∈A). As F is a strong neighbourhood consistent, RA?R1D, i.e.which contradicts that

(Sufficiency): If D1(xi,xj)={a}. By Definition 6, we haveand ?b ∈A-a, x?ib?xbj. So(xi,xj)∈RA-{a}and (xi,xj)?R1D. We can conclude RA-{a}■R1D,i.e. a is a core attribute of F.

Similarly, proof of other cases can be obtained.

Definition 8: Let F =(U,A,I,D,G) be a formal strong (weak)neighbourhood consistent decision context, Dq,q=1,2,xi,xj∈U,Dqd(xi,xj)≠?.

Denote

If no repetitive elements inq =1,2, are defined as the strong (weak)neighbourhood minimal disjunctive normal form of F.=}(k ≤p) is a strong (weak) neighbourhood reduction of F, respectively. Then, {Bk|k ≤p} is the set of all strong (weak) neighbourhood reductions.

The algorithm of strong neighbourhood reduction in strong neighbourhood consistent formal decision contexts is shown as Algorithm 1 (see Fig. 1).

Fig. 1 Algorithm 1. Strong neighbourhood reduction in strong neighbourhood consistent formal decision contexts

Example 1:Table 1 depicts an example of a formal decision context F1=(U,A,I,D,G) (see Fig. 2).

Since

It is easy to prove ?xi∈U(i=1,2,3,4,5) andSo, F1is a strong neighbourhood consistent. The strong neighbourhood discernibility matrix is shown in Table 2. The strong neighbourhood reductions of F1can be calculated by the Boolean function as follows:

Hence, F1has two strong neighbourhood reductions:B1={a1,a2,a3} and B2={a1,a3,a4}. a1, a3are strong core attributes of F1

Table 1 Formal decision context: F1

Fig. 2 Hasse diagram of the L(U,A,I) depicted in Table 1

Table 2 Strong discernibility matrix of F1

Table 3 Weak discernibility matrix of F1

Fig. 3 Hasse diagram of the L(U,A,I) depicted in Table 3

It can be obtained that ?xi∈U(i=1,2,3,4,5), x?i?F1is also a weak neighbourhood consistent. Since the weak neighbourhood discernibility matrix is shown in Table 3 (see Fig. 3). It is the same as Table 2. Also, the strong neighbourhood reductions of F1can be calculated as follows:

It is the same as M2. Through Example 1, we further verify the correctness of Theorem 1.

3.2 Rule acquisition in consistent formal decision contexts

Definition 9: Let F =(U,A,I,D,G) be a formal strong (weak)neighbourhood consistent decision context, for X ?UB ?A,(X,B)∈L(U,A,I) is called condition concept. Also,is named strong (weak) decision concept, where

If X ?Ns(xi) (X ?Ns(xi)), then a strong (weak) neighbourhood granular decision rule between the condition concept and the strong or weak decision concept can be obtained, denoted by

The extension of a concept is uniquely determined by its intension,so (X,B)?(Ns(xi),T) or (X,B)?T) can be simplified and denoted as B ?T.

Definition 10: A strong neighbourhood granular decision rule(X,B)?(Ns(xi),T) is said to be optimal if there is no other rule(X′,B′)?such that X ?X′or?Ns(xi).A weak neighbourhood granular decision rule (X,B)?is said to be optimal if there is no other rule(X′,B′)?such that X ?X′or

The algorithm of inducing all of strong neighbourhood granular decision rules is shown as follows. Similarly, the algorithm of inducing weak neighbourhood decision rules can be obtained by Algorithm 2 (see Fig. 4).

Example 2:In Example 1,all of the condition concepts of F1can be computed, results are as follows:

All of the decision concepts of F1: ({x1},{d1d3});({x2,x3},{d1,d2}); ({x1,x4,x5},{d3}).

All of the strong neighbourhood granular decision rules in F1can be observed as follows:

1. {a1,a3}?{d1,d3};

2. {a1,a3}?{d3};

3. {a1,a2,a4,a5}?{d1,d2};

4. {a1,a2,a5}?{d1,d2};

5. {a2,a3,a4}?{d3}.

Strong neighbourhood granular optimal decision rules in F1are 1,4,5.

B1={a1,a3,a2} is a reduction of F, we can induce granular decision rules from (U,B1,IB1,D,G) as follows:

(1) {a1,a3}?{d1,d3};

(2) {a1,a3}?{d3};

Fig. 4 Algorithm 2. Induction of strong neighbourhood granular decision rules

(3) {a1,a2,a5}?{d1,d2};

(4) {a2,a3,a4}?{d3}.

All of the strong neighbourhood granular optimal decision rules in(U,B1,IB1,D,G) are(1), (3), (4). They are the same as 1, 4, 5.

If F =(U,A,I,D,G) is a formal strong (weak) neighbourhood consistent decision context, then RA?RD, i.e. ?xi∈U, there

Let F =(U,A,I,D,G)be a formal strong(weak)neighbourhood consistent decision context, ?x ∈U, al∈A, define

?B ?A, and ?al∈A define

Definition 11: Let (P, ≤) be a poset, for E =(E1,E2, ...,Em),S =(S1,S2, ...,Sm)∈P, the inclusion degree is defined by

in which

By using the inclusion degree of set-valued vectors, the rule fusion method in formal decision contexts is proposed in Algorithm 3(see Fig. 5).

Example 3:A set of coverage about U can be obtained by the strong neighbourhood function

Denote

Thus, M1={({1},{0},{1},{0},{0})}, M2={({1},{1},{0},{1},{1}),({1},{1},{0},{0},{1})},M3={({1},{0},{1},{0},{0}),({0},{1},{1},{1},{0})}.Also, S1=({1},{0},{1},{0},{0}), S2=({1},{1},{0},{0,1},{1}), and S3=({0,1},{0,1},{1},{0,1},{0}).

For ({x2,x4,x5},{a2,a4})∈L(U,A,I), E =({0},{1},{0},{1},{0}), so

From the foregoing, it is easy to obtain the rule: ‘{a2,a4}?d3’(0.875), where 0.875 is the rule confidence. All rules can be obtained in the above way.

Fig. 5 Algorithm 3. Rule fusion method in formal decision contexts

{a1,a3}?{d1,d3}(1);{a1,a3}?{d3}(1); {a1,a2,a4,a5}?{d1,d2}(1); {a1,a2,a5}?{d1,d2}(1); {a2,a3,a4}?{d3}(1);{a1}?{d3}(0.875); {a3}?{d3}(1); {a2,a4}?{d3}(0.875);and {a2}?{d3}(0.875).

Obviously,the granular decision rules obtained in Section 3.2 are also deterministic rules in the rule fusion method.

4 Attribute reduction in inconsistent formal decision contexts

Let U be a finite and non-empty set and X,Y ?U,inclusion degree on 2Uis defined as follows:

|X| denotes the cardinality of X.

The strong neighbourhood generalised decision distribution functionand the weak neighbourhood generalised decision distribution functionare defined as

Denote as the strong maximum decision function and weak maximum decision function, respectively.

Definition 12: Let F =(U,A,I,D,G) be a formal inconsistent decision context and B ?A.Iffor all xi∈U, then B is referred to as a strong distribution consistent set of F.Furthermore, if no proper subset of B is a strong distribution consistent set of F, then B is referred to as a strong distribution reduction of F.

Definition 13: Let F =(U,A,I,D,G) be a formal inconsistent decision context and B ?A.

Theorem 4: Let F =(U,A,I,D,G) be a formal decision context,for B ?A, the following statements hold:

(1)If B is a strong distribution consistent set of F,then B must be a strong maximum distribution consistent set of F.

(2) If B is a weak distribution consistent set of F, then B must be a weak maximum distribution consistent set of F.

(3)If B is a strong distribution reduction of F,then B must be a strong maximum distribution consistent set of F.

(4)If B is a weak distribution reduction of F,then B must be a weak maximum distribution consistent set of F.

(5)If B is a strong distribution reduction of F,then B must be a weak distribution consistent set of F.

(6)If B is a strong maximum distribution reduction of F,then B must be a weak maximum distribution consistent set of F.

The proof is obvious by Definitions 12 and 13.

Definition 14: Let F =(U,A,I,D,G) be a formal inconsistent decision context, ?xi,xj∈U

Then we define

Dm(xi,xj)(m=1,2,3,4) are referred to as the strong distribution,the weak distribution, the strong maximum distribution, and the weak maximum distribution discernibility set between xiand xj,respectively.

Dm={Dm(xi,xj)|xi,xj∈U}(m=1,2,3,4) are referred to as the strong distribution, the weak distribution, the strong maximum distribution, and the weak maximum distribution discernibility matrix between xiand xj, respectively.

The followingproperties hold:

(1) Dm(xi,xi)=?;

(2) Dm(xi,xj)Dm(xj,xi)=?.

For B ?A, we denote that

Then, it is easy to prove

(1) RB?Thus, B is a strong distribution consistent set of F ?RB?

(2) RB?R?D2?Thus, B is a weak distribution consistent set of F ?RB?R?D2.

(3) RB?R?D3?Thus, B is a strong maximu m distribution consistent set of F ?RB?R?D3.

(4) RB?R?D4?Thus, B is a weak maximum distribution consistent set of F ?RB?

Theorem 5: Let F =(U,A,I,D,G) be a formal inconsistent decision context, B ?A, then the following statements hold:

(1) B is a strong distribution consistent set of F iff for all D1(xi,xj)≠?, BD1(xi,xj)≠?.

(2) B is a weak distribution consistent set of F iff for all D2(xi,xj)≠?, BD2(xi,xj)≠?.

(3)B is a strong maximum distribution consistent set of F iff for all D3(xi,xj)≠?, BD3(xi,xj)≠?.

(4) B is a weak maximum distribution consistent set of F iff for all D4(xi,xj)≠?, BD4(xi,xj)≠?.

Proof: For simplicity,we only prove(1), other cases can be proved by a similar way.

(Necessity): For all D1(xi,xj)≠?, by definition, we have μA(xi)≠and then (xi,xj)?Since B is a strong distribution consistent set of F, we have RB?Hence,(xi,xj)?RB. There must be an attribute b ∈B such that(xi,xj)?RB. So we have x?ix?j. Thus, b ∈D1(xi,xj), and then b ∈BD1(xi,xj). Consequently, BD1(xi,xj)≠?.

(Suffciiency): If for all D1(xi,xj)≠?, we have?thus (xi,xj)?. BD1(xi,xj)≠?, we can find an attribute b ∈B such that b ∈D1(xi,xj). By definition, we can observe that ,xj)?RB. Therefore, RB?, B is a strong distribution consistent set of F.

Theorem 6: Let F =(U,A,I,D,G) be a formal inconsistent decision context. a ∈A is a strong distribution, weak distribution,strong maximum distribution, and weak maximum distribution core attribute of F iff there exist xi,xj∈U, such that Dm(xi,xj)={a}, respectively.

Proof:We only prove the case of a strong distribution core attribute,other cases can be proved by a similar way.

(Necessity): If a ∈A is a strong distribution core attribute of F,then A-{a} is not a consistent set of F, i.e. RA-{a}There exist xi,xj∈U such thatand ?b ∈A-a,hold. It is easy to prove x?i{a}■x?j{a}, so by definition we have a ∈D1(xi,xj). Therefore, D1(xi,xj)={a}. (If x?{a}i?then from x?ib?xbj(?b ∈A-a), we have(?c ∈A).

Thus, Dm(xi,xj)=?, which contradicts that Dm(xi,xj)≠?.)

(Sufficiency): If D1(xi,xj)={a}. By definition, we haveand ?b ∈A-a, x?ib?xbj. So(xi,xj)∈RA-{a}and (xi,xj)?R?D1. We can conclude RA-{a}■R?D1, i.e. a is a core attribute of F. □

Definition 15: Let F =(U,A,I,D,G) be a formal inconsistent decision context, xi,xi∈U, D?m(xi,xi)≠?

Mm(m=1,2,3,4) are called the strong distribution, the weak distribution, the strong maximum distribution, and the weak maximum distribution discernibility functions, respectively.

The minimal disjunctive form of Mm(m=1,2,3,4) defined by

Denote Bmk={ams}(s=1,2, ...,qk), thus, {Bmk|k =1,2, ···,p}(m=1,2,3,4) are the sets of the strong distribution,the weak distribution, the strong maximum distribution, and the weak maximum distribution reductions of F, respectively.

The algorithm of the strong neighbourhood distribution reduction is shown in Algorithm 4 (see Fig. 6).

Example 4:Table 4 depicts an example of a formal decision context F2=(U,A,I,D,G).

Fig. 6 Algorithm 4. Strong distribution reduction in inconsistent formal decision contexts

Table 4 Formal decision context: F2 =(U,A,I,D,G)

Firstly, x?1A?So, we discuss the fact that F2is strong neighbourhood inconsistent formal decision contexts.

The strong distribution discernibility matrix is shown in Table 5.The strong distribution reductions of F2can be calculated as follows:

Hence, F2indeed has only one strong distribution reduction B1={a1,a2,a4,a5,a6,a7,a8}. Also, a1, a2, a4, a5, a6, a7, a8are strong distribution core elements of F2.

The strong maximum distribution discernibility matrix is shown in Table 6.

B3={a1,a2,a4,a5,a6,a7} is a strong maximum distribution reduction of F2. Also, a1, a2, a4, a5, a6, and a7are strong maximum distribution core elements of F2.

Table 5 Strong distribution discernibility matrix of F2 =(U,A,I,D,G)

Table 6 Strong maximum distribution discernibility matrix of F2 =(U,A,I,D,G)

Table 7 Weak distribution discernibility matrix of F2 =(U,A,I,D,G)

The weak distribution discernibility matrix is shown in Table 7.The weak distribution reductions of F2can be calculated as follows:

Hence, F2has only one weak distribution reduction B2={a1,a2,a4,a5,a6,a7,a8}. Also, a1, a2, a4, a5, a6, a7, and a8are weak distribution core elements of F2.

Through Example 4, the correctness of Theorem 4 is further verified. The relationship between those types of consistent sets and reduction in inconsistent formal decision contexts is explored further.

5 Conclusion

Attribute reduction is one of the key issues in formal concept analysis. Although there have been many attribute reduction approaches for formal decision contexts, little work has been done for neighbourhood system based formal decision context. The consistent set and reduction in neighbourhood systems based on formal consistent decision contexts have been studied. Inspired by attribute reduction theory in inconsistent information systems, this study developed approaches to attribute reduction in inconsistent formal decision contexts based on neighbourhood systems. It can be observed that all of the reduction referred in this study by the algorithms using Boolean reasoning is complex and timeconsuming. The simpler and more efficient attribute reduction method will be further studied.

6 Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61573127 and 61502144), the Natural Science Foundation of Hebei Province (no. F2018205196), the Science and Technology Research Program of Higher Education Institutions of Hebei Province (nos. BJ2019014 and QN2017095),and the Doctor Natural Science Foundation of Hebei Normal University (no. L2017B19).

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