relations between granular reduct and dominance reduct in formal contexts

Knowledge-Based Systems xxx (2014) xxx–xxx

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Knowledge-Based Systems

journal homepage: www.elsevier .com/ locate /knosys

Relations between granular reduct and dominance reduct in formalcontexts

http://dx.doi.org/10.1016/j.knosys.2014.03.0060950-7051/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (M.-W. Shao), [email protected].

hk (Y. Leung).

Please cite this article in press as: M.-W. Shao, Y. Leung, Relations between granular reduct and dominance reduct in formal contexts, Knowl. Base(2014), http://dx.doi.org/10.1016/j.knosys.2014.03.006

Ming-Wen Shao a,b, Yee Leung c,⇑a Computer Engineering Institute, Qingdao Technological University, Qingdao, Shandong 266520, PR Chinab College of Information Science and Technology, Shihezi University, Shihezi, Xinjiang 832003, PR Chinac Department of Geography and Resource Management, Institute of Future Cities, The Chinese University of Hong Kong, Hong Kong

a r t i c l e i n f o

Article history:Received 27 February 2013Received in revised form 5 March 2014Accepted 9 March 2014Available online xxxx

Keywords:Attribute reductionConcept latticeDominance relationFormal concept analysisRough set

a b s t r a c t

One of the key issues of knowledge discovery and data mining is knowledge reduction. Attribute reduc-tion of formal contexts based on the granules and dominance relation are first reviewed in this paper.Relations between granular reduts and dominance reducts are investigated with the aim to establish abridge between the two reduction approaches. We obtain meaningful results showing that granule-basedand dominance-relation-based attribute reducts and attribute characteristics are identical. Utilizingdominance reducts and attribute characteristics, we can obtain all granular reducts and attribute charac-teristics by the proposed approach. In addition, we establish relations between dominance classes andirreducible elements, and present some judgment theorems with respect to the irreducible elements.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The theory of formal concept analysis (FCA) proposed by Wille[13,54] has been formalized into an efficient methodology for dataanalysis and knowledge processing. FCA has become an importantand appealing theoretical undertaking in recent years. It has beenapplied to various domains such as data mining, informationretrieval, knowledge acquisition, software engineering, and data-base management systems [1–4,6,10,12,15,20,23–25,28,29,47,59,63,64].

Attribute reduction is an important issue in the discovery ofknowledge in information systems. In terms of formal contexts,attribute reduction is the search for a minimal attribute subset thatpreserves required concepts and their hierarchical structure bydeleting irrelevant attributes from the database. Rapid growth ofinterest in attribute reduction in FCA is evidenced in recent years[9,11,27,37–41,52,57,66].

The theory of rough set (RS), proposed by Pawlak [44], is anextension of the classical set theory. RS is a tool useful for dealingwith imprecision, vagueness and uncertainty of information. Theoriginal rough set theory does not consider attributes with prefer-ence-ordered domains. However, in reality, one often faces theordering problems of objects. For this reason, Greco et al. [16–18]

proposed a dominance-based rough sets approach (DRSA) to takeinto account the ordering properties of attributes and gave a gen-eral frame work. Greco et al. [19] presented a general model ofrough approximations based on ordinal properties of membershipfunctions of fuzzy sets, in which the classical rough set theory canbe considered as a particular case.

Attribute reduction is a major topic in rough set research. Anattribute reduct is a minimum subset of attributes that preservesrequired property of a given information system obtained underthe entire set of attributes by eliminating attributes that are notessential for the classification of objects or decision rules. Basedon different binary relation and different requirements, a largevariety of models and approaches to knowledge reduction havebeen proposed in the last two decades, see for examples[7,21,26,35,42,32–34,48,56,58,65].

Both FCA and RS are analyzed based on binary informationtables. Both deal with the problems of knowledge discovery andknowledge representation. Therefore, the relations between FCAand RS is an interesting research direction. This paper focuses onattribute reduts of FCA and RS, and investigate the relationsbetween FCA reduts and RS reducts in a formal context. The mean-ingful results have been obtained.

The paper is organized as follows. In the next section, therelated work is recalled. In Section 3, we first review some basicnotions of formal concept analysis. We then briefly recall thedefinitions of granular reduction and the corresponding reductiontheory in Wu et al. [57] that serves as a basis for subsequent

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analysis. In Section 4, we introduce dominance relation into formalcontexts and discuss the relations between dominance class andthe irreducible element. Based on dominance relation, attributereduction and attribute characteristics of formal contexts is pro-posed in Section 5. In Section 6, we discuss approaches to attributereduction in consistent formal decision contexts. We demonstrateapplications of the proposed method in Section 7. We then con-clude the paper with a summary and directions for future researchin Section 8.

2. Related work

Attribute reduction in FCA have been investigated in severalworks in the literatures. We recall, Gediga and Wille [13] first gavethe notions of reducible attributes and reducible objects by reduc-ing columns and rows in formal contexts. Cheung and Vogel [9] pro-posed to use the idea of quotient lattice to reduce the complexity ofa Term-Document Lattice. Zhang et al. [66] discussed attributereduction in classical formal contexts, and formulated the reductionapproach by using discernibility matrices and Boolean functions.Liu et al. [40] presented a reduction method for concept latticesbased on rough set theory. In [52], Wang and Zhang proposed thereduction by keeping the meet-irreducible elements. Wu et al.[57] studied knowledge reduction in formal contexts by keepinggranular structure of concept lattices. Liu et al. [39] showed an effi-cient post-processing method to prune redundant rules in virtue ofthe property of Galois connection, which inherently constrainsrules with respect to objects. Based on the Lukasiewicz implication,Elloumi et al. [11] gave a multi-level conceptual data reductionapproach via the reduction of the object sets by keeping only theminimal rows in a formal context. Belohlavek and Vychodil [4] for-mulated a method to reduce the number of formal fuzzy conceptsby only keeping the so-called crisply generated fuzzy concepts. Liand Zhang [37] introduced the notion of d-reducts in a fuzzy formalcontext, and gave some equivalent characterizations of d-consistentsets to determine d-reducts. Based on fuzzy K-means clustering,Kumar and Srinivas [27] presented a method to reduce the size ofa concept lattice by employing corresponding object-attributematrix. Li et al. [38] proposed a rule-acquisition oriented frame-work of knowledge reduction for real decision formal contextsand formulated a reduction method by constructing a discernibilitymatrix and its associated Boolean function.

Both FCA and RS deal with knowledge discovery and knowledgerepresentation on the basis of a binary relation between a set ofobjects and a set of attributes which is referred to as a formal con-text (in FCA theory) and an information system (in RS theory). Aswe know, FCA and RS are related and often complementary meth-odologies. For example, FCA can be introduced into RS theory bymeans of different types of formal concepts [14,61,62]. On theother hand, rough set approximation operators can be introducedinto FCA by means of different types of definability [22,45,46].The comparison and combination of FCA and RS theory may pro-vide new approaches to data analysis and knowledge discovery.To entertain such question, much effort has been made to compareand combine these two theories in recent years [30,31,36,50,55]. In[50], Wang and Liu proposed AFS formal concept via rough set andAFS algebra which is a generalization and development to mono-tone concept. Lai and Zhang [30] showed that every complete fuzzylattice can be represented as the concept lattice of a fuzzy contextbased on rough set theory if and only if the residuated latticeðL; �;1Þ satisfies the law of double negation. In [60], based on thenotion of homomorphisms of formal contexts, Yang and the firstauthor of this paper discussed the invariant characters of formalcontexts under homomorphisms, and reveal the relation of attri-butes characters and formal concepts between two formal contexts

Please cite this article in press as: M.-W. Shao, Y. Leung, Relations between gra(2014), http://dx.doi.org/10.1016/j.knosys.2014.03.006

under homomorphisms and isomorphisms. By using the idea ofknowledge reduction in rough set theory, Mi et al. [41] formulateda Boolean approach to calculate all reducts of a formal context viathe use of discernibility function. In order to find the similarexpressions with attribute reduction in rough set theory, Wang[51] developed notions and methods of attribute reduction in a for-mal context based on congruence relations.

For a formal context, one can consider two types of reduction:RS reduction (based on classification or decision rules) and FCAreduction (based on concepts and their hierarchy). However, whatexactly is the relation between FCA and RS remains an openquestion.

It is worth recalling that the relations between RS reduction andFCA reduction have been discussed in the literature. For instance,based on an equivalence relation, Qi et al. [43], Wang and Zhang[49], independently discussed the relation between partition andlattice structure for a certain database and showed that eachextension of formal concepts is the union of some equivalent clas-ses. Wang and Zhang [49] presented that every equivalent class isthe extent of a formal concept in an anti-chain formal context.Unfortunately, this result relies on the assumption that the formalcontext is anti-chain, in general, is a characteristic very difficult tomeet. Wei and Qi [53], Wang and Zhang [49] showed that concept-lattice consistent set is a rough-set consistent set, but the reversedoes not hold. These research results have enhanced our under-standing of the relation between FCA and RS. Although some con-nections between FCA and RS have been revealed, but the resultsare insufficient. For instance, in previous literature, RS reductionand RS classification are analyzed based on an equivalent relation,which is a binary relation between objects. However, in most situ-ations, the binary relation is partial order instead of equivalence.Hence, there are some problems need to be further consideredunder general relation. For instance,

1. What is the relation between classification (in partial orderrelation) and lattice structures for a formal context?

2. What is the relation between rough set reduction (in partialorder relation) and concept lattice reduction?

3. What is the relation between attribute characteristics (in partialorder relation) in rough set and concept lattice?

Inspired by the work of Qi [43], Wang and Zhang [49], Wei andQi [53], a combination of FCA and RS theory is undertaken in thepresent paper to give answers to the above three questions. Differ-ing from the previous studies in [43,49,53], we consider dominancerelation instead of equivalence relation in RS model. The derivedresults may provide a meaningful bridge between FCA and RS.

3. Preliminaries

In this section, basic notions of FCA are briefly reviewed first.Then, we summarize the definition and approach of granularreduct in [57], which serves as a basis for the present study.

A formal context is a triplet K ¼ ðU;A; IÞ, where U is a non-empty finite set of objects and A is a non-empty finite set of attri-butes, and I is a relation between U and A, which is a subset of theCartesian product U � A, where ðx; aÞ 2 I means that object x hasattribute a. For a pair of elements x 2 U and a 2 A, if ðx; aÞ 2 I, wewrite xIa and read it as ‘‘object x has attribute a’’, or ‘‘attribute ais possessed by object x’’.

Let ðU;A; IÞ be a formal context, with X # U and B # A. The oper-ator ⁄ is defined as:

X� ¼ fa 2 Aj8x 2 X; ðx; aÞ 2 Ig;B� ¼ fx 2 Uj8a 2 B; ðx; aÞ 2 Ig:

nular reduct and dominance reduct in formal contexts, Knowl. Based Syst.


M.-W. Shao, Y. Leung / Knowledge-Based Systems xxx (2014) xxx–xxx 3

For x 2 U and a 2 A, for simplicity, we denote x� ¼ fxg� anda� ¼ fag�.

A pair ðX;BÞ of two sets X # U and B # A is called a formal con-cept of the context ðU;A; IÞ if X ¼ B� and B ¼ X�, where X and B arecalled the extent and the intent of the concept, respectively. Espe-cially, for any x 2 U and a 2 A; ðx��; x�Þ and ða�; a��Þ are formal con-cepts, which are called object concept and attribute concept,respectively [13].

The concepts of a formal context ðU;A; IÞ are ordered by

ðX1;B1Þ 6 ðX2;B2Þ () X1 # X2 ðwhich is equivalent to B2 # B1Þ;

where, ðX1; B1Þ is called the sub-concept of ðX2; B2Þ, and ðX2;B2Þ iscalled the super-concept of ðX1;B1Þ.

The set of formal concepts forms a complete lattice which isdenoted by LðU;A; IÞ. The meet and join of the concepts are givenby:

ðX1;B1Þ _ ðX2;B2Þ ¼ ððX1 [ X2Þ��;B1 \ B2ÞðX1;B1Þ ^ ðX2;B2Þ ¼ ðX1 \ X2; ðB1 [ B2Þ��Þ:

Let K ¼ ðU;A; IÞ be a formal context and C # A, denoteIC ¼ I \ ðU � CÞ. We can obtain a formal context ðU;C; ICÞ, whichis called sub-context of ðU;A; IÞ. In sub-context ðU;C; ICÞ, withX # U and B # A, the operator �C is defined as:

X�C ¼ fa 2 C j 8x 2 X; ðx; aÞ 2 Ig;B�C ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 Ig:

Definition 1 [12]. Let L be a finite lattice and a 2 L. We denote

vH ¼_fx 2 Lj x < vg; vH ¼

^fx 2 Lj v < xg:

vH is called join-irreducible if v – vH; vH is called meet-irreducibleif v – vH.

Table 1A formal context ðU;A; IÞ.

a b c d e

Theorem 1 [12]. Let L be a finite lattice. Every element in L is a join(meet,respectively) of the join-irreducible (meet-irreducible, respec-tively) elements.

It should be noted that a formal concept ðX;BÞ in the conceptlattice LðU;A; IÞ can be represented as a join of the object conceptsof its extension or a meet of the attribute concepts of its intension(see [57]), that is,

ðX;BÞ ¼_x2X

ðx��; x�Þ;

ðX;BÞ ¼^b2B

ðb�; b��Þ:

Let K ¼ ðU;A; IÞ be a formal context, for any attribute set B # A,Wu et al. [57] define a binary relation JB on U � U as follows:

JB ¼ fðx; yÞ 2 U � U : x�B # y�Bg;

furthermore,

JBðxÞ ¼ fy 2 U : ðx; yÞ 2 JBg:

Similarly, we also define a binary relation JV on A� A for allV # U as follows:

JV ¼ fða; bÞ 2 A� A : a�V # b�V g;

and,

JV ðaÞ ¼ fb 2 A : ða; bÞ 2 JVg:

x1 0 1 0 1 0x2 1 0 1 0 1x3 1 1 0 0 1x4 0 1 1 1 0x5 1 0 0 0 1

Theorem 2 [53]. Let K ¼ ðU;A; IÞ be a formal context. Then,ðJAðxÞ; JAðxÞ

�Þ is a formal concept of ðU;A; IÞ and

JAðxÞ ¼ x��:


Theorem 3. Let K ¼ ðU;A; IÞ be a formal context. Then, ðJUðaÞ�; JUðaÞÞ

is a formal concept of ðU;A; IÞ and

JUðaÞ ¼ a��:

Definition 2 [53]. Let K ¼ ðU;A; IÞ be a formal context, and LðU;A; IÞbe the associated concept lattice. An attribute subset D # A is referredto as a granular consistent set of ðU;A; IÞ if x�D�D ¼ x�A�A for all x 2 U. IfD # A is a granular consistent set of ðU;A; IÞ and there is no propersubset E � D such that E is a granular consistent set, then D is referredto as a granular reduct of ðU;A; IÞ. The intersection of all granularreducts of ðU;A; IÞ is called the granular core of ðU;A; IÞ.

It should be noted that Wei and Qi’s [53] definition of attributereduction is required to preserve the extensions of all concepts, butthe definition in [57] is only preserving the extensions of objectconcepts. Since any concept ðX;BÞ in the concept lattice LðU;A; IÞcan be represented as a join of the object concepts of its extension,the two definitions are in essence equivalent.

Theorem 4 [53]. Let K ¼ ðU;A; IÞ be a formal context and C # A.Then C is a granular consistent set of ðU;A; IÞ iff

x�C�C # x�A�A ; 8 x 2 U:

4. Dominance relation in formal contexts

In this section, we introduce dominance relation in formal con-texts, and our aim is to reveal the relationship between dominanceclasses and extensions of concepts.

A formal context K ¼ ðU;A; IÞ is also an information systemðU;A; f Þ with the range f0;1g, namely, 8 x 2 U;8 a 2 A, if ðx; aÞ 2 I,then the value of object x under attribute a is 1 (i.e. f ðx; aÞ ¼ 1);otherwise, the value is 0 (i.e. f ðx; aÞ ¼ 0). Let ðU;A; f Þ be the infor-mation system that corresponds with the formal context K. ThenI ¼ fðx; aÞ : f ðx; aÞ ¼ 1g.

Let K ¼ ðU;A; IÞ be a formal context and B # A. Let ðU;A; f Þ be theinformation system corresponding to the formal context. A binaryrelation RB is defined by

RB ¼ fðx; yÞ 2 U � Ujf ðx; aÞ 6 f ðy; aÞ;8a 2 Bg;

RB is called a dominance relation on the object set, where, ðx; yÞ 2 RB

means that y is at least as good as x with respect to all attributes inB. It should be noted that RB ¼ \b2BRfbg.

The granules of knowledge induced by the dominance relationRB are: the set of objects dominating x,

½x�RB¼ fy 2 Uj ðx; yÞ 2 RBg ¼ fy 2 Uj f ðx; aÞ 6 f ðy; aÞ;8 a 2 Bg

which are called B� dominating set with respect to x 2 U.Let U=RB denotes the family set f½x�RB

j x 2 Ug. Any elementfrom U=RB will be called a dominance class.

Example 1. For illustration, we employ the formal context ðU;A; IÞin [57], which is described by Table 1, where,U ¼ fx1; x2; x3; x4; x5; x6g and A ¼ fa; b; c; d; e; fg. All formal concepts



Table 2All formal concepts in LðU;A; IÞ.

ci ðX;BÞc0 ð;;AÞc1 ðx4; bcdÞc2 ðx3; abeÞc3 ðx2; aceÞc4 ðx1x4; bdÞc5 ðx1x3x4; bÞc6 ðx2x4; cÞc7 ðx2x3x5; aeÞc8 ðU; ;Þ

Fig. 1. LðU;A; IÞ.


from LðU;A; IÞ are depicted by Table 2. The Hasse diagram ofconcept lattice LðU;A; IÞ is represented by Fig. 1. The dominanceclasses determined by RA are:

½x1�RA¼ fx1; x4g; ½x2�RA

¼ fx2g; ½x3�RA¼ fx3g;

½x4�RA¼ fx4g; ½x5�RA

¼ fx2; x3; x5g

Similarly, we can define a dominance relation on the attributeset.

Let ðU;A; IÞ be a formal context and V # U. Let ðU;A; f Þ be theinformation system corresponding to the formal context. The bin-ary relation RV is defined by

RV ¼ fða; bÞ 2 A� Ajf ðx; aÞ 6 f ðx; bÞ;8x 2 Vg;

where ða; bÞ 2 RV means that b is at least as good as a with respect toall objects in V.

For any a 2 A, the dominance class ½a�RVis defined by

½a�RV¼ fb 2 Aj ða; bÞ 2 RVg ¼ fb 2 Aj f ðx; aÞ 6 f ðx; bÞ;8 x 2 Vg;

and is called the V � dominating set with respect to a 2 A.

Example 2. The dominance classes determined by RU are:

½a�RU¼ fa; eg; ½b�RU

¼ fbg; ½c�RU¼ fcg;

½d�RU¼ fb; dg; ½e�RU

¼ fa; eg:

Dominance relation RV has the same properties presented inProperty 1.

Theorem 5. Let ðU;A; IÞ be a formal context and B # A; ðU;A; f Þ be theinformation system corresponding to the formal context. Then, RB ¼ JB.

Proof. For any ðx; yÞ 2 RB, we have


ðx; yÞ 2 RB () f ðx; bÞ 6 f ðy; bÞ; 8 b 2 B () f ðx; bÞ ¼ 1

) f ðy; bÞ ¼ 1; 8 b 2 B () xIb ) yIb; 8 b 2 B () x�B

) y�B () ðx; yÞ 2 JB:

Thus, RB ¼ JB. h

Theorem 6. Let ðU;A; IÞ be a formal context and x 2 U. Thenð½x�RA

; ð½x�RAÞ�Þ is a formal concept of ðU;A; IÞ and

½x�RA¼ x��:

Proof. Since RA ¼ JA, then ½x�RA¼ JAðxÞ. We obtain ð½x�RA

; ð½x�RAÞ�Þ ¼

ðJAðxÞ; JAðxÞ�Þ. By Theorem 2, ð½x�RA

; ð½x�RAÞ�Þ is obtained as a formal

concept and ½x�RA¼ x��. h

Theorem 7. Let ðU;A; IÞ be a formal context and a 2 A. Thenð½a��RV

; ð½a�RVÞÞ is a formal concept of ðU;A; IÞ and

½a�RV¼ a��:

Proof. It is similar to the proof of Theorem 6. h

Let ðU;A; IÞ be a formal context, we denote

PðUÞ ¼ fð½x�RB; ð½x�RB

Þ�Þj ½x�RB2 U=RBg;

QðAÞ ¼ fð½a��RV; ð½a�RV

ÞÞj ½a�RV2 U=RVg;

cðUÞ ¼ fðx��; x�Þj x 2 Ug;uðAÞ ¼ fða�; a��Þj a 2 Ag:

Corollary 1. Let ðU;A; IÞ be a formal context, x 2 U and a 2 A. ThenPðUÞ ¼ cðUÞ and QðAÞ ¼ uðAÞ.

Proof. It can easily be derived from Theorems 6 and 7. h

It should be noted that from Corollary 1 we can obtain all theobject concepts and the attribute objects by using the object dom-inance classes and the attribute dominance classes.

Example 3. In Examples 1 and 2, using Corollary 1 we get

cðUÞ ¼ fð½x1�RA; ð½x1�RA

Þ�Þ; ð½x2�RA; ð½x2�RA


Þ�Þ;¼ ð½x4�RA

; ð½x4�RAÞ�Þ; ð½x5�RA

; ð½x5�RAÞ�Þg;

uðAÞ ¼ fðð½a�RUÞ�; ½a�RU

Þ; ðð½b�RUÞ�; ½b�RU

Þ; ðð½c�RUÞ�; ½c�RU

Þ;¼ ðð½d�RU

Þ�; ½d�RUÞ; ðð½e�RU

Þ�; ½e�RUÞg:

Definition 3. Let ðU;6Þ be a partial order set, x; y 2 U and x < y. x iscalled the lower close neighbor of y, if there does not exist z 2 Usuch that x < z < y, where, y is also called the upper close neighborof x and is denoted by x � y.

In the sequel, we use symbol j � j to denote the cardinality of aset.

Theorem 8. Let ðU;A; IÞ be a formal context and x 2 U. Then theobject concept ð½x�RA

; ð½x�RAÞ�Þ is a join-irreducible element in LðU;A; IÞ

iff jfð½y�RA; ð½y�RA

Þ�Þ 2 LðU;A; IÞj ð½y�RA; ð½y�RA

Þ�Þ � ð½x�RA; ð½x�RA

Þ�Þgj 6 1.

Proof. ð)Þ Suppose that

jfð½y�RA; ð½y�RA



Þ�ÞgjP 2:

Let ð½z�RA; ð½z�RA

Þ�Þ; ð½u�RA; ð½u�RA

Þ�Þ 2 fð½y�RA; ð½y�RA

Þ�Þ 2 LðU;A; IÞjð½y�RA;

ð½y�RAÞ�Þ � ð½x�RA

; ð½x�RAÞ�Þg. We are going to prove that ð½z�RA

;

ð½z�RAÞ�Þ _ ð½u�RA

; ð½u�RAÞ�Þ ¼ ð½x�RA

; ð½x�RAÞ�Þ. We have




ð½x�RA; ð½x�RA

Þ�ÞP ð½z�RA; ð½z�RA

Þ�Þ _ ð½u�RA; ð½u�RA

Þ�Þ¼ ðð½z�RA

[ ½u�RAÞ��; ð½z�RA

Þ� \ ð½u�RAÞ�Þ

> ð½z�RA; ð½z�RA

Þ�Þ and ð½u�RA; ð½u�RA

Þ�Þ:

If

ðð½z�RA[ ½u�RA

Þ��; ð½z�RAÞ� \ ð½u�RA

Þ�Þ < ð½x�RA; ð½x�RA

Þ�Þ;

then it is in conflict with Definition 3. Thus,

ðð½z�RA[ ½u�RA

Þ��; ð½z�RAÞ� \ ð½u�RA

Þ�Þ ¼ ð½x�RA; ð½x�RA

Þ�Þ:

It also conflicts with the assumption that ð½x�RA; ð½x�RA

Þ�Þ is join-irreducible element. Consequently, we conclude that




Þ�Þgj 6 1:

ð(Þ Assume that




Þ�Þgj 6 1:

If jfð½y�RA; ð½y�RA



Þ�Þgj ¼ 0,then ð½x�RA

; ð½x�RAÞ�Þ does not contain non-zero sub-concept and it

has only one sub-concept ð;;AÞ. Hence, by Definition 1, ð½x�RA;

ð½x�RAÞ�Þ is obtained as a join-irreducible element.

If jfð½y�RA; ð½y�RA



Þ�Þgj ¼1, then ð½x�RA

; ð½x�RAÞ�Þ have only one maximum sub-concept, and we

denote as ð½u�RA; ð½u�RA

Þ�Þ. It is evident that ð½u�RA; ð½u�RA

Þ�Þ <ð½x�RA

; ð½x�RAÞ�Þ. By Definition 1, we conclude that ð½x�RA

; ð½x�RAÞ�Þ is a

join-irreducible element. h

From Theorem 8, in essence, an object concept is join-irreduc-ible iff it has only one maximum sub-concept.

Corollary 2. Let ðU;A; IÞ be a formal context and x 2 U. Then theobject concept ð½x�RA

; ð½x�RAÞ�Þ is a join-irreducible element in LðU;A; IÞ

iff jf½y�RA2 U=RAj ½y�RA

� ½x�RAgj 6 1.

Proof. It can easily be proved from Theorem 8. h

Example 4. In Example 1, since

jf½y�RA2U=RAj ½y�RA

� ½x1�RAgj ¼ 1; jf½y�RA

2U=RAj ½y�RA� ½x2�RA

gj ¼ 0;

jf½y�RA2U=RAj ½y�RA

� ½x3�RAgj ¼ 0; jf½y�RA


gj ¼ 0;

using Corollary 2 we obtain that ð½x1�RA; ð½x1�RA


Þ�Þ;ð½x3�RA

; ð½x3�RAÞ�Þ, and ð½x4�RA

; ð½x4�RAÞ�Þ are join-irreducible elements

in LðU;A; IÞ.Since jf½y�RA


gj¼ jfx2;x3gj¼2;ð½x5�RA;ð½x5�RA

Þ�Þis not a join-irreducible element.

Theorem 9. Let ðU;A; IÞ be a formal context and a 2 A. Then the attri-bute concept ð½a�RU

Þ�; ½a�RUÞ is a meet-irreducible element in LðU;A; IÞ

iff jfðð½g�RUÞ�; ½g�RU

Þ 2 LðU;A; IÞj ðð½a�RUÞ�; ½a�RU

Þ � ðð½g�RUÞ�; ½g�RU

Þgj 6 1.


From Theorem 9, in fact, an attribute concept is meet-irreduc-ible iff it has only one minimum sup-concept.

Corollary 3. Let ðU;A; IÞ be a formal context and a 2 A. Then theattribute concept ð½a�RU

Þ�; ½a�RUÞ is a meet-irreducible element in

LðU;A; IÞ iff jf½g�RU2 A=RU j ½a�RU

� ½g�RUgj 6 1.

Proof. It can easily be proved from Theorem 9. h


Example 5. In Example 2, since

jf½g�RU2 A=RU j ½a�RU

� ½g�RUgj ¼ 1; jf½g�RU

2 A=RU j ½b�RU� ½g�RU

gj ¼ 1;

jf½g�RU2 A=RU j ½c�RU

� ½g�RUgj ¼ 1; jf½g�RU

2 A=RU j ½d�RU� ½g�RU

gj ¼ 1;

using Corollary 3 we have thatðð½a�RU

Þ�; ½a�RUÞ; ðð½b�RU

Þ�; ½b�RUÞ; ðð½c�RU

Þ�; ½c�RUÞ, and ðð½d�RU

Þ�; ½d�RUÞ are

meet-irreducible elements in LðU;A; IÞ.

5. Attribute reduction of formal contexts based on dominancerelation

In this section, we discuss attribute reduction and attributecharacteristics of formal contexts from the viewpoint of domi-nance relation and study the relationship between RS reductionand FCA reduction. In what follows, we first introduce the defini-tions of dominance consistent set and dominance reduct.

Definition 4. Let K ¼ ðU;A; IÞ be a formal context. An attributesubset C # A is called a dominance consistent set (based ondominance relation) of K if RC ¼ RA. Furthermore, if RC�fcg – RA

for all c 2 C, then C is called a dominance reduct (based ondominance relation) of K.

Theorem 10. Let K ¼ ðU;A; IÞ be a formal context and C # A. Then, Cis a granular consistent set of K iff RC ¼ RA.

Proof. ð)Þ Let C be a granular consistent set of K. Then x�C�C ¼ x�A�A

for all x 2 U. Using Theorem 2 we get JCðxÞ ¼ JAðxÞ for all x 2 U,which means JC ¼ JA. By Theorem 5, we obtain RC ¼ JC ¼ JA ¼ RA.ð(Þ Suppose RA ¼ RC . Using Theorems 2 and 5, one can easily

conclude that C is a granular consistent set of K. h

Corollary 4. Let K ¼ ðU;A; IÞ be a formal context and D # A. Then, Dis a granular reduct of K iff D is a dominance reduct of K.

Proof. By routine verification using Definition 4 andTheorem 10. h

By Theorem 10 and Corollary 4, we can obtain the granularreducts of K by computing the dominance reducts of K.

We denote the set of all reducts of K ¼ ðU;A; IÞ as RedðKÞ.According to the significance of attribute, the attribute set isdivided into two parts:

1. indispensable (core) attribute set C : C ¼ \RedðKÞ;2. dispensable attribute set K : K ¼ A� \RedðKÞ.

Theorem 11. Let K ¼ ðU;A; IÞ be a formal context and a 2 A. Then, ais an indispensable attribute in K iff RA – RA�fag.

Proof. ð)Þ Let a be an indispensable attribute in K. Then a 2 D for allD 2 RedðKÞ. If RA ¼ RA�fag, i.e. RA�fag \ Ra ¼ RA�fag, then RA�fag # Ra. ByTheorem 5, we get JA�fag# Ja. Hence we have JA�fag ¼ JA, which impliesx�ðA�fagÞ�ðA�fagÞ ¼ x�� for all x 2 U. Therefore, A� fag is a granular consis-tent set of K, and there must exists a reduct D of K such thatD # A� fag, which contradicts the assumption that a is an indis-pensable attribute. Consequently, we have RA – RA�fag.ð(Þ If a is not an indispensable attribute, then there exists

D # A� fag such that D is a reduct of K. Using Theorem 10, we obtainRD ¼ RA. On the other hand, since D # A� fag# A, thenRA # RA�fag# RD. Hence, we have RA ¼ RA�fag, which contradicts theassumption RA – RA�fag. Therefore, a is an indispensable attribute inK. h




Corollary 5. Let K ¼ ðU;A; IÞ be a formal context and a 2 A. Then, a isa dispensable attribute in K iff RA ¼ RA�fag.

Proof. It can easily be derived from Theorem 11. h

g;

Example 6. In Example 1, by computing we have

RA ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx3; x3Þ; ðx4; x4Þ; ðx5; x2Þ; ðx5; x3Þ; ðx5; x5ÞRA�feg ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx3; x3Þ; ðx4; x4Þ; ðx5; x2Þ;

ðx5; x3Þ; ðx5; x5Þg;RA�fbg ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx3; x2Þ; ðx3; x3Þ; ðx3; x5Þ;

ðx4; x4Þ; ðx5; x2Þ; ðx5; x3Þ; ðx5; x5Þg:

Since RA�fbg – RA and RA�feg ¼ RA, using Theorem 11 and Corollary 5we conclude that b is an indispensable attribute and e is a dispens-able attribute in K.

Definition 5. Let K ¼ ðU;A; IÞ be a formal context, and x; y 2 U. Wedefine

Dðx; yÞ ¼ fa 2 A j f ðx; aÞ > f ðy; aÞg:

Dðx; yÞ is referred to as the discernibility attribute set with respectto the dominance relation RA, and D ¼ ðDðx; yÞ : x; y 2 UÞ is calledthe discernibility matrix with respect to the dominance relation RA.

We denote

D0 ¼ fDðx; yÞj Dðx; yÞ – 0 ðx; y 2 UÞg:

Theorem 12. Let K ¼ ðU;A; IÞ be a formal context and C # A. Then, Cis a granular consistent set iff C \ Dðx; yÞ– ; for all Dðx; yÞ 2 D0.

Proof. ð)Þ Let C be a granular consistent set. Using Theorem 10we have RC ¼ RA. For any Dðx; yÞ 2 D0, by Definition 5 we getðx; yÞ R RA. Hence ðx; yÞ R RC , i.e. there exists c 2 C such thatf ðx; cÞ > f ðy; cÞ. It is obvious that c 2 Dðx; yÞ. Therefore,C \ Dðx; yÞ – ; for all Dðx; yÞ 2 D0.ð(Þ Let C \ Dðx; yÞ– ; for all Dðx; yÞ 2 D0. For any x; y 2 U, if

ðx; yÞ R RA, i.e. there exists a 2 A such that f ðx; aÞ > f ðy; aÞ, we haveDðx; yÞ– 0, hence C \ Dðx; yÞ – ;. Thus, there exists c 2 C such thatf ðx; cÞ > f ðy; cÞ, which means ðx; yÞ R RC . Therefore, we can con-clude that RC # RA. It should be noted that RA # RC is a routine,combining the previous proof we obtain RA ¼ RC . This proves that Cis a granular consistent set of K. h.

Table 3The discernibility matrix M.

x=y x1 x2 x3 x4 x5

x1 ; bd d ; bd

Theorem 13. Let K ¼ ðU;A; IÞ be a formal context and a 2 A. Then,a is an indispensable attribute in K iff there exists ðx; yÞ 2 U � Usuch that Dðx; yÞ ¼ fag.

x2 ace ; c ae cx3 ae b ; ae bx4 c bd cd ; bcdx5 ae ; ; ae ;

Table 4A formal decision context ðU;A; I;D; JÞ.

a b c d e f g h

x1 0 1 0 1 0 1 0 0x2 1 0 1 0 1 0 1 1x3 1 1 0 0 1 0 1 0x4 0 1 1 1 0 1 0 0x5 1 0 0 0 1 0 1 0

Proof. ð)Þ Suppose a is an indispensable attribute (core) in K.Using Theorem 11 we get RðA�fagÞ – RA. Hence RA � RðA�fagÞ. Namely,there exists ðx; yÞ 2 U � U such that ðx; yÞ 2 RðA�fagÞ and ðx; yÞ R RA,which means f ðx; bÞ 6 f ðy; bÞ ð8 b 2 A� fagÞ and f ðx; aÞ > f ðy; aÞ.By Definition 5 we have Dðx; yÞ ¼ fag.ð(Þ If there exists ðx; yÞ 2 U � U such that Dðx; yÞ ¼ fag, then, by

Definition 5 we obtain f ðx; bÞ 6 f ðy; bÞ ð8 b 2 A� fagÞ andf ðx; aÞ > f ðy; aÞ. Thus, RA�fag – RA. Therefore, a is an indispensableattribute in K. �

LetW

Dðx; yÞ be a Boolean expression which is equal to 1, ifWDðx; yÞ ¼ ;. Otherwise,

WDðx; yÞ is a disjunction of variables cor-

responding to attributes contained in Dðx; yÞ.


Let M ¼Vðx; yÞ2 U�U

WDðx; yÞ;MðxÞ ¼

Vy2 U

WDðx; yÞ. M is called

the discernibility function for formal context K, and MðxÞ is calledthe discernibility function for object x. Where, a relative reductfor objects x is a minimal attribute subset that distinguishes objectconcept ð½x�RA

; ð½x�RAÞ�Þ from other object concepts. Discernibility

functions are monotonic Boolean functions and their prime impli-cations uniquely determine the reducts.

Example 7. In Example 1, the discernibility matrix with respect todominance relation RA is represented as Table 3. From Table 3,using discernibility function we have:

M ¼^

ðx; yÞ2 U�U

_Dðx; yÞ

¼ b ^ c ^ d ^ ða _ eÞ ^ ðb _ dÞ ^ ðc _ dÞ ^ ða _ c _ eÞ ^ ðb _ c _ dÞ¼ b ^ c ^ d ^ ða _ eÞ ¼ ða ^ b ^ c ^ dÞ _ ðb ^ c ^ d ^ eÞ

Mðx1Þ ¼ d ^ ðb _ dÞ ¼ d;

Mðx2Þ ¼ c ^ ða _ eÞ ^ ða _ c _ eÞ ¼ ða ^ cÞ _ ðc ^ eÞ;

Mðx3Þ ¼ b ^ ða _ eÞ ¼ ða ^ bÞ _ ðb ^ eÞ;

Mðx4Þ ¼ c ^ ðb _ dÞ ^ ðc _ dÞ ^ ðb _ c _ dÞ ¼ ðb ^ cÞ _ ðc ^ dÞ;

Mðx5Þ ¼ a _ e:

Therefore, fa; b; c;dg and fb; c; d; eg are two reducts of K; b; c and dare indispensable attributes.

Object x1 has one relative reduct fdg; object x2 has two relativereducts fa; cg and fc; eg; object x3 has two relative reducts fa; bgand fb; eg; object x4 has two relative reducts fb; cg and fc; dg; andobject x5 has two relative reducts fag and feg.

In Example 7, we obtain the same reduction results as Example5 in [57].

The algorithm of knowledge reduction based on dominancerelation is described as Algorithm 1. The running time of line 1and line 2 is jUj2, the running time of line 4 is jAj. Then, the runningtime from line 1 to line 10 is jAjjUj2. The maximum running time ofline 17 is 2jAj. Thus, the maximum time complexity of Algorithm 1is t ¼ 2jAj þ jAjjUj2. But, the maximum time complexity of the algo-rithm of knowledge reduction presented in [57] ist ¼ 2jAj þ jAjjUj2 þ jAjjUj. The algorithm of relative reduction ofobject xi is described as Algorithm 2. The running time from line1 to line 8 is jAjjUj, and the maximum running time of line 13 is2jAj. Then, the maximum time complexity of Algorithm 2 ist ¼ 2jAj þ jAjjUj.




Algorithm 1

Fig. 2. LðU;D; JÞ.

Algorithm 2.

6. Knowledge reduction in consistent formal decision contexts

Similarly, in this section, we consider attribute reduction andattribute characteristics of formal decision contexts from the per-spective of dominance relation, and discuss the relationshipbetween RS reduction and FCA reduction. Let us first recall somedefinitions.

Definition 6 [53]. A formal decision context is a quintupleS ¼ ðU;A; I;D; JÞ, where ðU;A; IÞ and ðU;D; JÞ are formal contexts,A \ D ¼ ;;C and D are the conditional attribute set and decisionattribute set, respectively.

Let ðU;A; I;D; JÞ be a formal decision context, X # U and B # A. Toavoid confusion, the operator � in contexts ðU;A; IÞ and ðU;D; JÞ aredenoted as �B and �D respectively, and are defined by

X�B ¼ fa 2 Bj8x 2 X; ðx; aÞ 2 Ig;


X�D ¼ fa 2 Dj8x 2 X; ðx; aÞ 2 Jg:

Example 8. For illustration, we adopt the formal decision contextðU;A; I;D; JÞ in [57], which is depicted by Table 4. Fig. 2 is the Hassediagram of concept lattice LðU;D; JÞ.

Definition 7 [53]. Let S ¼ ðU;A; I;D; JÞ be a formal decision con-text. S is said to be consistent if x�A�A # x�D�D for all x 2 U. Otherwise,it is said to be inconsistent.

Definition 8 [53]. Let S ¼ ðU;A; I;D; JÞ be a consistent formal deci-sion context and C # A. If x�C�C # x�D�D for all x 2 U, then C is referredto as a granular consistent set of S. If C is a granular consistent setof S and no proper subset of C is a granular consistent set of S, thenC is referred to as a granular reduct of S. The intersection of allgranular reducts of S is referred to as a granular core of S.

Similarly, a formal decision context S ¼ ðU;A; I;D; JÞ can also betaken as a decision information system, where, A is referred to ascondition attribute set and D is referred to as decision attribute set.

Similar to the definition of dominance relation RB, the domi-nance relation RD with respect to decision attributes is defined by

RD ¼ fðx; yÞ 2 U � Ujf ðx;dÞ 6 f ðy;dÞ;8d 2 Dg:

Definition 9. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decisioncontext and C # A. An attribute subset C # A is called a dominanceconsistent set (based on dominance relation) of S if RC # RD.Furthermore, if RC�fcg – RD for all c 2 C, then C is called adominance reduct (based on dominance relation) of S.

From Definition 8 and Theorem 5, we obtain that the formaldecision context S ¼ ðU;A; I;D; JÞ is consistent iff RA # RD.

Theorem 14. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decisioncontext and C # A. Then, C is a granular consistent set of S iff RC # RD.

Proof. It should be noted that:

C is a granular consistent set () x�C�C # x�D�D ;8 x

2 U () JCðxÞ# JDðxÞ;8 x 2 U () RCðxÞ# RDðxÞ;8 x

2 U () RC # RD;

which proves the theorem. h

Corollary 6. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decision con-text and D # A. Then, D is a granular reduct of S iff D is a dominancereduct of S.



Table 5The discernibility matrix MS .


Proof. It can easily be proved from Definition 9 andTheorem 14. h

x=y x1 x2 x3 x4 x5

x1 ; bd d ; bdx2 ace ; c ae cx3 ae ; ; ae ;x4 ; bd cd ; bcd

Theorem 15. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decisioncontext and a 2 A. Then, a is an indispensable attribute (core) in S

iff RA�fag� RD.
x5 ae ; ; ae ;

Corollary 7. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decision con-text and a 2 A. Then, a is a dispensable attribute in S iff RA�fag # RD.

Proof. It can easily be derived from Theorem 15. h

Fig. 3. LðU;C1; IC1 Þ.

Fig. 4. LðU;C2; IC2 Þ.

Example 9. In Example 8, by computing we have

RD ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx3; x2Þ; ðx3; x3Þ; ðx3; x5Þ; ðx4; x1Þ;ðx4; x4Þ; ðx5; x2Þ; ðx5; x3Þ; ðx5; x5Þg;

RA�fbg ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx3; x2Þ; ðx3; x3Þ; ðx3; x5Þ; ðx4; x4Þ;ðx5; x2Þ; ðx5; x3Þ; ðx5; x5Þg;

RA�fcg ¼ fðx1; x1Þ; ðx1; x4Þ; ðx2; x2Þ; ðx2; x3Þ; ðx2; x5Þ; ðx3; x3Þ; ðx4; x1Þ;ðx4; x4Þ; ðx5; x2Þ; ðx5; x3Þ; ðx5; x5Þg:

Since RA�fbg # RD and RA�fcg � RD, using Theorem 15 and Corollary 7we conclude that c is an indispensable attribute and b is a dispens-able attribute in S.

Let S ¼ ðU;A; I;D; JÞ be a consistent formal decision context, andx; y 2 U. We denote

DSðx; yÞ ¼fa 2 A : f ðx; aÞ > f ðy; aÞg; f ðx;dÞ > f ðy; dÞ ð9d 2 DÞ;;; f ðx;dÞ 6 f ðy;dÞ ð8d 2 DÞ:

�

DSðx; yÞ is referred to as the discernibility attribute set of x and ywith respect to dominance relations RA and RD, andDS ¼ ðDSðx; yÞ : x; y 2 UÞ is referred to as the discernibility matrixof S with respect to dominance relations RA and RD.

We denote

DS0 ¼ fD

Sðx; yÞj DSðx; yÞ – 0 ðx; y 2 UÞg:

Theorem 16. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decisioncontext and C # A. Then, C is a granular consistent set iffC \ DSðx; yÞ – ; for all DSðx; yÞ 2 DS

0.


Theorem 17. Let S ¼ ðU;A; I;D; JÞ be a consistent formal decisioncontext and a 2 A. Then, a is an indispensable attribute in S iff thereexists ðx; yÞ 2 U � U such that DSðx; yÞ ¼ fag.

Proof. It is similar to the proof of Theorem 13. �

Let MS ¼

Vðx; yÞ2 U�U

WDSðx; yÞ;MSðxÞ ¼

Vy2U

WDSðx; yÞ. M

S iscalled the discernibility function for the formal decision contextS, and MSðxÞ is called the discernibility function for object x. Theirprime implications uniquely determine all reducts.

Example 10. In Example 8, the discernibility matrix with respectto the formal decision context S is represented as Table 5. FromTable 5, using discernibility function we have:


MS ¼

^ðx; yÞ2 U�U

_DSðx; yÞ

¼ c ^ d ^ ða _ eÞ ^ ðb _ dÞ ^ ða _ c _ eÞ ^ ðb _ c _ dÞ¼ c ^ d ^ ða _ eÞ ¼ ða ^ c ^ dÞ _ ðc ^ d ^ eÞ

MSðx1Þ ¼ d ^ ðb _ dÞ ¼ d;

MSðx2Þ ¼ c ^ ða _ eÞ ^ ða _ c _ eÞ ¼ ða ^ cÞ _ ðc ^ eÞ;M

Sðx3Þ ¼ a _ e;

MSðx4Þ ¼ ðb _ dÞ ^ ðc _ dÞ ^ ðb _ c _ dÞ ¼ d _ ðb ^ cÞ;M

Sðx5Þ ¼ a _ e:

Therefore, C1 ¼ fa; c;dg and C2 ¼ fc;d; eg are two reducts of S; c andd are indispensable attributes. Figs. 3 and 4 are the Hasse diagramsof concept lattices LðU;C1; IC1 Þ and LðU;C2; IC2 Þ.

Object x1 has one relative reduct fdg; object x2 has two relativereducts fa; cg and fc; eg; object x3 has two relative reducts fag andfeg; object x4 has two relative reducts fdg and fb; cg; and object x5

has two relative reducts fag and feg.

In Example 10, we obtain the same reduction results as Example7 in [57].

The algorithm of knowledge reduction of formal decision con-texts based on dominance relation is described as Algorithm 3,



Table 6Documents and terms.

Documents (n ¼ 7) Terms (m ¼ 9)

D1 = Infant and Toddler First Aid Baby

D2 = Babies and Children’s Room (For Your Home) Child

D3 = Child Safety at Home Guide

D4 = Your Baby’s Health and Safety: From Infant to Toddler Health

D5 = Baby Proofing Basics Home


and its maximum time complexity is t ¼ 2jCj þ jCjjUj2. But, themaximum time complexity of the algorithm of knowledge reduc-tion in formal decision contexts presented in [57] ist ¼ 2jCj þ jCjjUj2 þ jCjjUj. The algorithm of relative reduction ofobject xi in formal decision contexts is described as Algorithm 4,and its maximum time complexity is t ¼ 2jCj þ jCjjUj.

Algorithm 3.

D6 = Your Guide to Easy Rust Proofing Infant

D7 = Beanie Babies Collector’s Guide Proofing

SafetyToddler

Table 7Formal context for the document-terms.

Doc/term

Baby Child Guide Health Home Infant Proofing Safety Toddler

D1 0 0 0 0 0 1 0 0 1D2 1 1 0 0 1 0 0 0 0D3 0 1 0 0 1 0 0 1 0D4 1 0 0 1 0 1 0 1 1D5 1 0 0 0 0 0 1 0 0D6 0 0 1 0 0 0 1 0 0D7 1 0 1 0 0 0 0 0 0

Algorithm 4.

Table 8The discernibility matrix M.

D1 D2 D3 D4 D5 D6 D7

D1 ; IT IT ; IT IT ITD2 BCO ; B CO CO BCO COD3 COS S ; CO COS CO COD4 BHS HIST BHIT ; HIST BHIST HISTD5 BP P BP P ; B PD6 GP GP GP GP G ; PD7 BG G BG G G B ;

7. Applications

In this section we examine application of the proposed attributereduction method in application domains of information retrieval(IR) and UCI machine learning.


Example 11. We take a standard information retrieval exampleavailable in Berry and Brown [5] to demonstrate the application ofproposed method for formal context reduction. The followingTable 6 shows the documents and corresponding key terms. Thedocuments are a collection of seven book titles. Table 7 shows theformal context for these documents and terms. In IR applicationsthe documents serve as objects and the index terms serve asattributes.

From Table 7, the discernibility matrix is represented as Table 8,where, term Baby, Child, Guide, Health, Home, Infant, Proofing,Safety and Toddler are denoted as B;C;G;H;O; I; P, S and Trespectively. From Table 8, using discernibility function we have:

M ¼ B ^ G ^ P ^ S ^ ðI _ TÞ ^ ðC _ OÞ¼ ðB ^ G ^ P ^ S ^ I ^ CÞ _ ðB ^ G ^ P ^ S ^ I ^ OÞ_ ðB ^ G ^ P ^ S ^ T ^ CÞ _ ðB ^ G ^ P ^ S ^ T ^ OÞ:

Therefore, fB;G; P; S; I;Cg; fB;G; P; S; I;Og; fB;G; P; S; T;Cg andfB;G; P; S; T;Og are four reducts of the document-terms context;

B;G; P and S are indispensable attributes.

The number of columns in the document-term matrix isreduced from nine to six there by making the size of the docu-ment-term matrix as 7� 6. One of reduced formal context isshown in Table 9 and we can observe that the reduced formal con-text is quite smaller than original one.



Fig. 5. Concept lattice of the context presented in Table 7.

Table 9A reduced formal context of the document-terms.

Doc/term Baby Guide Home Infant Proofing Safety

D1 0 0 0 1 0 0D2 1 0 1 0 0 0D3 0 0 1 0 0 1D4 1 0 0 1 0 1D5 1 0 0 0 1 0D6 0 1 0 0 1 0D7 1 1 0 0 0 0

Table 10The number of calculations of attribute reduction.

No. Datasets Objects Attribute Shao Wu Zhang

1 House 435 17 3,282,361 3,289,756 1.7592e+142 Vehicle 846 19 13,860,748 13,876,822 3.5135e+153 Lymph 148 19 678,320 681,132 3.2908e+124 Credit-a 690 16 7,650,368 7,661,408 9.2845e+145 Diabetes 768 9 5,308,672 5,315,584 2.5361e+146 Australian 690 15 7,157,884 7,168,234 7.6502e+147 Vote 435 17 3,282,361 3,289,756 1.7592e+148 Hepatitis 155 20 1,004,788 1,007,888 4.6176e+129 Autos 205 26 34,647,082 34,652,412 3.1041e+13


Compared with Wu et al.’s [57] method, the proposed methodavoid computing all (Di)⁄ (i 2 f1; . . . 7gÞ. In Zhang et al.’s [66]method, all formal concepts needs to be calculated in advance(Fig. 5 presents the concept lattice constructed from the contextpresented in Table 7) and the computational cost is very high (itstime complexity is t ¼ 2jCj þ jCj3jUj4). Compared with Zhanget al.’s [66] method, the proposed method avoid computing all for-mal concepts and the computational cost of discernibility matrix isgreatly reduced.

Example 12. This example illustrates the comparison of compu-tational efficiency of proposed method and the current methods onsome real-life data sets. Nine real-life data sets available from theUCL Repository of Machine Learning Database at University OfCalifornia [8] are used. The characteristics of the data sets aresummarized by the first four columns of Table 10. In Table 10,‘‘Shao’’ is the proposed method, ‘‘Wu’’ and ‘‘Zhang’’ refer to Wu’smethod and Zhang’s method respectively. The number of calcula-tions of attribute reduction are represented form the fifth columnto the seventh column in Table 10.

The two examples show that the proposed method works fasteron such data sets than the others do. Moreover, the proposedmethod is more readable and easier to implement.


8. Conclusion

In this paper, the relations between granular reduts and domi-nance reducts in formal contexts has been established. The follow-ing meaningful conclusions have been obtained:

1. For any object, its extension of object concept is identical withits dominance class.

2. A dominance consistent set (reduts) is also a granular consis-tent set (reduts), and vice versa.

3. Attribute characteristics in RS is identical to attribute character-istics in FCA.

Using the above assertions one can get all granular reducts andattribute characteristics via dominance reducts and attribute char-acteristics. We believe that the proposed approaches we offer herewill turn out to be more useful for practical applications of formalcontext reduction.

Compared with [43,49,53], this paper has more general conclu-sions and meaningful assertions. Conclusion 1 above does not holdin literatures [43,49]. Investigation in [49,53] does not discuss therelation between rough set attribute characteristics and conceptlattice attribute characteristics, and the reverse of conclusion 2above does not hold in general. One main difference between theprevious approaches and the one proposed in this paper is thechoice of binary relation. Dominance relation is considered in thispaper instead of equivalence relation. The derived results may pro-vide a meaningful bridge between FCA and RS, and will help us togain much more insights into the two theories. The relationbetween rough set attribute reduction and concept lattice attributereduction based on fuzzy formal contexts and interval-valued for-mal contexts should be an issue for further research.

Acknowledgements

The authors wish to thank both referees and editor for theirinvaluable advice in revising this paper. This research was sup-ported by the Geographical Modeling and Geocomputation Pro-gram under the focused investment Scheme of The ChineseUniversity of Hong Kong. This work was supported by grants fromthe National Natural Science Foundation of China (Nos. 60963006,61173181, and 61363056), the Humanities and Social Sciencefunds Project of Ministry of Education of China (Nos. 09YJCZH082and 11XJJAZH001), and the Science and Technology Project ofQingdao (No. 12-1-4-4-(9)-jch).

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