knowledge reduction in formal decision contexts based on an order-preserving mapping
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This article was downloaded by: [UQ Library]On: 18 November 2014, At: 14:40Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of GeneralSystemsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/ggen20
Knowledge reduction in formal decisioncontexts based on an order-preservingmappingJinhai Li a , Changlin Mei a & Yuejin Lv ba School of Science, Xi'an Jiaotong University , Xi'an , Shaanxi ,710049 , P.R. Chinab School of Mathematics and Information Sciences, GuangxiUniversity , Nanning , Guangxi , 530004 , P.R. ChinaPublished online: 18 Nov 2011.
To cite this article: Jinhai Li , Changlin Mei & Yuejin Lv (2012) Knowledge reduction in formaldecision contexts based on an order-preserving mapping, International Journal of General Systems,41:2, 143-161, DOI: 10.1080/03081079.2011.634410
To link to this article: http://dx.doi.org/10.1080/03081079.2011.634410
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Knowledge reduction in formal decision contexts based on anorder-preserving mapping
Jinhai Lia*, Changlin Meia and Yuejin Lvb
aSchool of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China; bSchool ofMathematics and Information Sciences, Guangxi University, Nanning, Guangxi 530004, P.R. China
(Received 28 May 2010; final version received 19 October 2011)
Knowledge reduction is one of the basic issues in knowledge presentation and datamining. In this study, an order-preserving mapping between the set of all the extensionsof the conditional concept lattice and that of the decision concept lattice is defined toclassify formal decision contexts into consistent and inconsistent categories. Then,methods of knowledge reduction for both the consistent and the inconsistent formaldecision contexts are formulated by constructing proper discernibility matrices andtheir associated Boolean functions. For the consistent formal decision contexts, theproposed reduction method can avoid redundancy subject to maintaining consistency,while for the inconsistent formal decision contexts, the reduction method can make theset of all the compact non-redundant decision rules complete in the initial formaldecision context.
Keywords: formal concept analysis; formal context; formal decision context; conceptlattice; knowledge reduction; order-preserving mapping
1. Introduction
Formal concept analysis (FCA), proposed by Wille (1982), is an effective mathematical
theory of data analysis using formal contexts and concept lattices. In FCA, the data are
described by a formal context (Wille 1982) or a formal decision context (Zhang and Qiu
2005). A basic notion in this theory is formal concept and the set of all the formal concepts
of a formal context forms a complete lattice, called a concept lattice (Wille 1982), to
reflect the relationship between generalization and specialization among the formal
concepts. FCA has been applied extensively in information retrieval (Cole et al. 2003,
Carpineto and Romano 2004), machine learning (Carpineto and Romano 1993, 1996),
knowledge discovery (Stumme et al. 1998, Bastide et al. 2000, Valtchev et al. 2004,
Zaki 2004), and many other aspects (Godin et al. 1995, Ho 1995, Nguyen and Corbett
2006, Zhang et al. 2007, Wang and Zhang 2008a, 2008b, Arevalo et al. 2009,
Belohlavek et al. 2009).
In FCA, much attention has been paid to the issue of knowledge reduction.
For instance, Ganter and Wille (1999) proposed a knowledge reduction method by
removing the reducible objects and attributes of a formal context. Elloumi et al. (2004) put
forward a multilevel reduction approach to reduce the size of the initial fuzzy context
under the condition that the association rules extracted from reduced databases are
identical at the given precision level. Zhang et al. (2005) presented a knowledge reduction
ISSN 0308-1079 print/ISSN 1563-5104 online
q 2012 Taylor & Francis
http://dx.doi.org/10.1080/03081079.2011.634410
http://www.tandfonline.com
*Corresponding author. Email: [email protected]
International Journal of General Systems
Vol. 41, No. 2, February 2012, 143–161
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method to avoid redundancy in the representation of a formal context while maintaining
hierarchy structure of the concept lattice of the initial context. From the point of view of
rough set theory, Liu et al. (2007) discussed the problem of knowledge reduction in formal
contexts. In addition, methods of knowledge reduction for formal decision contexts were
also explored in recent years. For example, Wang and Zhang (2008a) developed a method
to generate such kinds of reducts from a consistent formal decision context that can make
each image in the decision concept lattice have at least one preimage in the conditional
concept lattice. Wei et al. (2008) investigated the issue of knowledge reduction in
consistent formal decision contexts under two partial order relations between concept
lattices. Wu et al. (2009) and Li et al. (2011) discussed knowledge reduction in consistent
formal decision contexts from the perspectives of granular computing and implication rule
extraction, respectively.
In the existing knowledge reduction methods in FCA, formal decision contexts are
generally classified into consistent and inconsistent categories by constructing a proper
partial order between the conditional concept lattice and the decision concept lattice, and
the consequent reduction approaches only focus on the consistent formal decision
contexts. In general, however, an inconsistent formal decision context appears more
frequently than a consistent one no matter how a partial order between the conditional
concept lattice and the decision concept lattice is constructed. In this paper, by defining an
order-preserving mapping between the set of all the extensions of the conditional concept
lattice and that of the decision concept lattice, we still divide formal decision contexts into
consistent and inconsistent formal decision contexts. However, unlike the existing
literature in FCA, we not only propose a knowledge reduction method for the consistent
formal decision contexts, but also investigate the issue of knowledge reduction in the
inconsistent formal decision contexts. For the consistent formal decision contexts, the
proposed reduction method can avoid redundancy and, at the same time, can maintain the
consistency, whereas for the inconsistent formal decision contexts, the reduction method
can make the set of all the compact non-redundant decision rules complete in the initial
formal decision context.
This paper is organized as follows. In the next section, we briefly review preliminary
definitions to be used throughout the remainder of the paper. In Section 3, an order-
preserving mapping between the set of all the extensions of the conditional concept lattice
and that of the decision concept lattice is defined to classify formal decision contexts into
consistent and inconsistent categories. We then introduce the notions of a reduct and a
consistent mapping reduct in the consistent formal decision contexts, and propose an
approach of computing consistent mapping reducts by constructing a proper discernibility
matrix and its associated Boolean function. Furthermore, we prove that for a consistent
formal decision context, all of its reducts can be obtained based on its consistent mapping
reducts. In Section 4, we put forward a knowledge reduction method for the inconsistent
formal decision contexts. Knowledge discovery in formal decision contexts is investigated
in Section 5. Finally, the paper is concluded with a brief summary and an outlook for
further research.
2. Preliminaries
To make the paper self-contained, we briefly review in this section some basic notions and
results related to FCA.
A formal context is a triple ðU;A; IÞ, where U ¼ {x1; x2; . . . ; xn}, called the universe
of discourse, is a non-empty and finite set of objects, A ¼ {a1; a2; . . . ; am} is a non-empty
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and finite set of attributes, and I is a binary relation on U £ A with (x,a) [ I denoting that
the object x has the attribute a.
Let K ¼ (U, A, I) be a formal context. For X # U and B # A, two operators are defined
as follows (Wille 1982):
X * ¼ {a [ Aj;x [ X; ðx; aÞ [ I};
BS ¼ {x [ Uj;a [ B; ðx; aÞ [ I}:
That is, X* is the maximal family of the attributes that all the objects in X have in common,
and BS is the maximal family of the objects shared by all the attributes in B.
Let P(U) and P(A) denote the power sets of U and A, respectively. It can easily be
verified that X # BS () B # X * for any ðX;BÞ [ PðUÞ £ PðAÞ. Thus, the pair ð*;SÞ of
the mappings * : PðUÞ! PðAÞ and S : PðAÞ! PðUÞ forms a Galois connection between
the posets (P(U), # ) and (P(A), # ).
Definition 2.1. (Ganter and Wille 1999). Let K ¼ ðU;A; IÞ be a formal context. A pair
ðX;BÞ with X # U, B # A, X * ¼ B and BS ¼ X is called a formal concept (or simply a
concept) of K. Here, X and B are termed as the extension and the intension of (X, B),
respectively.
Let K ¼ ðU;A; IÞ be a formal context. For two concepts ðXi;BiÞ and ðXj;BjÞ of K, if
Xi # Xjð,Bj # Bi), then ðXi;BiÞ is called a sub-concept of ðXj;BjÞ, or equivalently,
ðXj;BjÞ is called a super-concept of ðXi;BiÞ, which is denoted by ðXi;BiÞ W ðXj;BjÞ. The set
of all the concepts of K together with the partial order relation W forms a complete lattice
and it is called a concept lattice (Wille 1982) denoted by BðU;A; IÞ.In addition, the set of all the extensions of BðU;A; IÞ is denoted by UðU;A; IÞ. The meet
and join in BðU;A; IÞ are, respectively, defined by
ðX1;B1Þ ^ ðX2;B2Þ ¼ ðX1 > X2; ðB1 < B2ÞS*Þ;
ðX1;B1Þ _ ðX2;B2Þ ¼ ððX1 < X2Þ*S;B1 > B2Þ:
Definition 2.2 (Wu et al. 2009). Let K ¼ ðU;A; IÞ be a formal context. For E # A and
IE ¼ I > ðU £ EÞ, the formal context ðU;E; IEÞ is called a sub-context of K.
Let ðU;E; IEÞ be a sub-context of K ¼ ðU;A; IÞ and P(E) be the power set of E. For
X [ PðUÞ and B [ PðEÞ, two operators are defined as follows:
X *E ¼ {a [ Ej;x [ X; ðx; aÞ [ IE};
BSE ¼ {x [ Uj;a [ B; ðx; aÞ [ IE}:
In fact, the above operators *E : PðUÞ! PðEÞ and SE : PðEÞ! PðUÞ are the restriction
of the operators * : PðUÞ! PðAÞ and S : PðAÞ! PðUÞ on the sub-context ðU;E; IEÞ.
A pair ðX;BÞ is a formal concept of ðU;E; IEÞ if and only if X *E ¼ B and BSE ¼ X. We
denote the concept lattice of ðU;E; IEÞ by BðU;E; IEÞ and the set of all the extensions of
BðU;E; IEÞ by UðU;E; IEÞ. Then, similar to the case in the initial context K ¼ ðU;A; IÞ,
we have the following properties for ðU;E; IEÞ.
Proposition 2.1. Let ðU;E; IEÞ be a sub-context of K ¼ ðU;A; IÞ. For X;X1;X2 # U and
B;B1;B2 # E, the following statements hold:
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(1) X *E ¼ X * > E, BSE ¼ BS.
(2) X1 # X2 ) X*E
2 # X*E
1 ; B1 # B2 ) BSE
2 # BSE
1 .
(3) X # X *ESE ; B # BSE*E .
(4) ðX *ESE , X *E), (B SE, BSE*E ) [ B(U, E, IE).
(5) UðU;E; IEÞ # UðU;A; IÞ.
Definition 2.3. (Hungerford 1974). Let ðM;#Þ and ðN;#Þ be two posets. The mapping
h : M ! N is called an order-preserving mapping if for any x; y [ M,
x # y ) hðxÞ # hðyÞ.
Definition 2.4. (Hungerford 1974). Let ðM;#Þ be a poset and N # M. y [ N is called a
minimal element of N if for all x [ N, x # y implies x ¼ y.
3. Knowledge reduction in consistent formal decision contexts
Definition 3.1. (Zhang and Qiu 2005). A formal decision context is a quintuple
ðU;A; I;D; JÞ, where ðU;A; IÞ and ðU;D; JÞ are two formal contexts, and A and D are,
respectively, called the conditional attribute set and the decision attribute set with
A > D ¼ Y.
For a formal decision context F ¼ ðU;A; I;D; JÞ, if ðU;E; IEÞ is a sub-context of
ðU;A; IÞ, we say that ðU;E; IE;D; JÞ is a formal decision sub-context (or simply a sub-
context) of F.
Let F ¼ ðU;A; I;D; JÞ be a formal decision context. According to the definition in
Section 2, UðU;A; IÞ is the set of all the extensions of BðU;A; IÞ, i.e.
UðU;A; IÞ ¼ {XjðX;BÞ [ BðU;A; IÞ}:
In order to facilitate our subsequent discussion, we denote by
UðU;D; JÞ ¼ {YjðY ;CÞ [ BðU;D; JÞ}
the set of all the extensions of BðU;D; JÞ, where BðU;D; JÞ is the concept lattice of the
formal context ðU;D; JÞ. It can easily be observed that UðU;A; IÞ and UðU;D; JÞ are
two subsets of the power set of U. Noting that UðU;D; JÞ preserves the > -operator
(Ganter and Wille 1999), we have >t[T Yt [ UðU;D; JÞ for Yt [ UðU;D; JÞ (t [ T),
where T is an index set.
For a formal decision context F ¼ ðU;A; I;D; JÞ, it can be known from the above
discussion that ðUðU;A; IÞ;#Þ and ðUðU;D; JÞ;#Þ are both posets. We define a mapping
f : UðU;A; IÞ! UðU;D; JÞ as follows:
f ðXÞ ¼\
X#YY[UðU;D;JÞ
Y; X [ UðU;A; IÞ:
Based on Definition 2.3, it can easily be verified that f : UðU;A; IÞ! UðU;D; JÞ is an
order-preserving mapping between ðUðU;A; IÞ;#Þ and ðUðU;D; JÞ;#Þ. Now, we use the
mapping f to classify formal decision contexts into consistent and inconsistent formal
decision contexts by the definition to follow.
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Definition 3.2. Let F ¼ ðU;A; I;D; JÞ be a formal decision context. If
f ðUðU;A; IÞÞ ¼ UðU;D; JÞ, then F is said to be consistent; otherwise, F is said to be
inconsistent.
Definition 3.3. Let F ¼ ðU;A; I;D; JÞ be a consistent formal decision context and E # A.
If f(U(U,E,IE)) ¼ U(U,D,J), then E is called a consistent set of F. Furthermore, if E is a
consistent set and there is no proper subset H , E such that H is a consistent set of F, then
E is called a reduct of F. The intersection of all the reducts of F is called the core of F.
It is perhaps interesting to mention that the notion of reduct introduced in Definition
3.3 has something to do with that of the minimal generator (Bastide et al. 2000, Hamrouni
et al. 2008) or the free set (Boulicaut et al. 2000, 2003). Specifically, both of them can
abstractly be described as ‘a minimal set under some given mathematical properties’. By
the way, the minimal generator and the free set have been applied, respectively, in mining
non-redundant association rules (Zaki 2004, Hamrouni et al. 2008) and in studying
condensed representations for frequent sets (Calders et al. 2005).
According to Definition 3.3, a reduct E of a consistent formal decision context is such a
minimal conditional attribute subset that makes ðU;E; IE;D; JÞ consistent. Thus, for the
consistent formal decision contexts, the aim of knowledge reduction is to avoid
redundancy while maintaining consistency.
In what follows, we aim at developing an approach of computing all reducts of a
consistent formal decision context. To this end, we first introduce the notions of a
consistent mapping and a consistent mapping reduct, and then discuss the issue of
calculating consistent mapping reducts. Finally, we show that all the reducts of a
consistent formal decision context can be obtained based on the consistent mapping
reducts.
3.1 Consistent mapping reducts and their computation
Definition 3.4. Let F ¼ ðU;A; I;D; JÞ be a consistent formal decision context under the
order-preserving mapping f. An injective mapping w : UðU;D; JÞ! UðU;A; IÞ is called a
consistent mapping of F if f ðwðYÞÞ ¼ Y for all Y [ UðU;D; JÞ.
It can be known from Definition 3.4 that there must exist at least one consistent
mapping for any consistent formal decision context.
Proposition 3.1. For a consistent formal decision context F ¼ ðU;A; I;D; JÞ, the number
of all the consistent mappings of F isQ
Y[UðU;D;JÞjYf j, where Yf ¼ {X [UðU;A; IÞj f ðXÞ ¼ Y} and jYf j denotes the cardinality of Yf.
Proof. The proof is immediate from Definition 3.4. A
Definition 3.5. For a consistent formal decision context F ¼ ðU;A; I;D; JÞ, let w be a
consistent mapping of F. E # A is called a w-consistent set of F if w(U(U,D,J)) #U(U, E, IE). Furthermore, if E is a w-consistent set and there is no proper subset H , E
such that H is a w-consistent set of F, then E is called a consistent mapping w-reduct
(or simply w-reduct) of F.
As well known in rough set theory, reduct computation can be translated into the
calculation of the prime implicants of a Boolean function (Skowron and Rauszer 1992,
Skowron 1993). We now employ this approach to compute consistent mapping reducts in
the consistent formal decision contexts.
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Definition 3.6. Let F ¼ ðU;A; I;D; JÞ be a consistent formal decision context and w be a
consistent mapping of F. For (Xi, Bi), (Xj, Bj) [ B(U, A, I), define
DwððXi;BiÞ; ðXj;BjÞÞ ¼Bi 2 Bj; if Xi [ wðUðU;D; JÞÞ and Xi , Xj;
Y; otherwise;
(
where Bi 2 Bj denotes the set difference of Bi and Bj. We call D w((Xi, Bi), (Xj, Bj)) the
w-discernibility attribute set of ðXi;BiÞ and ðXj;BjÞ in F, and
Dw ¼ {DwððXi;BiÞ; ðXj;BjÞÞjðXi;BiÞ; ðXj;BjÞ [ BðU;A; IÞ}
the w-discernibility matrix of F.
Theorem 3.1. Let F ¼ ðU;A; I;D; JÞ be a consistent formal decision context and w be a
consistent mapping of F. Then, E # A is a w-consistent set of F if and only if E > D w((Xi,
Bi), (Xj, Bj)) – Y for any non-empty set D w((Xi, Bi), (Xj, Bj)) [ Dw.
Proof ( ) ). For any non-empty set D w((Xi, Bi), (Xj, Bj)) [ Dw, it can be known from
Definition 3.6 that Xi [ wðUðU;D; JÞÞ and Xi , Xj. Since E # A is a w-consistent set of F,
we have w(U(U, D,J)) # U(U, E, IE) according to Definition 3.5 and consequently
Xi [ U(U, E, IE). Thus,
ðBi > EÞS ¼ ðX*i > EÞS ¼ X*ESE
i ¼ Xi , Xj ¼ BSj # ðBj > EÞS;
yielding Bi > E – Bj > E. Based on Xi , Xj and Bi >E – Bj >E, we have
Bj > E , Bi > E. Hence,
E > DwððXi;BiÞ; ðXj;BjÞÞ ¼ ðBi 2 BjÞ> E ¼ Bi > B,j > E ¼ ðBi > EÞ> B,
j
¼ ðBi > EÞ> ðB,j < E,Þ ¼ ðBi > EÞ> ðBj > EÞ,
¼ ðBi > EÞ2 ðBj > EÞ – Y;
where B,j and E, are the complements of Bj and E in A, respectively.
( ( ) Since E > D w((Xi, Bi), (Xj, Bj)) – Y for any D w((Xi, Bi), (Xj, Bj)) – Y, we have
E > (Bi 2 Bj) – Y, which implies Bi > E – Bj > E. In what follows, we prove w(U(U, D,
J)) # U(U, E, IE).
In fact, for any X [ wðUðU;D; JÞÞ, we have ðX;X *Þ [ BðU;A; IÞ since
wðUðU;D; JÞÞ # UðU;A; IÞ. If ðX * > EÞSE ¼ X, we have ðX;X * > EÞ [ BðU;E; IEÞ
because X *E ¼ X * > E. Suppose ðX * > EÞSE – X. Then, X , ðX * > EÞS holds due to
X ¼ X *S # ðX * > EÞS ¼ ðX * > EÞSE . Combining X [ wðUðU;D; JÞÞ with
X , ðX * > EÞS, we get D w((X, X *), ((X *>E)S, (X *>E)S*)) – Y. Moreover, according
to the conclusion that Bi > E – Bj > E for any D w((Xi, Bi), (Xj, Bj)) – Y, we obtain
X *>E – (X *>E)S*>E. However, based on
X , ðX * > EÞS ) ðX * > EÞS* # X * ) ðX * > EÞS* > E # X * > E
and
ðX * > EÞ # ðX * > EÞS* ) X * > E ¼ ðX * > EÞ> E # ðX * > EÞS* > E;
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we have X *>E ¼ (X *>E)S*>E, which is in contradiction with X *>E – (X *>E)S*>E.
Therefore, (X, X *>E) [ B(U, E, IE) and consequently X [ U(U, E, IE). A
Definition 3.7. For a consistent formal decision context F ¼ (U, A, I, D, J), let w be a
consistent mapping of F. We call
Fw ¼ ^D wððXi;BiÞ;ðXj;BjÞÞ–Y
D w ððXi ;BiÞ;ðXj ;Bj ÞÞ[D w_DwððXi;BiÞ; ðXj;BjÞÞn o
the w-discernibility function of F.
Theorem 3.2. For a consistent formal decision context F ¼ ðU;A; I;D; JÞ, let w be a
consistent mapping of F and the minimal disjunctive normal form of Fw be
Fw ¼_k
t¼1^
rt
s¼1as
!;
where^rt
s¼1 as, t # k, are all the prime implicants of Fw. Then Et ¼ {asjs # rt}, t # k,
are all the w-reducts of F.
Proof. The proof is immediate from Theorem 3.1 and the definition of the minimal
disjunctive normal form of a Boolean function (Skowron 1993). A
Theorem 3.2 provides an approach of calculating consistent mapping reducts for the
consistent formal decision contexts.
3.2 Reduct computation in the consistent formal decision contexts
Based on the method of computing the consistent mapping reducts in Section 3.1, we are
now ready to discuss how to generate all the reducts of a consistent formal decision context.
Proposition 3.2. Let F ¼ ðU;A; I;D; JÞ be a consistent formal decision context. E # A is
a consistent set of F if and only if there exists a consistent mapping w of F such that E is a
w-consistent set of F.
Proof ( ) ). It is immediate from Definitions 3.3, 3.4, and 3.5.
( ( ) If there exists a consistent mapping w of F such that E is a w-consistent set of F,
then wðUðU;D; JÞÞ # UðU;E; IEÞ and UðU;D; JÞ ¼ f ðwðUðU;D; JÞÞÞ # f ðUðU;E; IEÞÞ #UðU;D; JÞ. Thus, f ðUðU;E; IEÞÞ ¼ UðU;D; JÞ and consequently E is a consistent set
of F. A
Theorem 3.3. For a consistent formal decision context F ¼ ðU;A; I;D; JÞ and E # A, let
M ¼ <w[MF
Rw, where MF denotes the set of all the consistent mappings of F and Rw denotes
the set of all the w-reducts of F. Then, E is a reduct of F if and only if E is a minimal
element of ðM;#Þ.
Proof ( ) ). If E is a reduct of F, then it follows from Definition 3.3 that f ðUðU;E; IEÞÞ ¼
UðU;D; JÞ and f ðUðU;E 2 {e}; IE2{e}ÞÞ , UðU;D; JÞ for any e [ E. Thus, according to
Proposition 3.2, there exists w [ MF such that E is a w-reduct of F, yielding E [ M.
Assume that E is not a minimal element of ðM;#Þ. Then, based on Definition 2.4, there
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exists E0 [ M with E0 , E such that E0 is a w0-reduct of F, where w0 [ MF. From
Proposition 3.2, E0 is a consistent set of F, which is in contradiction with the assumption
that E is a reduct of F.
( ( ) Suppose that E is a minimal element of ðM;#Þ. Then, there exists w [ MF such
that E is a w-reduct of F. Based on Proposition 3.2, E is a consistent set of F. Note that E
consists of at least one reduct of F. So there must exist E0 # E such that E0 is a reduct of F.
Also there exists w0 [ MF such that E0 is a w0-reduct of F, which implies E0 [ M. Since E
is a minimal element of ðM;#Þ, we have E0 ¼ E. A
Theorem 3.3 says that for a consistent formal decision context, all of its reducts are
just the minimal elements of the set of all its consistent mapping reducts. So, the
procedure of computing all the reducts of a consistent formal decision context F ¼
ðU;A; I;D; JÞ is as follows: Firstly, compute all the consistent mappings of F according to
Proposition 3.1. Furthermore, for each of the consistent mappings, compute its
corresponding consistent mapping reducts by Theorem 3.2. Lastly, find the minimal
elements of the set of all the consistent mapping reducts. Based on Theorem 3.3, these
minimal elements are just all the reducts of F. Now, we use an example to illustrate the
implementation of this procedure.
Example 3.1. Table 1 presents a formal decision context F ¼ ðU;A; I;D; JÞ, where
U ¼ {1; 2; 3; 4; 5}, A ¼ {a; b; c; d; e; f }, and D ¼ {d1; d2; d3}. For each ðx; rÞ [ U £ A,
we use numbers 1 and 0 to denote ðx; rÞ [ I and ðx; rÞ � I, respectively, and for each
ðy; tÞ [ U £ D, the same notations are used for ðy; tÞ [ J and ðy; tÞ � J. The Hasse
diagrams of the conditional concept lattice BðU;A; IÞ and the decision concept lattice
BðU;D; JÞ are depicted in Figures 1 and 2, respectively. For brevity, a non-empty set of
objects (or attributes) in a formal concept is denoted by listing its elements in sequence.
For example, {2,4} and {b; d} are simply denoted by 24 and bd, respectively.
Based on Figures 1 and 2, we have f ðUðU;A; IÞÞ ¼ {12345; 1235; 2345; 235; 4; Y} ¼
UðU;D; JÞ. Thus, F is consistent according to Definition 3.2. It can be known from
Definition 3.4 and Proposition 3.1 that F has six consistent mappings:
w1 : {12345 7! 12345; 1235 7! 135; 2345 7! 245; 235 7! 35; 4 7! 4; Y 7! Y};
w2 : {12345 7! 12345; 1235 7! 135; 2345 7! 245; 235 7! 3; 4 7! 4; Y 7! Y};
w3 : {12345 7! 12345; 1235 7! 135; 2345 7! 245; 235 7! 5; 4 7! 4; Y 7! Y};
w4 : {12345 7! 12345; 1235 7! 135; 2345 7! 24; 235 7! 35; 4 7! 4; Y 7! Y};
w5 : {12345 7! 12345; 1235 7! 135; 2345 7! 24; 235 7! 3; 4 7! 4; Y 7! Y};
w6 : {12345 7! 12345; 1235 7! 135; 2345 7! 24; 235 7! 5; 4 7! 4; Y 7! Y}:
Table 1. A formal decision context F ¼ ðU;A; I;D; JÞ.
U a b c d e f d1 d2 d3
1 1 0 0 0 0 0 1 0 02 0 1 0 1 0 0 1 1 03 1 0 1 0 1 0 1 1 04 0 1 0 1 0 1 0 1 15 1 1 1 0 0 0 1 1 0
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Moreover, the w1-discernibility matrix of F is given in Table 2. By Theorem 3.2, we can
compute all the w1-reducts of F as follows:
Fw1 ¼ a ^ b ^ ða _ cÞ ^ c ^ ðb _ d _ f Þ ^ ðd _ f Þ ^ f ^ ðb _ c _ d _ e _ f Þ^
ða _ c _ d _ e _ f Þ ^ ðb _ d _ e _ f Þ ^ ða _ c _ e _ f Þ ^ ðd _ e _ f Þ ^ ða _ c _ eÞ
¼ a ^ b ^ c ^ f :
Similarly, we can obtain that Fw2 ¼ a ^ b ^ e ^ f , Fw3 ¼ a ^ b ^ f , Fw4 ¼ a ^ c ^ d ^ f ,
Fw5 ¼ a ^ d ^ e ^ f , and Fw6 ¼ a ^ b ^ d ^ f .
Hence, M ¼ {abcf ; abef ; abf ; acdf ; adef ; abdf } is the set of all the consistent mapping
reducts of F. Since {a; b; f }, {a; c; d; f }, and {a; d; e; f } are the minimal elements of M,
( ,abcdef)
(3,ace) (5,abc) (4,bdf )
(35,ac)
(135,a)
(24,bd)
(245,b)
(12345, )
Figure 1. BðU;A; IÞ.
( ,d1d2d3)
(235,d1d2) (4,d2d3)
(1235,d1) (2345,d2)
(12345, )
Figure 2. BðU;D; JÞ.
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it follows from Theorem 3.3 that F has three reducts: E1 ¼ {a; b; f }, E2 ¼ {a; c; d; f }, and
E3 ¼ {a; d; e; f }, and the core of F is {a; f }. Using the reduct E1 ¼ {a; b; f }, we can obtain
the reduced sub-context ðU;E1; IE1;D; JÞ which is also consistent. The Hasse diagram of
BðU;E1; IE1Þ is given in Figure 3 and the Hasse diagram of BðU;D; JÞ is the same as that in
Figure 2.
4. Knowledge reduction in the inconsistent formal decision contexts
In the previous section, we have discussed knowledge reduction in the consistent formal
decision contexts under the order-preserving mapping f defined at the beginning of
Section 3. In this section, we investigate the issue of knowledge reduction in the
inconsistent formal decision contexts.
With the same notations as those in the former sections, the definitions of a
pseudo-consistent set (we here use the word pseudo to distinguish it from the consistent set
defined in the consistent formal decision contexts) and a reduct in the inconsistent formal
decision contexts are given below.
( ,abf)
(5,ab) (4,bf)
(135,a) (245,b)
(12345, )
Figure 3. BðU;E1; IE1Þ.
Table 2. The w1-discernibility matrix Dw1 of F ¼ ðU;A; I;D; JÞ.
ðU; YÞ (135, a) (245, b) (35, ac) (24, bd) (3, ace) (5, abc) (4, bdf) ðY;AÞ
ðU; YÞ Y Y Y Y Y Y Y Y Y(135, a) a Y Y Y Y Y Y Y Y(245, b) b Y Y Y Y Y Y Y Y(35, ac) ac c Y Y Y Y Y Y Y(24, bd) Y Y Y Y Y Y Y Y Y(3, ace) Y Y Y Y Y Y Y Y Y(5, abc) Y Y Y Y Y Y Y Y Y(4, bdf) bdf Y df Y f Y Y Y YðY;AÞ abcdef bcdef acdef bdef acef bdf def ace Y
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Definition 4.1. Let F ¼ ðU;A; I;D; JÞ be an inconsistent formal decision context under
the order-preserving mapping f and E # A. If f ðX *ESE Þ ¼ f ðXÞ for all X [ UðU;A; IÞ,
then E is called a pseudo-consistent set of F. Furthermore, if E is a pseudo-consistent
set and there is no proper subset H , E such that H is a pseudo-consistent set of F,
then E is called a reduct of F. The intersection of all the reducts of F is called the
core of F.
Definition 4.2. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, a [ A is
called a dispensable attribute of F if A 2 {a} is a pseudo-consistent set of F. Otherwise, a
is called an indispensable attribute of F.
Definition 4.3. Let F ¼ ðU;A; I;D; JÞ be an inconsistent formal decision context. For
ðXi;BiÞ; ðXj;BjÞ [ BðU;A; IÞ, define
D7ððXi;BiÞ; ðXj;BjÞÞ ¼ðBi < BjÞ2 ðBi > BjÞ; if f ðXiÞ – f ðXjÞ;
Y; otherwise;
(
we call D7((Xi, Bi), (Xj, Bj)) the discernibility attribute set of ðXi;BiÞ and ðXj;BjÞ in F,
and call
D7 ¼ {D7ððXi;BiÞ; ðXj;BjÞÞjðXi;BiÞ; ðXj;BjÞ [ BðU;A; IÞ}
the discernibility matrix of F.
Proposition 4.1. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, let
ðXi;BiÞ, ðXj;BjÞ; ðXk;BkÞ [ BðU;A; IÞ. Then,
(1) D7((Xi, Bi), (Xi, Bi)) ¼ Y;
(2) D7((Xi, Bi), (Xj, Bj)) ¼ D7((Xj, Bj),(Xi, Bi));
(3) D7ððXi;BiÞ, ðXj;BjÞÞ # D7ððXi;BiÞ, ðXk;BkÞÞ< D7ððXk;BkÞ, ðXj;BjÞÞ if f ðXiÞ,
f ðXjÞ and f ðXkÞ are pairwise unequal.
Proof. The proofs of the first two items follow immediately from Definition 4.3. The
remainder is to prove the third item.
For any c [ D7((Xi, Bi),(Xj, Bj)), we have that c [ Bi and c � Bj, or c � Bi and
c [ Bj.
If c [ Bi, c � Bj, and c � Bk, then c [ D7ððXi;BiÞ, ðXk;BkÞÞ; if c [ Bi, c � Bj, and
c [ Bk, then c [ D7ððXk;BkÞ, ðXj;BjÞÞ. Thus, c [ D7((Xi, Bi), (Xk, Bk)) < D7((Xk, Bk),
(Xj, Bj)).
If c � Bi and c [ Bj, it can analogously be proved that c [ D7((Xi, Bi), (Xk,
Bk)) < D7((Xk, Bk), (Xj, Bj)). A
Theorem 4.1. Let F ¼ ðU;A; I;D; JÞ be an inconsistent formal decision context. Then
E # A is a pseudo-consistent set of F if and only if E > D7((Xi, Bi), (Xj, Bj)) – Y for any
non-empty set D7((Xi, Bi), (Xj, Bj)) [ D7.
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Proof ( ) ). For any non-empty set D7((Xi, Bi), (Xj, Bj)) [ D7, it follows from Definition
4.3 that f(Xi) – f(Xj). Note that
E > D7ððXi;BiÞ; ðXj;BjÞÞ ¼ E > ððBi < BjÞ2 ðBi > BjÞÞ ¼ E > ðBi < BjÞ> ðBi > BjÞ,
¼ E > ðBi < BjÞ> ðB,i < B,
j Þ
¼ ðE > Bi > B,j Þ< ðE > Bj > B,
i Þ
¼ ððBi > EÞ2 ðBj > EÞÞ< ððBj > EÞ2 ðBi > EÞÞ:
Thus, to prove E > D7((Xi, Bi), (Xj, Bj)) – Y, it is sufficient to show Bi > E – Bj > E. In
fact, if Bi > E ¼ Bj > E, then
X*E
i ¼ X*i > E ¼ Bi > E ¼ Bj > E ¼ X*
j > E ¼ X*E
j ;
which implies f(X*ESE
i ) ¼ f(X*ESE
j ). Since E is a pseudo-consistent set of F, we get
f(Xi) ¼ f(X*ESE
i ) ¼ f(X*ESE
j ) ¼ f(Xj), which is in contradiction with f ðXiÞ – f ðXjÞ.
( ( ) To prove that E is a pseudo-consistent set of F, it is sufficient to show f ðX *ESE Þ ¼
f ðXÞ for all X [ UðU;A; IÞ.
Suppose that there exists X0 [ UðU;A; IÞ such that f ðX*ESE
0 Þ – f ðX0Þ. Then, it can be
known from Proposition 2.1 and Definition 4.3 that D7((X0, X*0), (X*ESE
0 , X*ESE*0 )) – Y.
Since E > D7((Xi, Bi), (Xj, Bj)) – Y for any non-empty set D7((Xi, Bi), (Xj, Bj)) [ D7, we
have E > D7((X0, X*0), (X*ESE
0 , X*ESE*0 )) – Y, which implies X*
0 > E – X*ESE*0 >E.
However, based on
X*0 > E # X*
0)ðX*0 > EÞS $ X*S
0 ¼ X0
)ðX*0 > EÞS* # X*
0
)ðX*0 > EÞS* > E # X*
0 > E
and
X*0 > E # ðX*
0 > EÞS* ) X*0 > E ¼ ðX*
0 > EÞ> E # ðX*0 > EÞS* > E;
we obtain X*0 > E ¼ (X*
0 > E)S*>E, or equivalently, X*0 > E ¼ X*ESE*
0 >E, which is in
contradiction to X*0 > E – X*ESE*
0 > E. A
Proposition 4.2. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, c [ A is
an indispensable attribute of F if and only if there exists D7((Xi, Bi), (Xj, Bj)) [ D7 such
that D7((Xi, Bi), (Xj, Bj)) ¼ {c}.
Proof ( ) ). If c [ A is indispensable in F, it follows from Definition 4.2 that A 2 {c} is
not a pseudo-consistent set of F. According to Theorem 4.1, there exists a non-empty set
D7((Xi, Bi), (Xj, Bj)) [ D7 such that (A 2 {c}) > D7((Xi, Bi), (Xj, Bj)) ¼ Y, yielding
D7((Xi, Bi), (Xj, Bj)) ¼ {c}.
( ( ) If there exists D7((Xi, Bi), (Xj, Bj)) [ D7 such that D7((Xi, Bi), (Xj, Bj)) ¼ {c},
then (A 2 {c}) > D7((Xi, Bi), (Xj, Bj)) ¼ Y. Based on Theorem 4.1, we have that A 2 {c}
is not a pseudo-consistent set of F, which implies that c is indispensable in F. A
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DEFINITION 4.4. Let F ¼ ðU;A; I;D; JÞ be an inconsistent formal decision context. We call
F7 ¼^
D 7ððXi;BiÞ;ðXj;BjÞÞ–YD 7 ððXi ;BiÞ;ðXj ;BjÞÞ[D7
_D7ððXi;BiÞ; ðXj;BjÞÞn o
the discernibility function of F.
Theorem 4.2. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, let the
minimal disjunctive normal form of the discernibility function of F be
F7 ¼_k
t¼1^
rt
s¼1as
!;
where^rt
s¼1 as, t # k, are all the prime implicants of F7. Then Et ¼ {asjs # rt}, t # k,
are all the reducts of F.
Proof. By the definition of the minimal disjunctive normal form of a Boolean
function (Skowron 1993), the proof follows immediately based on Theorem 4.1 and
Proposition 4.2. A
Theorem 4.2 provides an approach to compute all reducts of an inconsistent formal
decision context. Now, we present an example to illustrate this approach.
Example 4.1. Table 3 depicts a formal decision context F ¼ ðU;A; I;D; JÞ, where
U ¼ {1; 2; 3; 4; 5}, A ¼ {a; b; c; d; e; f }, and D ¼ {d1; d2; d3}. The Hasse diagram of
BðU;A; IÞ is the same as that in Figure 1 and the Hasse diagram of BðU;D; JÞ is given in
Figure 4.
According to Figures 1 and 4, f ðUðU;A; IÞÞ ¼ {12345; 234; 5; Y} , UðU;D; JÞ. Thus,
it can be known from Definition 3.2 that F is inconsistent. Moreover, the discernibility
matrix of F is shown in Table 4. Based on Theorem 4.2, we can compute all the reducts of
F as follows:
F7 ¼^
D 7ððXi;BiÞ;ðXj;BjÞÞ–YD 7 ððXi ;BiÞ;ðXj ;Bj ÞÞ[D7
_D7ððXi;BiÞ; ðXj;BjÞÞn o
¼ d ^ e ^ ða _ cÞ ^ b
¼ ða ^ b ^ d ^ eÞ _ ðb ^ c ^ d ^ eÞ:
Thus, F has two reducts: R1 ¼ {a; b; d; e}, R2 ¼ {b; c; d; e}, and the core of F is {b; d; e}.
By using the reduct R1 ¼ {a; b; d; e}, we obtain the reduced sub-context ðU;R1; IR1;D; JÞ.
Table 3. A formal decision context F ¼ ðU;A; I;D; JÞ.
U a b c d e f d1 d2 d3
1 1 0 0 0 0 0 1 0 02 0 1 0 1 0 0 0 1 03 1 0 1 0 1 0 0 1 04 0 1 0 1 0 1 0 1 05 1 1 1 0 0 0 0 0 1
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The Hasse diagram of BðU;R1; IR1Þ is given in Figure 5, and the Hasse diagram of
BðU;D; JÞ is the same as that in Figure 4.
It can easily be checked that the formal decision context in Example 4.1 is also
inconsistent under the knowledge reduction frameworks proposed by, for example, Wang
and Zhang (2008a), Wei et al. (2008), Wu et al. (2009), Li et al. (2011). Thus, the
knowledge reduction methods in such literature exclude this kind of formal decision
contexts.
5. Induction of decision rules in formal decision contexts
As we know, FCA provides an appropriate framework for knowledge discovery in
databases. For instance, Wille (1982) introduced implication rule mining in formal
contexts. Ganter and Wille (1999) and Valtchev et al. (2004) gave a further investigation
of implication rules based on FCA. Bastide et al. (2000) and Zaki (2004) discussed the
issue of mining non-redundant association rules by making use of the closure operator of
the Galois connection. Wu et al. (2009) put forward granular rule extraction in consistent
( ,d1d2d3)
(1,d1) (234,d2) (5,d3)
(12345, )
Figure 4. BðU;D; JÞ.
Table 4. The discernibility matrix D7 of F ¼ ðU;A; I;D; JÞ.
ðU; YÞ (135, a) (245, b) (35, ac) (24, bd) (3, ace) (5, abc) (4, bdf) ðY;AÞ
ðU; YÞ Y Y Y Y bd ace abc bdf abcdef(135, a) Y Y Y Y abd ce bc abdf bcdef(245, b) Y Y Y Y d abce ac df acdef(35, ac) Y Y Y Y abcd e b abcdf bdef(24, bd) bd abd d abcd Y Y acd Y acef(3, ace) ace ce abce e Y Y be Y bdf(5, abc) abc bc ac b acd be Y acdf def(4, bdf) bdf abdf df abcdf Y Y acdf Y aceðY;AÞ abcdef bcdef acdef bdef acef bdf def ace Y
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formal decision contexts. In this section, we discuss knowledge discovery in formal
decision contexts.
Definition 5.1 (Li et al. 2011). Let F ¼ ðU;A; I;D; JÞ be a formal decision context,
E # A, ðX;BÞ [ BðU;E; IEÞ and ðY;CÞ [ BðU;D; JÞ. If X # Y , and X, B, Y and C are
non-empty, we say that a decision rule can be generated between ðX;BÞ and ðY;CÞ and
denote it by B ! C, where B and C are called the premise and the conclusion of B ! C,
respectively. We denote by RðE;DÞ ¼ {B ! CjðX;BÞ [ BðU;E; IEÞ; ðY;CÞ [BðU;D; JÞ} the set of all the decision rules between the formal concepts of BðU;E; IEÞ
and those of BðU;D; JÞ.
Definition 5.2. Let F ¼ ðU;A; I;D; JÞ be a formal decision context and E # A. B ! C [RðE;DÞ is said to be redundant in RðE;DÞ if there exists B0 ! C0 [ RðE;DÞ such that
B0 , B and C # C0, or B0 # B and C , C0. Otherwise, B ! C is said to be non-redundant
in RðE;DÞ. We denote by RðE;DÞ the set of all the non-redundant decision rules of
RðE;DÞ.
For a formal decision context, its non-redundant decision rules are more appealing
than its redundant ones since the redundant decision rules can be implied by the others.
Proposition 5.1. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, if E # A
is a pseudo-consistent set of F, then ðB > EÞ! C [ RðE;DÞ for any B ! C [ RðA;DÞ.
Proof. The proof follows immediately from Definitions 4.1, 5.1, and 5.2. A
Based on Proposition 5.1, if E is a pseudo-consistent set of an inconsistent formal
decision context F ¼ ðU;A; I;D; JÞ, then for any B ! C [ RðA;DÞ, there exists B0 !
C0 [ RðE;DÞ (e.g. ðB > EÞ! C) such that B0 ! C0 implies B ! C. Thus, for an
inconsistent formal decision context F, the knowledge reduction can make it avoid
redundancy in the attributes and, at the same time, preserve the information of the non-
redundant decision rules provided by F since each non-redundant decision rule derived
( ,abde)
(3,ae)
(5,ab)
(24,bd)
(135,a) (245,b)
(12345, )
Figure 5. BðU;R1; IR1Þ.
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from F can be implied by one of the non-redundant decision rules derived from the reduced
formal decision context.
Proposition 5.2. For an inconsistent formal decision context F ¼ ðU;A; I;D; JÞ, let
{Rtjt [ T} be the set of all the reducts of F, where T is an index set. Then all the compact
non-redundant decision rules derived from all the reduced sub-contexts ðU;Rt; IRt;D; JÞ
(t [ T) imply all the decision rules derived from F.
Proof. For any decision rule B ! C derived from F, it can be known from Definition 5.2
that there exists a non-redundant decision rule B0 ! C0 [ RðA;DÞ such that B0 ! C0
implies B ! C. According to Proposition 5.1, there exists t [ T such that
ðB0 > RtÞ! C0 [ RðRt;DÞ. Note that ðB0 > RtÞ! C0 implies B0 ! C0. Then, it follows
from Definition 5.2 that ðB0 > RtÞ! C0 can imply B ! C as well. A
That is to say, we can draw a conclusion from Proposition 5.2 that for the inconsistent
formal decision contexts, the set of all the compact non-redundant decision rules derived
from all the reduced sub-contexts (determined by all the reducts) is complete in the initial
formal decision context (see, e.g. Ganter and Wille 1999 for the notion of completeness of
a rule set). However, such completeness satisfied by the inconsistent formal decision
contexts is not held by the consistent formal decision contexts. We use the following
counterexample to confirm this issue.
Example 5.1. Let F ¼ ðU;A; I;D; JÞ be the formal decision context that is adopted partly
from Table 1 with U ¼ {1; 2; 3; 4; 5}, A ¼ {a; b; c; e; f }, and D ¼ {d1; d2; d3}. In other
words, F is generated by removing the data in the fourth column of Table 1. The Hasse
diagram of BðU;A; IÞ is shown in Figure 6 and the Hasse diagram of BðU;D; JÞ is the
same as that in Figure 2. Based on Definition 3.2, it can easily be checked that F is
consistent.
( ,abcef)
(3,ace) (5,abc) (4,bf)
(35,ac)
(135,a) (245,b)
(12345, )
Figure 6. BðU;A; IÞ.
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According to the knowledge reduction method in Section 3, we find that F only has
one reduct R ¼ {a; b; f }. So, to confirm that the completeness satisfied by the
inconsistent formal decision contexts is not held by the consistent formal decision
contexts, it is sufficient to find a decision rule derived from F which cannot be implied by
any compact non-redundant decision rule derived from the reduced sub-context
ðU;R; IR;D; JÞ.
Note that for the reduced sub-context ðU;R; IR;D; JÞ, the Hasse diagrams of the
concept lattices BðU;R; IRÞ and BðU;D; JÞ are the same as those in Figures 3 and 2,
respectively. Therefore, all the compact non-redundant decision rules derived from
ðU;R; IR;D; JÞ are as follows:
r1: If a, then d1;
r2: If b, then d2;
r3: If a and b, then d1 and d2;
r4: If b and f, then d2 and d3.
However, the following decision rule
r01: If a and c, then d1 and d2
derived from the initial formal decision context F cannot be implied by any non-redundant
decision rule rk (k [ {1; 2; 3; 4}).
6. Conclusion
Knowledge reduction is one of the important issues in FCA. Although there have been a
few knowledge reduction approaches for formal decision contexts, these methods mainly
focus on the consistent formal decision contexts under a given partial order relation
between concept lattices, while the inconsistent formal decision contexts are generally
excluded.
In this paper, we have defined an order-preserving mapping between the set of all
the extensions of the conditional concept lattice and that of the decision concept lattice
to classify formal decision contexts into consistent and inconsistent formal decision
contexts. By constructing the suitable discernibility matrices and Boolean functions, we
have proposed knowledge reduction methods for both the consistent and the inconsistent
formal decision contexts. For the consistent formal decision contexts, the proposed
reduction method can avoid redundancy and at the same time maintain consistency,
whereas for the inconsistent formal decision contexts, the reduction method can make
the set of all the compact non-redundant decision rules derived from all the reduced
sub-contexts (determined by all the reducts) complete in the initial formal decision
context.
From the computational point of view, heuristic methods need to be developed
in further research to speed up the process of generating reducts, since computing
all the reducts of a formal decision context by Boolean reasoning is an NP-hard
problem.
Acknowledgements
The authors are grateful to the anonymous reviewers for their constructive comments andsuggestions that led to a significant improvement of the paper. This work was supported by theNational Natural Science Foundation of China (Nos. 10971161 and 70861001).
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Notes on contributors
Jinhai Li received his BSc degree in 2006 from Qiongzhou University and
MSc degree in 2009 from Guangxi University, China. He is currently
working towards the PhD degree in Xi’an Jiaotong University, China. He has
published more than 10 papers in international journals and book chapters.
His current research interests include rough sets, FCA, and algorithm
analysis.
Changlin Mei received his BSc, MSc, and PhD degrees from Xi’an
Jiaotong University, China, in 1983, 1989, and 2000, respectively. From
2001 to 2003, he was a postdoctoral fellow of Peking University, China. He
is currently a professor of School of Science, Xi’an Jiaotong University,
China. He has published more than 50 papers in international journals. His
current research interests include data mining, spatial data analysis, and
regression analysis.
Yuejin Lv received his BSc degree in 1982 from Guangxi University,
China. He is currently a professor of School of Mathematics and
Information Sciences, Guangxi University, China. He has published more
than 40 papers in international journals and book chapters. His current
research interests include data mining, concept lattices, and information
management.
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