Knowledge reduction in real decision formal contexts

Information Sciences 189 (2012) 191–207

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Knowledge reduction in real decision formal contexts Jinhai Li a,⇑, Changlin Mei a, Yuejin Lv b a b

School of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China School of Mathematics and Information Sciences, Guangxi University, Nanning, Guangxi 530004, PR China

a r t i c l e

i n f o

Article history: Received 8 July 2010 Received in revised form 6 September 2011 Accepted 29 November 2011 Available online 4 December 2011 Keywords: Real formal context Real decision formal context Concept lattice Knowledge reduction Rule acquisition

a b s t r a c t Rule acquisition is one of the main purposes in the analysis of real decision formal contexts. In general, the decision rules derived directly from a real decision formal context are not concise or compact. In order to derive more compact decision rules, this study proposes a rule acquisition oriented framework of knowledge reduction for real decision formal contexts and formulates a corresponding reduction method by constructing a discernibility matrix and its associated Boolean function. The proposed reduction method is applicable to any real decision formal contexts and with the reduced real decision formal contexts, we can obtain more compact decision rules that can imply all the decision rules derived from the initial real decision formal context. Some numerical experiments are conducted to assess the efficiency of the proposed method. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Formal concept analysis (FCA), proposed by Wille [23], is one of the effective mathematical tools for conceptual data analysis and knowledge processing. Two basic notions in this theory are formal contexts and formal concepts. A formal context is a triple (U, A, I) consisting of a set U of objects, a set A of attributes, and a crisp binary relation I on U  A to specify each object of U has what attributes in A. A formal concept is a pair (X, B) of a set of objects and a set of attributes with X containing exactly those objects shared by all the attributes in B and B containing exactly those attributes that all the objects in X have in common. The set of all the formal concepts of a formal context forms a complete lattice, called the concept lattice [12], which reflects the relationship of generalization and specialization among the formal concepts. Nowadays, FCA has been applied to a variety of fields such as information retrieval, machine learning and knowledge discovery [6–9,20]. Knowledge reduction is one of the key issues in FCA and there have been many studies on this topic. For example, Ganter and Wille [12] discussed the knowledge reduction issue by removing the reducible objects and attributes of a formal context. Zhang et al. [28] proposed a knowledge reduction method in formal contexts from the viewpoint of lattice isomorphism. Liu et al. [16] put forward two kinds of knowledge reduction approaches for formal contexts based on the object-oriented concept lattice [26,27] and the property-oriented concept lattice [10]. Wei et al. [22] investigated the problem of knowledge reduction in decision formal contexts by defining two partial orders between the conditional and the decision concept lattices. Based on granular computing, Wu et al. [24] presented a novel reduction method for decision formal contexts. In the classical formal contexts, the relationship between the objects and the attributes is described by a two-valued form that can only specify whether or not an attribute is possessed by an object. In many real-world applications, however, the relationship may be fuzzy-valued or interval-valued. Therefore, some studies have recently been devoted to the generalized formal contexts. For instance, Burusco and Fuentes-González [4,5] studied the concept lattice in L-fuzzy contexts based on some implication operators. Popescu [17] gave a general approach for building fuzzy concepts of a fuzzy context. Be˘lohlávek ⇑ Corresponding author. E-mail addresses: [email protected] (J. Li), [email protected] (C. Mei), [email protected] (Y. Lv). 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.11.041

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[1–3] defined a fuzzy Galois connection based on the complete residuated lattice and further analyzed the structure of the corresponding fuzzy concept lattice. From the point of view of fuzzy logic, Georgescu and Popescu [13] generalized the work in [1] and put forward a so-called non-commutative fuzzy Galois connection. Zhang et al. [29] presented variable threshold concept lattice in fuzzy contexts. Jaoua and Elloumi [14] extended the classical Galois connection, formal concept and concept lattice to the case of real binary relations. Knowledge reduction is still of great importance in the analysis of the generalized formal contexts. Some attention has recently been paid to this issue. For example, Elloumi et al. [11] put forward a multi-level data reduction approach for the fuzzy contexts. Li and Zhang [15] reconsidered the reduction issue in the fuzzy contexts by replacing the Lukasiewicz implication in [11] with T-implication. Particularly, for the real decision formal contexts, an extension of the classical decision formal contexts, Yang et al. [25] proposed a knowledge reduction method to avoid redundancy in the attributes conditional on maintaining the decision consistency. For this purpose, Yang et al. [25] defined an implication mapping between the conditional real concept lattice and the decision real concept lattice to classify real decision formal contexts into consistent and inconsistent categories. And the knowledge reduction method was proposed only for the consistent real decision formal contexts. Generally speaking, however, an inconsistent real decision formal context appears more often than a consistent one no matter how an implication mapping between the conditional and the decision real concept lattices is defined. Thus, how to develop some knowledge reduction methods that are applicable to any real decision formal contexts rather than to the consistent ones only, is worth being investigated. Furthermore, rule acquisition is always one of the main purposes in the analysis of all kinds of formal contexts (see e.g. [12,20,21,24,25]), and the decision rules derived directly from a real decision formal context are in general not concise or compact. Therefore, it is of practical importance to propose some knowledge reduction approaches that not only are applicable to any real decision formal contexts but also can derive more compact decision rules with the reduced real decision formal context. In this paper, we formulate a framework of knowledge reduction in real decision formal contexts from the perspective of rule acquisition and derive a corresponding reduction method by constructing a discernibility matrix and its associated Boolean function. The proposed reduction method is applicable to any real decision formal contexts and with the reduced real decision formal contexts, more compact decision rules can be obtained and they can imply all the decision rules derived from the initial real decision formal context. The rest of this paper is organized as follows. We briefly recall in Section 2 some basic notions of the real formal contexts to facilitate our subsequent discussion. In Section 3, we introduce the notions of real sub-contexts and their concept lattices. In Section 4, we first discuss how to derive decision rules from a real decision formal context and then formulate a reduction framework for the real decision formal contexts. A reduction method is given in Section 5. Some numerical experiments are conducted in Section 6 to assess the efficiency of the proposed reduction procedure. The paper is then concluded with a brief summary and an outlook for further research.

2. Preliminaries 2.1. Real sets Definition 1 ([14]). A real interval on the set R of real numbers, denoted by I ¼ ½u; v ; u; v 2 R, represents the set of real numbers delimited by u and v, where u and v are called the lower and the upper bounds of I, respectively. If u > v, I is said to be empty and is denoted by [,]. f I2 Þ, if Let I1 = [u1, v1] and I2 = [u2, v2] be two real intervals. I1 and I2 are called overlapping intervals, denoted by ðI1 OV max(u1, u2) 6 min(v1, v2). The intersection of I1 and I2 is defined by

InterðI1 ; I2 Þ ¼ ½maxðu1 ; u2 Þ; minðv 1 ; v 2 Þ: Furthermore, the closure operator is defined by

( ClosureðfI1 ; I2 gÞ ¼

f I2 Þ; f½minðu1 ; u2 Þ; maxðv 1 ; v 2 Þg; if ðI1 OV fI1 ; I2 g;

otherwise:

This closure operator can be applied recursively to a set of n real intervals by Closure ({I1, I2, . . . , In}) = Closure(Closure({I1, I2}), I3, . . . , In}). n o Let E = {I1, I2, . . . , Ip} and E0 ¼ I01 ; I02 ; . . . ; I0q be two sets of p and q real intervals, respectively. The union and the intersection of E and E0 are, respectively, defined by

n o E [ E0 ¼ Closure I1 ; I2 ; . . . ; Ip ; I01 ; I02 ; . . . ; I0q ; n   o E \ E0 ¼ Closure Inter Ii ; I0j ji ¼ 1; . . . ; p; j ¼ 1; . . . ; q :

ð1Þ ð2Þ

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Definition 2 [14]. Let U = {x1, x2, . . . , xn} be a set of n objects, V be the set of all real intervals on R, and P(V) be the set of all e of U is defined by its characteristic function l : U ! PðVÞ, where l ðxÞ, a set of real intervals, partitions on V. A real set X e e X X e The real set X e is denoted by indicates the possible values that can be chosen for x in the real set X.

e¼ X



leX ðx1 Þ leX ðx2 Þ x1

;

x2

 l ðxn Þ : ; . . . ; eX xn

ð3Þ

Without lost of generality, we suppose henceforth that the intervals in each le ðxÞðx 2 UÞ are pairwise disjoint. For brevity, X e we write le ðxÞ as XðxÞ and denote the real empty set of U by X

;e ¼



 f½; g f½; g f½; g : ; ;...; x2 xn x1

In what follows, we introduce two kinds of intersection and union of real sets (see [14] for details). e ðxÞ, if for any ½u; v  2 Y e and Y e be two real sets of U. X e ðxÞ is said to be largely less than Y e ðxÞ, denoted by X e ðxÞ6L Y e ðxÞ, Let X e ðxÞ such that u0 6 u and v 0 P v ; X e is said to be largely included in Y e , denoted by X e # LY e , if XðxÞ6 e e there exists ½u0 ; v 0  2 X L Y ðxÞ e and Y e are, respectively, defined by for any x 2 U. The large intersection and the large union of X

e ÞðxÞ ¼ X e ðxÞ [ Y e ðxÞ; ð X e [L Y e ÞðxÞ ¼ XðxÞ e e ðxÞ: e \L Y \Y ðX

ð4Þ

e e e ðxÞ, denoted by XðxÞ6 e e On the other hand, XðxÞ is said to be strictly less than Y S Y ðxÞ, if for any ½u; v  2 Y ðxÞ, there exists e ðxÞ such that u 6 u0 and v P v 0 ; X e is said to be strictly included in Y e , denoted by X e # SY e , if XðxÞ6 e e ½u0 ; v 0  2 X S Y ðxÞ for any e and Y e are, respectively, defined by x 2 U. The strict intersection and the strict union of X

e ÞðxÞ ¼ X e ðxÞ \ Y e ðxÞ; ð X e [S Y e ÞðxÞ ¼ XðxÞ e e ðxÞ: e \S Y [Y ðX

ð5Þ

2.2. Real formal contexts and real concept lattices Let U be a set of objects and A be a set of attributes. A real binary relation eI is such a mapping that is defined on U  A and takes its each value to be a set of real intervals. That is, for each ðx; aÞ 2 U  A; eIðx; aÞ is a set of real intervals. Definition 3 [25]. A real formal context is a triple ðU; A; eIÞ, where U is a set of objects, A is a set of attributes, and eI is a real binary relation on U  A. Example 1. Table 1, taken from [14], shows a real formal context ðU; A; eIÞ with U = {x1,x2,x3,x4} and A = {a, b, c, d}. Let ðU; A; eIÞ be a real formal context and X # U. For a given a 2 A, define

f ðaÞ ¼

[

eIðx; aÞ;

gðaÞ ¼

x2X

\

eIðx; aÞ:

ð6Þ

x2X

That is, f(a) is a set of real intervals obtained by the union of eIðx; aÞðx 2 XÞ according to Eq. (1) and g(a), the intersection of eIðx; aÞðx 2 XÞ according to Eq. (2), is a set of all the real intervals shared by all eIðx; aÞðx 2 XÞ. Definition 4 ([14]). Let eI be a real binary relation on U  A; PðUÞ be the power set of U, and RðAÞ be the set of all real sets of e 2 RðAÞ, the operators ";  : PðUÞ ! RðAÞ and #; } : RðAÞ ! PðUÞ are defined as follows: A. For X 2 PðUÞ and B

(

"

X ¼

)  [ f ðaÞ  e a 2 A; f ðaÞ ¼ Iðx; aÞ ; a  x2X

e # ¼ fx 2 Uj8a 2 A; BðaÞ6 e e B L Iðx; aÞg; ( )  \ gðaÞ   e a 2 A; gðaÞ ¼ Iðx; aÞ ; X ¼ a  x2X

ð7Þ

e } ¼ fx 2 Uj8a 2 A; BðaÞ6 e e B S Iðx; aÞg:

Table 1 A real formal context ðU; A; eIÞ. U

a

b

c

d

x1 x2 x3 x4

{[5, 7], [8, 9]} {[6, 8]} {[10, 11]} {[10, 11]}

{[10, 11]} {[7, 9], [11, 12]} {[8, 10]} {[9, 10]}

{[7, 9], [11, 14]} {[9, 12], [14, 17]} {[10, 12]} {[9, 10]}

{[5, 8]} {[4, 4], [7, 8]} {[1, 5]} {[1, 5]}

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e # is a maximal set of the objects In fact, X" is a real set of A with each value of its characteristic function being f(a)(a 2 A). B e that make each BðaÞða 2 AÞ be largely less than eIðx; aÞ. The other two operators can be explained analogously. According to [14], two types of real concepts, i.e., the large real concept and the strict real concept, can be defined based on e with X 2 PðUÞ and B e 2 RðAÞ is called a large real concept of ðU; A; eIÞ if the above four operators. Specifically, a pair ðX; BÞ e and B e # ¼ X, and it is called a strict real concept of ðU; A; eIÞ if X  ¼ B e and B e } ¼ X. In both types of the real concepts, X" ¼ B e are termed as the extension and the intension of ðX; BÞ, e respectively. Hereinafter, we denote the set of all the large X and B real concepts of ðU; A; eIÞ by BL ðU; A; eIÞ and that of all the strict real concepts of ðU; A; eIÞ by BS ðU; A; eIÞ. e B e1; B e 2 2 RðAÞ, the following properties Proposition 1 [14]. Let eI be a real binary relation on U  A. For X; X 1 ; X 2 2 PðUÞ and B; hold: (i) (ii) (iii) (iv) (v) (vi)

e2 ) B e1 # L B e# # B e# ; X 1 # X 2 ) X "2 # L X "1 ; B 2 1 e #" ; e # LB X # X "# ; B e#; B e #" Þ 2 BL ðU; A; eIÞ; ðX "# ; X " Þ; ð B  e e} e} e X1 # X2 ) X2 # S X 1 ; B1 # S B2 ) B2 # B1 ; e } ; e # SB X # X } ; B e}; B e } Þ 2 BS ðU; A; eIÞ. ðX } ; X  Þ; ð B

e i Þ and ðX j ; B e j Þ, if Xi # Xj, then ðX i ; B e i Þ is called a sub-concept of ðX j ; B e j Þ, or ðX j ; B ejÞ For two large (or strict) real concepts ðX i ; B e i Þ. We denote this relation by ðX i ; B e i Þ L ðX j ; B e j Þ for the large real concepts and by is called a super-concept of ðX i ; B e i Þ S ðX j ; B e j Þ for the strict real concepts. ðX i ; B The set of all the large real concepts of a real formal context ðU; A; eIÞ together with the relation L forms a complete lattice e i Þ; ðX j ; B e j Þ 2 BL ðU; A; eIÞ, the which is called the large real concept lattice [14] of ðU; A; eIÞ and is denoted by BL ðU; A; eIÞ. For ðX i ; B V W meet ( L) and the join ( L) in BL ðU; A; eIÞ are respectively defined by

0 !#" 1 ^ \ [ ej e i Þ ðX j ; B e j Þ ¼ @X i X j ; B ei A; B ðX i ; B L

L



_ e i Þ ðX j ; B ejÞ ¼ ðX i ; B

Xi

[

Xj

"#

\

ei ;B

L

ð8Þ

! ej : B

L

Similarly, the set of all the strict real concepts of ðU; A; eIÞ together with the relation S also forms a complete lattice which e i Þ; ðX j ; B e j Þ 2 BS ðU; A; eIÞ, the meet is called the strict real concept lattice [14] of ðU; A; eIÞ and is denoted by BS ðU; A; eIÞ. For ðX i ; B V W e ( S) and the join ( S) in BS ðU; A; IÞ are respectively defined by

 ^  ei ej ¼ Xi; B Xj; B

Xi

\

ei Xj; ðB

[

S

! e j Þ} ; B

S

 _  ei ej ¼ Xi; B Xj; B



Xi

[

Xj

}

ei ;B

\

S

! e Bj :

ð9Þ

S

3. Real sub-contexts and their concept lattices Before embarking on constructing a knowledge reduction framework in real decision formal contexts from the perspective of rule acquisition, we introduce the notions of real sub-contexts and their concept lattices and discuss some related issues on the real sub-contexts. For two crisp sets X = {x1, x2, . . . , xm} and Y = {y1, y2, . . . , yn} with X \ Y = ;, let U = X [ Y, and let

(

e¼ X

e ðx1 Þ X x1

e ðxm Þ X

;...;

xm

)

;

(

e¼ Y

e ðy Þ Y 1 y1

;...;

e ðy Þ Y n

)

yn

and

( e ¼ U

e 1Þ Uðx x1

;...;

e m Þ Uðy e Þ e Þ Uðx Uðy 1 n ; ;...; xm y1 yn

)

e and Y e is defined by be the real sets of X,Y and U, respectively. The mergence of X

(

erY e¼ X

e ðx1 Þ X x1

;...;

e ðxm Þ Y e ðy Þ e ðy Þ X Y 1 n ; ;...; xm y1 yn

)

ð10Þ

J. Li et al. / Information Sciences 189 (2012) 191–207

195

e on X is defined by and the restriction of U

(

e ¼ Uj X

e 1Þ Uðx x1

;...;

e mÞ Uðx xm

)

ð11Þ

:

Furthermore, for a real formal context ðU; A; eIÞ and E # A, the restriction of eI on U  E, denoted by eI E , is defined by eI E ðx; eÞ ¼ eIðx; eÞ for any (x, e) 2 U  E. Definition 5. Let ðU; A; eIÞ be a real formal context. For any E # A, the real formal context ðU; E; eI E Þ is called a real sub-context of ðU; A; eIÞ. Definition 6. Let ðU; A; eIÞ be a real formal context and RðEÞðE # AÞ be the set of all the real sets of E. For X 2 PðUÞ and e 2 RðEÞ, we define the operators " ; E : PðUÞ ! RðEÞ and # ; }E : RðEÞ ! PðUÞ as follows: B E E

(

)  [ f ðaÞ  e X ¼ a 2 E; f ðaÞ ¼ I E ðx; aÞ ; a  x2X e e e #E ¼ fx 2 Uj8a 2 E; BðaÞ6 B L I E ðx; aÞg; ( )  \ gðaÞ  E e a 2 E; gðaÞ ¼ X ¼ I E ðx; aÞ ; a  x2X e e e }E ¼ fx 2 Uj8a 2 E; BðaÞ6 B S I E ðx; aÞg: "E

ð12Þ

In fact, the above operators are the restriction of the four operators in Definition 4 on the real sub-context ðU; E; eI E Þ. e a large real concept of ðU; E; eI E Þ if X "E ¼ B e and B e #E ¼ X, and a strict real concept Based on the above operators, we call ðX; BÞ E }E e e e of ðU; E; I E Þ if X ¼ B and B ¼ X. The sets of all the large real concepts and all the strict real concepts of ðU; E; eI E Þ are denoted by BL ðU; E; eI E Þ and BS ðU; E; eI E Þ, respectively. And the large and the strict real concept lattices of ðU; E; eI E Þ are, respectively, denoted by BL ðU; E; eI E Þ and BS ðU; E; eI E Þ and called the sub-lattices of BL ðU; A; eIÞ and BS ðU; A; eIÞ. Furthermore, we denote the sets of all the extensions and the sets of all the intensions of BL ðU; E; eI E Þ and BS ðU; E; eI E Þ respectively by

e 2 BL ðU; E; eI E Þg; UL ðU; E; eI E Þ ¼ fXjðX; BÞ e e 2 BS ðU; E; eI E Þg; US ðU; E; I E Þ ¼ fXjðX; BÞ e e 2 BL ðU; E; eI E Þg; BÞ IL ðU; E; eI E Þ ¼ f BjðX; e e 2 BS ðU; E; eI E Þg: BÞ IS ðU; E; eI E Þ ¼ f BjðX;

ð13Þ

e 2 RðAÞ, the following properties Proposition 2. Let ðU; A; eIÞ be a real formal context, E  A and F = A  E. For X 2 PðUÞ and B hold: e # ¼ ð Bj e Þ#E \ ð Bj e Þ#F ; (i) X " ¼ X "E rX "F ; B E F }E  E F } e e e Þ}F ; (ii) X ¼ X rX ; B ¼ ð Bj Þ \ ð Bj (iii) (iv) (v) (vi)

X " E ¼ X " jE ; X E ¼ X  jE ; X "# ¼ X "E #E \ X "F #F ; X } ¼ X E }E \ X F }F .

E

F

Proof. The proofs of (i) and (ii) follow immediately from Definitions 4 and 6. (iii) (iv) (v) (vi)

It can be proved from (i) that X " jE ¼ ðX "E r X "F ÞjE ¼ X "E . It follows from (ii) that X  jE ¼ ðX E r X F ÞjE ¼ X E . Based on (i), we have X "# ¼ ðX "E r X "F Þ# ¼ ððX "E r X "F ÞjE Þ#E \ ððX "E r X "F ÞjF Þ#F ¼ ðX "E Þ#E \ ðX "F Þ#F ¼ X "E #E \ X "F #F : The proof is analogous to that of (v). h

Proposition 3. Let ðU; A; eIÞ be a real formal context and E # A. Then both UL ðU; E; eI E Þ # UL ðU; A; eIÞ and US ðU; E; eI E Þ # US ðU; A; eIÞ hold. e 2 IL ðU; E; eI E Þ such that ðX; BÞ e 2 BL ðU; E; eI E Þ. Thus X "E #E ¼ ðX "E Þ#E ¼ B e #E ¼ X. It can Proof. For any X 2 UL ðU; E; eI E Þ, there exists B be known from (ii) and (iii) in Proposition 1 and (v) in Proposition 2 that X # X "# ; ðX "# ; X " Þ 2 BL ðU; A; eIÞ and X "# # X "E #E ¼ X, which yields X"; = X. Therefore X 2 UL ðU; A; eIÞ, and consequently, UL ðU; E; eI E Þ # UL ðU; A; eIÞ. Analogously, we can prove US ðU; E; eI E Þ # US ðU; A; eIÞ. h

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4. A rule acquisition oriented knowledge reduction framework for real decision formal contexts In this section, we develop a knowledge reduction framework that is applicable to any real decision formal contexts and can derive more compact decision rules from the reduced real decision formal context. Definition 7 [25]. A real decision formal context is a quintuple ðU; A; eI; D; eJÞ, where ðU; A; eIÞ and ðU; D; eJÞ are two real formal contexts, and A and D are respectively called the conditional attribute set and the decision attribute set with A \ D = ;. As discussed in the former sections, we can construct two kinds of real concept lattices, i.e., the large real concept lattice and the strict real concept lattice, for a real formal context. In what follows, we mainly focus on the large real concept lattice to formulate the knowledge reduction framework in real decision formal contexts. This framework is also applicable to the strict real concept lattice as long as the operators and notations associated to the large real concept lattice are respectively replaced with the corresponding operators and notations associated to the strict real concept lattice. Firstly, we discuss how to derive decision rules from a real decision formal context based on the large real concept lattice. e Þ 2 BL ðU; D; eJÞ, where e 2 BL ðU; E; eI E Þ, and ðY; C Definition 8. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context, E # A; ðX; BÞ e e e e e e e X – ;; Y – ;; B – ; and C – ;. If X # Y, we say that an L decision rule, denoted by B ! C , can be generated between ðX; BÞ e Þ. The real sets B e are called the premise and the conclusion of the L decision rule B e , respectively. e and C e!C and ðY; C The L decision rules are in fact a kind of ‘‘If-then’’ rules that are defined from the intension set of a sub-lattice of the conditional large real concept lattice to the intension set of the decision large real concept lattice. Specifically, an L decision rule e with B e 2 IL ðU; D; eJÞ means that for any x 2 U, if the value eI E ðx; aÞ for each a 2 E is included in BðaÞ, e!C e 2 IL ðU; E; eI E Þ and C e B e e then the value Jðx; dÞ for each d 2 D is included in C ðdÞ. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ and E # A, let

e e 2 BL ðU; D; eJÞg e ! CjðX; e 2 BL ðU; E; eI E Þ; ðY; CÞ RL ðE; DÞ ¼ f B BÞ

ð14Þ

be the set of all the L decision rules generated between the large real concepts in BL ðU; E; eI E Þ and those in BL ðU; D; eJÞ. e 0 2 RL ðE; DÞ and e0 ! C Definition 9. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context and E # F # A. For B 00 00 0 00 00 0 00 00 0 0 e e e e e e e e e e B ! C 2 RL ðF; DÞ, if B # L B jE and C # L C , we say that B ! C can be implied by B ! C and denote this implication e 0 )L B e 00 . e 00 ! C e0 ! C relationship by B Definition 10. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context, E # F # A; XE # RL ðE; DÞ, and XF # RL ðF; DÞ. If for any e 0 2 XE such that B e 0 )L B e , we say that XF can be implied by XE and denote this relae 2 XF , there exists B e0 ! C e0 ! C e!C e!C B tionship by XE )L XF. Definition 11. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ; E # A is called an L consistent set of K if RL ðE; DÞ )L RL ðA; DÞ; otherwise, E is called an L inconsistent set of K. Furthermore, if E is an L consistent set and any F  E is an L inconsistent set of K, then E is called an L reduct of K. The intersection of all the L reducts of K is called the L core of K. In fact, an L reduct E of a real decision formal context K ¼ ðU; A; eI; D; eJÞ is such a minimal attribute subset of A that can make all the L decision rules derived from K be implied by those derived from the reduced real decision formal context ðU; E; eI E ; D; eJÞ. Definition 12. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ; c 2 A is said to be an L dispensable attribute of K if A  {c} is an L consistent set of K; otherwise, c is called an L indispensable attribute of K. e 2 RL ðE; DÞ is said to be redundant in e!C Definition 13. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context and E # A. B 0 0 0 0 e e is said to be non-redune e e e e!C e e e RL ðE; DÞ if there exists B ! C 2 RL ðE; DÞ  f B ! C g such that B ! C )L B ! C . Otherwise, B dant in RL ðE; DÞ. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context. For E # A, denote

e UL ðU; E; eI E Þ ¼ fX 2 UL ðU; E; eI E ÞjX – ;; X "E – ;g; "  e D U ðU; D; eJÞ ¼ fY 2 UL ðU; D; eJÞjY – ;; Y – ;g:

ð15Þ

L

Here, "D denotes the operator " in ðU; D; eJÞ. For X 2 UL ðU; E; eI E Þ and Y 2 UL ðU; D; eJÞ, two mappings E ; E : UL ðU; E; eI E Þ UL ðU; D; eJÞ ! f0; 1g are defined as follows:

(

X E Y ¼

1; if X # Y; and X  X 0 ) X 0  Y for 8X 0 2 UL ðU; E; eI E Þ; 0;

otherwise;

ð16Þ

J. Li et al. / Information Sciences 189 (2012) 191–207

( X E Y ¼

1; if X # Y; and Y 0  Y ) X  Y 0 for 8Y 0 2 UL ðU; D; eJÞ; 0;

otherwise:

197

ð17Þ

Based on the above mappings, we can obtain a sufficient and necessary condition of justifying whether or not an L decision rule is redundant. e Þ 2 BL ðU; D; eJÞ and e 2 BL ðU; E; eI E Þ; ðY; C Theorem 1. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ and E # A, let ðX; BÞ e is non-redundant in e . Then B e is redundant in RL ðE; DÞ if and only if X E Y = 0 or X E Y = 0. Equivalently, B e!C e!C e!C B RL ðE; DÞ if and only if X E Y = 1 and X E Y = 1. e Þ 2 BL ðU; D; eJÞ and B e , it follows from Definition 8 that X – ;; Y – ;; e!C e 2 BL ðU; E; eI E Þ; ðY; C Proof. ()) Since ðX; BÞ e and Y "D ¼ C e which yields that X 2 U ðU; E; eI E Þ and Y 2 U ðU; D; eJÞ. e – ;, e – ;, X "E ¼ B L L e 0 2 RL ðE; DÞ  f B e g such that e is redundant in RL ðE; DÞ, then according to Definition 13, there exists B e0 ! C e!C e!C If B 0 0 0 0 0 0 0 0 e , which implies that B e # LC e )L B e # LC e–C e . Thus, there exist e , or B e and C e!C e # L B; e B e –B e C e !C e and C e # L B; B e 0 Þ 2 BL ðU; D; eJÞ such that X  X0 and Y0 # Y, or e 0 Þ 2 BL ðU; E; eI E Þ and Y 0 2 U ðU; D; eJÞ with ðY 0 ; C X 0 2 UL ðU; E; eI E Þ with ðX 0 ; B L 0 0 X # X and Y  Y. Therefore, X E Y = 0 or X E Y = 0. e 0 Þ 2 BL ðU; E; eI E Þ such that X  X0 and X0 # Y, which implies that (Ü) If X E Y = 0, there exists ðX 0 ; B 0 0 0 e e is redundant in RL ðE; DÞ because B e 2 RL ðE; DÞ  f B e g and e e e!C e0 ! C e!C e e e B ! C 2 RL ðE; DÞ; B # L B and B – B. Thus, B 0 e e e e B ! C )L B ! C . e 0 Þ 2 BL ðU; D; eJÞ such that Y0  Y and X # Y0 , which yields that Similarly, if X E Y = 0, then there exists ðY 0 ; C e # LC e–C e 0 . Therefore, B e is redundant in RL ðE; DÞ since B e 0 2 RL ðE; DÞ  f B e g and e 0 and C e 0 2 RL ðE; DÞ; C e!C e!C e!C e!C B 0 e e e e B ! C )L B ! C . h For a real decision formal context K ¼ ðU; A; eI; D; eJÞ and E # A, let

UL ðU; E; eI E Þ ¼ fX 2 UL ðU; E; eI E Þj9Y 2 UL ðU; D; eJÞ; X E Y ¼ 1; X E Y ¼ 1g

ð18Þ

e 2 RL ðE; DÞj B e #D ¼ 1; B e #D ¼ 1g; e!C e #E  E C e #E E C RL ðE; DÞ ¼ f B

ð19Þ

and

where denotes the operator in ðU; D; eJÞ. It can be known from Theorem 1 that RL ðE; DÞ is just the set of all the non-redundant L decision rules of RL ðE; DÞ. ;D

;

Proposition 4. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context and E # A. Then RL ðE; DÞ )L RL ðE; DÞ and RL ðE; DÞ )L RL ðE; DÞ. Proof. RL ðE; DÞ )L RL ðE; DÞ follows immediately from Definition 10. The remainder is to prove RL ðE; DÞ )L RL ðE; DÞ. e 2 RL ðE; DÞ, if B e is non-redundant in RL ðE; DÞ, then B e 2 RL ðE; DÞ and B e )L B e ; if e!C e!C e!C e!C e!C In fact, for any B e e e e e e e e e e e e B ! C is redundant in RL ðE; DÞ, then there exists B 1 ! C 1 2 RL ðE; DÞ  f B ! C g such that B 1 ! C 1 )L B ! C . If B 1 ! C 1 is e; e 1 2 RL ðE; DÞ e 1 )L B e!C e1 ! C e1 ! C otherwise, there exists non-redundant in RL ðE; DÞ, then B and B e e e e e e e e e e B 2 ! C 2 2 RL ðE; DÞ  f B ! C ; B 1 ! C 1 g such that B 2 ! C 2 )L B ! C . Since RL ðE; DÞ is a finite set of the L decision rules, e n 2 RL ðE; DÞ  f B e; B e 1; . . . ; B e n1 g such that B e n is non-redundant in RL ðE; DÞ en ! C e!C e1 ! C e n1 ! C en ! C there must exist B e, e. e n )L B e n 2 RL ðE; DÞ e n )L B e!C en ! C e!C en ! C en ! C which implies that B Consequently, and B and B RL ðE; DÞ )L RL ðE; DÞ. h Theorem 2. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ; E # A is an L consistent set of K if and only if UL ðU; E; eI E Þ ¼ UL ðU; A; eIÞ. Proof. ()) If E is an L consistent set of K, it follows from Definition 11 that RL ðE; DÞ )L RL ðA; DÞ. According to Proposition 4, we have RL ðE; DÞ )L RL ðE; DÞ and RL ðA; DÞ )L RL ðA; DÞ. Therefore, RL ðE; DÞ )L RL ðA; DÞ. Firstly, we prove UL ðU; A; eIÞ # UL ðU; E; eI E Þ. For any X 2 UL ðU; A; eIÞ, it follows from Eqs. (13), (18) and (19) that there e Þ 2 BL ðU; D; eJÞ such that X A Y = 1, X A Y = 1 and B e 2 RL ðA; DÞ. Since RL ðE; DÞ )L RL ðA; DÞ, e 2 IL ðU; A; eIÞ and ðY; C e!C exist B 0 e0 0 e0 0 0 e e e there exists B ! C 2 RL ðE; DÞ with ðX ; B Þ 2 BL ðU; E; I E Þ and ðY ; C Þ 2 BL ðU; D; eJÞ such that X 0 2 UL ðU; E; eI E Þ and e0 ) B e . Thus, B e # LC e 0 , which implies that X0 # Y0 # Y and X ¼ B e and C e0 ! C e!C e 0 # L Bj e # # ð Bj e Þ#E # ð B e 0 Þ#E ¼ X 0 . Therefore, B E E 0 0 X is not a proper subset of X due to X A Y = 1. Nothing that X is a subset of X but is not a proper subset of X0 , we have X ¼ X 0 2 UL ðU; E; eI E Þ.

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e Þ 2 BL ðU; D; eJÞ e 2 IL ðU; E; eI E Þ and ðY; C Secondly, we prove UL ðU; E; eI E Þ # UL ðU; A; eIÞ. For any X 2 UL ðU; E; eI E Þ, there exist B 0 e0 e e e e e such that ðX; BÞ 2 BL ðU; E; I E Þ and B ! C 2 RL ðE; DÞ. Furthermore, there exists ðX ; B Þ 2 BL ðU; A; IÞ such that X # X0 and e 2 RL ðA; DÞ. Since RL ðE; DÞ )L RL ðA; DÞ, there exists B e 0 2 RL ðE; DÞ with ðX 0 ; B e0 ! C e 0 Þ 2 BL ðU; E; eI E Þ such that X0 = X0 e0 ! C B 0" " " 0 0 e0 ) B e . Therefore, B e # LC e0 ) B e . Since e 0 , which implies B e0 ! C e !C e0 # L B e and C e0 ! C e!C e j ¼ X j # LX j ¼ X E ¼ B and B E E E e is non-redundant in RL ðE; DÞ, we obtain from Definition 13 that B e 0 is just B e . Consequently, e!C e0 ! C e!C B X ¼ X 0 ¼ X 0 2 UL ðU; A; eIÞ. e 2 RL ðA; DÞ with ðX; BÞ e Þ 2 BL ðU; D; eJÞ, there exists e!C e 2 BL ðU; A; eIÞ and ðY; C (Ü) If UL ðU; A; eIÞ ¼ UL ðU; E; eI E Þ, then for any B e 2 RL ðE; DÞ with ðX 0 ; B e 0 Þ 2 BL ðU; E; eI E Þ such that X0 = X. It can be known from (iii) in Proposition 2 that e0 ! C B e)B e , which means that e . Therefore, according to Definition 9 we obtain B e0 ! C e!C e 0 ¼ X "E ¼ X " j ¼ X " j ¼ Bj B 0

0 E

E

E

RL ðE; DÞ )L RL ðA; DÞ. Noting that RL ðE; DÞ )L RL ðE; DÞ and RL ðA; DÞ )L RL ðA; DÞ, we have RL ðE; DÞ)L RL ðA; DÞ. Hence, it follows from Definition 11 that E is an L consistent set of K. h

It should be pointed out that our reduction framework is different from the one in [25]. The main differences are as follows: (i) The main purpose of our reduction framework is to derive more compact decision rules that can imply all the decision rules derived from the initial real decision formal context, while the end of the reduction framework in [25] is to avoid redundancy in the attributes of a real decision formal context subject to preserving the decision consistency. (ii) Our reduction framework is applicable to any real decision formal contexts, while the reduction framework in [25] is only applicable to the consistent real decision formal contexts under the given implication mapping between the conditional and the decision real concept lattices. Example 2. Table 2 shows a real decision formal context K ¼ ðU; A; eI; D; eJÞ with U = {x1, x2, x3, x4}, A = {a, b, c, d} and D = {d1, d2}. All the large real concepts of ðU; A; eIÞ are given in Table 3 and those of ðU; D; eJÞ are shown in Table 4. Based on Definition 8 and Theorem 1, we can derive all the non-redundant L decision rules from K as follows:     f½5; 14g f½20; 29g f½9; 17g f½1; 6g f½2; 9g f½4; 12g r1: If ; ; ; , then ; . a b c d d1 d2     f½7; 14g f½20; 25; ½27; 29g f½9; 10; ½11; 17g f½2; 6g f½3; 9g f½4; 6; ½9; 12g ; ; ; , then ; . r2: If a b c d d1 d2     f½5; 13g f½20; 27g f½10; 16g f½1; 4g f½2; 6g f½4; 9g r3: If . ; ; ; , then ; a b c d d1 d2     f½5; 14g f½20; 23; ½25; 29g f½9; 12; ½13; 17g f½1; 6g f½2; 9g f½4; 5; ½6; 12g ; ; ; , then ; . r4: If a b c d d1 d2     f½7; 14g f½23; 25; ½27; 29g f½9; 10; ½12; 17g f½2; 6g f½3; 9g f½5; 6; ½9; 12g r5: If ; ; ; , then ; . a b c d d1 d2     f½8; 14g f½20; 25; ½27; 29g f½9; 10; ½11; 17g f½3; 6g f½4; 9g f½4; 6; ½9; 12g ; ; ; , then ; . r6: If a b c d d1 d2     f½7; 13g f½20; 25g f½11; 15g f½2; 4g f½3; 6g f½4; 6g r7: If ; ; ; , then ; . a b c d d1 d2     f½8; 14g f½27; 29g f½9; 10; ½14; 17g f½3; 6g f½4; 9g f½9; 12g ; ; ; , then ; . r8: If a b c d d1 d2     f½5; 9g f½20; 23; ½25; 27g f½10; 12; ½13; 16g f½1; 3g f½2; 4g f½4; 5; ½6; 9g r9: If ; ; ; , then ; . a b c d d1 d2     f½7; 13g f½23; 25g f½12; 15g f½2; 4g f½3; 6g f½5; 6g ; ; ; , then ; . r10: If a b c d d1 d2     f½4; 5g f½4; 6g f½8; 13g f½20; 25g f½11; 15g f½3; 4g ; ; ; , then ; . r11: If a b c d d1 d2 It can easily be checked that the above real decision formal context K is inconsistent under the reduction framework in [25]. Therefore, the method in [25] is not applicable to this real decision formal context for knowledge reduction. Table 2 A real decision formal context K ¼ ðU; A; eI; D; eJÞ. U

a

b

c

d

d1

d2

x1 x2 x3 x4

{[8, 14]} {[5, 9]} {[7, 13]} {[8, 13]}

{[27, 29]} {[20, 23],[25, 27]} {[23, 25]} {[20, 25]}

{[9, 10], [14, 17]} {[10, 12], [13, 16]} {[12, 15]} {[11, 15]}

{[3, 6]} {[1, 3]} {[2, 4]} {[3, 4]}

{[4, 9]} {[2, 4]} {[3, 6]} {[4, 5]}

{[9, 12]} {[4, 5], [6, 9]} {[5, 6]} {[4, 6]}

J. Li et al. / Information Sciences 189 (2012) 191–207

199

Table 3 All the large real concepts of ðU; A; eIÞ. Label

Large real concept    f½5; 14g f½20; 29g f½9; 17g f½1; 6g fx1 ; x2 ; x3 ; x4 g; ; ; ; a b c d    f½7; 14g f½20; 25; ½27; 29g f½9; 10; ½11; 17g f½2; 6g fx1 ; x3 ; x4 g; ; ; ; a b c d    f½5; 13g f½20; 27g f½10; 16g f½1; 4g fx2 ; x3 ; x4 g; ; ; ; a b c d    f½5; 14g f½20; 23; ½25; 29g f½9; 12; ½13; 17g f½1; 6g fx1 ; x2 g; ; ; ; a b c d    f½7; 14g f½23; 25; ½27; 29g f½9; 10; ½12; 17g f½2; 6g fx1 ; x3 g; ; ; ; a b c d    f½8; 14g f½20; 25; ½27; 29g f½9; 10; ½11; 17g f½3; 6g fx1 ; x4 g; ; ; ; a b c d    f½7; 13g f½20; 25g f½11; 15g f½2; 4g fx3 ; x4 g; ; ; ; a b c d    f½8; 14g f½27; 29g f½9; 10; ½14; 17g f½3; 6g fx1 g; ; ; ; a b c d    f½5; 9g f½20; 23; ½25; 27g f½10; 12; ½13; 16g f½1; 3g fx2 g; ; ; ; a b c d    f½7; 13g f½23; 25g f½12; 15g f½2; 4g fx3 g; ; ; ; a b c d    f½8; 13g f½20; 25g f½11; 15g f½3; 4g fx4 g; ; ; ; a b c d    f½; g f½; g f½; g f½; g ;; ; ; ; a b c d

LC0 LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11

Table 4 All the large real concepts of ðU; D; eJÞ. Label LC00 LC01 LC02 LC03 LC04 LC05 LC06 LC07 LC08 LC09 LC010 LC011 LC012

Large real concept    f½2; 9g f½4; 12g fx1 ; x2 ; x3 ; x4 g; ; d1 d2    f½3; 9g f½4; 6; ½9; 12g fx1 ; x3 ; x4 g; ; d1 d2    f½2; 6g f½4; 9g fx2 ; x3 ; x4 g; ; d1 d2    f½2; 9g f½4; 5; ½6; 12g fx1 ; x2 g; ; d1 d2    f½3; 9g f½5; 6; ½9; 12g fx1 ; x3 g; ; d1 d2    f½4; 9g f½4; 6; ½9; 12g fx1 ; x4 g; ; d1 d2    f½3; 6g f½4; 6g fx3 ; x4 g; ; d1 d2    f½2; 5g f½4; 9g fx2 ; x4 g; ; d1 d2    f½4; 9g f½9; 12g fx1 g; ; d1 d2    f½2; 4g f½4; 5; ½6; 9g fx2 g; ; d1 d2    f½3; 6g f½5; 6g fx3 g; ; d1 d2    f½4; 5g f½4; 6g fx4 g; ; d1 d2    f½; g f½; g ;; ; d1 d2

5. Implementation of the knowledge reduction in real decision formal contexts Based on the proposed theoretical reduction framework in Section 4, we give in this section a method to implement the knowledge reduction of real decision formal contexts. As well known in the rough set theory, reduct computation can be translated into the calculation of the prime implicants of a Boolean function [18,19], and discernibility matrix and Boolean function are two basic tools for the computation. Recently, this approach has been used extensively in formal concept analysis (see e.g. [16,22,24,25,28]). We now employ this approach to compute all the L (or S) reducts of a real decision formal context.

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e i Þ; ðX j ; B e j Þ 2 BL ðU; A; eIÞ, define Definition 14. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context. For ðX i ; B

( e i Þ; ðX j ; B e j ÞÞ ¼ D ððX i ; B L

e j ; if X i  X j 2 UL ðU; A; eIÞ; or X j  X i 2 UL ðU; A; eIÞ; ei  B B

ð20Þ

otherwise;

;;

ei  B e j ¼ fa 2 Aj B e i ðaÞ – B e j ðaÞg. We call DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ the L discernibility attribute set of ðX i ; B e i Þ and ðX j ; B e j Þ in K, where B and

e i Þ; ðX j ; B e j ÞÞjðX i ; B e i Þ; ðX j ; B e j Þ 2 BL ðU; A; eIÞg DL ¼ fDL ððX i ; B

ð21Þ

the L discernibility matrix of K. e i Þ; ðX j ; B e j Þ; ðX k ; B e k Þ 2 BL ðU; A; eIÞ. Then Proposition 5. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context and ðX i ; B e i Þ; ðX i ; B e i ÞÞ ¼ ;; (i) DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ ¼ DL ððX j ; B e j Þ; ðX i ; B e i ÞÞ; (ii) DL ððX i ; B L L e e e e e k Þ; ðX j ; B e j ÞÞ holds if X i ; X j ; X k 2 UL ðU; A; eIÞ. (iii) D ððX i ; B i Þ; ðX j ; B j ÞÞ # D ððX i ; B i Þ; ðX k ; B k ÞÞ [ DL ððX k ; B Proof. The proofs of (i) and (ii) are trivial. e i Þ; ðX j ; B e j ÞÞ, it follows from Definition 14 that B e i ðaÞ – B e j ðaÞ. If (iii) Let X i ; X j ; X k 2 UL ðU; A; eIÞ. For any a 2 DL ððX i ; B e e e e B i ðaÞ – B k ðaÞ, then Xi – Xk, which implies that X i  X k 2 UL ðU; A; IÞ or X k  X i 2 UL ðU; A; IÞ. As a result, e i Þ; ðX k ; B e k ÞÞ. On the other hand, if B e i ðaÞ ¼ B e k ðaÞ, then B e k ðaÞ – B e j ðaÞ. Analogously, we can prove a 2 DL ððX i ; B L L L e k Þ; ðX j ; B e j ÞÞ. To sum up, a 2 D ððX i ; B e i Þ; ðX k ; B e k ÞÞ [ D ððX k ; B e k Þ; ðX j ; B e j ÞÞ. h a 2 D ððX k ; B Theorem 3. Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context and E # A. Then E is an L consistent set of K if and only if e i Þ, ðX j ; B e j ÞÞ – ; for any nonempty set DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ 2 DL . E \ DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ 2 DL , it follows from Definition 14 that X i 2 UL ðU; A; eIÞ and Xj  Xi, or Proof. ()) For any nonempty set DL ððX i ; B e e i Þ; ðX j ; B e j ÞÞ – ;, we first prove B eij – B ejj . X j 2 UL ðU; A; IÞ and Xi  Xj. In order to prove E \ DL ððX i ; B E E If X i 2 UL ðU; A; eIÞ and Xj  Xi, then it can be known from Theorem 2 that X i 2 UL ðU; E; eI E Þ. According to (iii) in Proposition " e i j . As a result, ðX i ; B e i j Þ 2 BL ðU; E; eI E Þ. If B eij ¼ B e j j , it follows from (i) in Proposition 2 that 2, we have that X i E ¼ X "i jE ¼ B E E E E #E #E # e e e e e j j . On the other hand, if X j 2 UL ðU; A; eIÞ X j ¼ B j # ð B j jE Þ ¼ ð B i jE Þ ¼ X i , which is in contradiction with Xj  Xi. Thus, B i jE – B E eij – B ejj . and Xi  Xj, we can analogously prove B E E eij – B e j j , there exists c 2 E such that ð B e i j ÞðcÞ – ð B e j j ÞðcÞ, which yields B e i ðcÞ – B e j ðcÞ. Therefore, Based on B E E E E e i Þ; ðX j ; B e j ÞÞ ¼ E \ fa 2 Aj B e i ðaÞ – B e j ðaÞg ¼ fa 2 Ej B e i ðaÞ – B e j ðaÞg – ;. E \ DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ 2 DL , if E \ DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ – ;, we have B eij – B e j j . According to (Ü) For any nonempty set DL ððX i ; B E E e Theorem 2, it is sufficient to prove UL ðU; E; I E Þ ¼ UL ðU; A; eIÞ. e 2 IL ðU; A; eIÞ such that ðX; BÞ e 2 BL ðU; A; eIÞ. From (iii) in Proposition 1, we On one hand, for any X 2 UL ðU; A; eIÞ, there exists B have ðX "E #E ; X "E Þ 2 BL ðU; E; eI E Þ. Since UL ðU; E; eI E Þ # UL ðU; A; eIÞ, we conclude that ðX "E #E ; ðX "E #E Þ" Þ 2 BL ðU; A; eIÞ. If X "E #E – X, then we have X "E #E  X due to X # X "E #E . Combining X "E #E  X with X 2 UL ðU; A; eIÞ, we can obtain from Definition 14 that e ðX "E #E ; ðX "E #E Þ" Þ – ;. From the conclusion that B eij – B e j j for any nonempty set DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ 2 DL , we have DL ððX; BÞ; E E "E #E " " "E "E #E "E "E #E " e e e Bj – ðX Þ j . However, Bj ¼ X j ¼ X ¼ X ¼ ðX Þ j , which is in contradiction to Bj – ðX "E #E Þ" j . Therefore, E

E

E

E

E

E

E

X ¼ X "E #E 2 UL ðU; E; eI E Þ, and consequently, UL ðU; A; eIÞ # UL ðU; E; eI E Þ. On the other hand, for any X 2 UL ðU; E; eI E Þ, there exists Y 2 UL ðU; D; eJÞ such that X E Y = 1 and X E Y = 1. According to X E Y = 1, we have X A Y = 1. Thus, to prove X 2 UL ðU; A; eIÞ, it is sufficient to check X A Y = 1. If X A Y = 0, then there exists X 0 2 UL ðU; A; eIÞ such that X  X0 and X0 # Y. Noting that UL ðU; A; eIÞ # UL ðU; E; eI E Þ, we conclude X 0 2 UL ðU; E; eI E Þ. As a result, X E Y = 0, which is in contradiction with X E Y = 1. Therefore, UL ðU; E; eI E Þ # UL ðU; A; eIÞ. h Proposition 6. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ; c 2 A is an L indispensable attribute of K if and only if there e i Þ; ðX j ; B e j ÞÞ 2 DL such that DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ ¼ fcg. exists DL ððX i ; B Proof. If c 2 A is an L indispensable attribute of K, then A  {c} is an L inconsistent set of K. From Theorem 3, there exists a e i Þ; ðX j ; B e j ÞÞ 2 DL such that ðA  fcgÞ \ DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ ¼ ;, which yields DL ððX i ; B e i Þ; ðX j ; B e j ÞÞ ¼ fcg. nonempty set DL ððX i ; B L L L e e e e e i Þ; Conversely, if there exists D ððX i ; B i Þ; ðX j ; B j ÞÞ 2 D such that D ððX i ; B i Þ; ðX j ; B j ÞÞ ¼ fcg, then ðA  fcgÞ \ DL ððX i ; B e j ÞÞ ¼ ;. According to Theorem 3, we can conclude that A  {c} is an L inconsistent set of K, and consequently, c is an ðX j ; B L indispensable attribute of K. h

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Definition 15. Let DL be the L discernibility matrix of a real decision formal context K ¼ ðU; A; eI; D; eJÞ. We call

n_

^

FL ¼

o e i Þ; ðX j ; B e j ÞÞ DL ððX i ; B

ð22Þ

DL ððX i ;e B i Þ;ðX j ;e B j ÞÞ2DL DL ððX i ;e B i Þ;ðX j ;e B j ÞÞ–;

the L discernibility function of K. Theorem 4. For a real decision formal context K ¼ ðU; A; eI; D; eJÞ, let L

F ¼

rt k _ ^ t¼1

!

as

ð23Þ

s¼1

be the minimal disjunctive normal form of the L discernibility function of K, where Then Et = {asjs 6 rt}(t 6 k) are all the L reducts of K.

Vr t

s¼1 as ðt

6 kÞ are all the prime implicants of FL.

Proof. According to the definition of the minimal disjunctive normal form of a Boolean function [19], the proof follows immediately from Theorem 3 and Proposition 6. h

Based on Theorem 4, we can compute all the L reducts of a real decision formal context by the following algorithm: Input: A real decision formal context K ¼ ðU; A; eI; D; eJÞ. Output: All the L reducts of K. Step 1: Compute BL ðU; A; eIÞ; BL ðU; D; eJÞ and UL ðU; A; eIÞ according to Definition 4 and Eq. (18). Step 2: Compute the L discernibility matrix DL of K by Eqs. (20) and (21). Step 3: Compute the L discernibility function FL of K by Eq. (22).

W V t Step 4: Translate FL into its minimal disjunctive normal form in Eq. (23), i.e., FL ¼ kt¼1 rs¼1 as . Step 5: Output Et = {asjs 6 rt}(t 6 k).

It should be pointed out that the above algorithm can also be applied to computing all the S reducts of a real decision formal context by replacing the operators and notations associated to the large real concept lattice with the corresponding operators and notations associated to the strict real concept lattice. In what follows, we take the real decision formal context in Table 2 as an example to demonstrate the application of the above algorithm. Example 3. For the real decision formal context K ¼ ðU; A; eI; D; eJÞ in Table 2, it follows from Eqs. (13), (15) and (18) that UL ðU; A; eIÞ ¼ UL ðU; A; eIÞ  f;g, where UL ðU; A; eIÞ is shown in Table 3. Furthermore, the L discernibility matrix DLof K is given in Table 5. According to Definition 15 and Theorem 4, we can obtain the L discernibility function FL of K as

FL ¼ ða _ b _ c _ dÞ ^ ðb _ cÞ ^ ða _ dÞ ¼ ða ^ cÞ _ ða ^ bÞ _ ðb ^ dÞ _ ðc ^ dÞ: Thus, K has four L reducts: E1 = {a, c}, E2 = {a, b}, E3 = {b, d}, and E4 = {c, d}.

Table 5 The L discernibility matrix DL of ðU; A; eI; D; eJÞ.

LC0 LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11

LC0

LC1

LC2

LC3

LC4

LC5

LC6

LC7

LC8

LC9

LC10

LC11

; A A bc A A A A A A A ;

A ; A A bc ad A A A A A ;

A A ; A A A A A A A A ;

bc A A ; A A A A A A A ;

A bc A A ; A A A A A A ;

A ad A A A ; A bc A A A ;

A A A A A A ; A A bc ad ;

A A A A A bc A ; A A A ;

A A A A A A A A ; A A ;

A A A A A A bc A A ; A ;

A A A A A A ad A A A ; ;

; ; ; ; ; ; ; ; ; ; ; ;

202

J. Li et al. / Information Sciences 189 (2012) 191–207 Table 6 All the large real concepts of ðU; E1 ; eI E1 Þ. Label

Large real concept    f½5; 14g f½9; 17g fx1 ; x2 ; x3 ; x4 g; ; a c    f½7; 14g f½9; 10; ½11; 17g fx1 ; x3 ; x4 g; ; a c    f½5; 13g f½10; 16g fx2 ; x3 ; x4 g; ; a c    f½5; 14g f½9; 12; ½13; 17g fx1 ; x2 g; ; a c    f½7; 14g f½9; 10; ½12; 17g fx1 ; x3 g; ; a c    f½8; 14g f½9; 10; ½11; 17g fx1 ; x4 g; ; a c    f½7; 13g f½11; 15g fx3 ; x4 g; ; a c    f½8; 14g f½9; 10; ½14; 17g fx1 g; ; a c    f½5; 9g f½10; 12; ½13; 16g fx2 g; ; a c    f½7; 13g f½12; 15g fx3 g; ; a c    f½8; 13g f½11; 15g fx4 g; ; a c    f½; g f½; g ;; ; a c

LC0 LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8 LC9 LC10 LC11

By using the L reduct E1 = {a, c}, we can obtain the reduced real decision formal context ðU; E1 ; eI E1 ; D; eJÞ. All the large real concepts of ðU; E1 ; eI E1 Þ are given in Table 6 and all the large real concepts of ðU; D; eJÞ are the same as those in Table 4. Based on Theorem 1, we can derive all the non-redundant L decision rules from ðU; E1 ; eI E1 ; D; eJÞ as follows:

r 01 : If

r 02 : If

r 03 : If

r 04 : If

r 05 : If

r 06 : If

r 07 : If



   f½5; 14g f½9; 17g f½2; 9g f½4; 12g : ; ; then ; a c d1 d2



   f½3; 9g f½4; 6; ½9; 12g f½7; 14g f½9; 10; ½11; 17g : ; ; then ; a c d1 d2



   f½5; 13g f½10; 16g f½2; 6g f½4; 9g : ; ; then ; a c d1 d2



   f½5; 14g f½9; 12; ½13; 17g f½2; 9g f½4; 5; ½6; 12g : ; ; then ; a c d1 d2



   f½7; 14g f½9; 10; ½12; 17g f½3; 9g f½5; 6; ½9; 12g : ; ; then ; a c d1 d2



   f½8; 14g f½9; 10; ½11; 17g f½4; 9g f½4; 6; ½9; 12g : ; ; then ; d1 a c d2



   f½7; 13g f½11; 15g f½3; 6g f½4; 6g : ; ; then ; a c d1 d2

r 08 : If f

r 09 : If

r 010 : If

f½8; 14g f½9; 10; ½14; 17g ; g; then a c





 f½4; 9g f½9; 12g ; : d1 d2

   f½5; 9g f½10; 12; ½13; 16g f½2; 4g f½4; 5; ½6; 9g : ; then ; ; a c d1 d2



   f½7; 13g f½12; 15g f½3; 6g f½5; 6g : ; ; then ; a c d1 d2

J. Li et al. / Information Sciences 189 (2012) 191–207

r 011 : If



203

   f½8; 13g f½11; 15g f½4; 5g f½4; 6g : ; ; then ; a c d1 d2

It can be verified that the L decision rules r1–r11 obtained in Example 2 can be implied by the L decision rules r 01 —r011 . Moreover, it is easy to observe that the L decision rules derived from the reduced real decision formal context are more compact. Similarly, all the strict real concepts of ðU; A; eIÞ and those of ðU; D; eJÞ are shown in Tables 7 and 8, respectively. The S discernibility matrix DS of K is given in Table 9, and the S discernibility function FS of K is

FS ¼ ða _ b _ c _ dÞ ^ ða _ dÞ ^ ðb _ cÞ ¼ ða ^ bÞ _ ða ^ cÞ _ ðb ^ dÞ _ ðc ^ dÞ:

Table 7 All the strict real concepts of ðU; A; eIÞ. Label

Strict real concept    f½5; 14g f½20; 29g f½9; 17g f½1; 6g ;; ; ; ; a b c d    f½8; 14g f½27; 29g f½9; 10; ½14; 17g f½3; 6g fx1 g; ; ; ; a b c d    f½5; 9g f½20; 23; ½25; 27g f½10; 12; ½13; 16g f½1; 3g fx2 g; ; ; ; a b c d    f½7; 13g f½23; 25g f½12; 15g f½2; 4g fx3 g; ; ; ; a b c d    f½8; 13g f½20; 25g f½11; 15g f½3; 4g fx4 g; ; ; ; a b c d    f½8; 9g f½27; 27g f½10; 10; ½14; 16g f½3; 3g fx1 ; x2 g; ; ; ; a b c d    f½7; 9g f½23; 23; ½25; 25g f½12; 12; ½13; 15g f½2; 3g fx2 ; x3 g; ; ; ; a b c d    f½8; 13g f½23; 25g f½12; 15g f½3; 4g fx3 ; x4 g; ; ; ; a b c d    f½8; 13g f½; g f½14; 15g f½3; 4g fx1 ; x3 ; x4 g; ; ; ; a b c d    f½8; 9g f½23; 23; ½25; 25g f½12; 12; ½13; 15g f½3; 3g fx2 ; x3 ; x4 g; ; ; ; a b c d    f½8; 9g f½; g f½14; 15g f½3; 3g fx1 ; x2 ; x3 ; x4 g; ; ; ; a b c d

SC0 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10

Table 8 All the strict real concepts of ðU; D; eJÞ. Label SC00 SC01 SC02 SC03 SC04 SC05 SC06 SC07 SC08 SC09 SC010 SC011

Strict real concept    f½2; 9g f½4; 12g ;; ; d1 d2    f½4; 9g f½9; 12g fx1 g; ; d1 d2    f½2; 4g f½4; 5; ½6; 9g fx2 g; ; d1 d2    f½3; 6g f½5; 6g fx3 g; ; d1 d2    f½4; 5g f½4; 6g fx4 g; ; d1 d2    f½4; 4g f½9; 9g fx1 ; x2 g; ; d1 d2    f½3; 4g f½5; 5; ½6; 6g fx2 ; x3 g; ; d1 d2    f½4; 6g f½; g fx1 ; x3 g; ; d1 d2    f½4; 5g f½5; 6g fx3 ; x4 g; ; d1 d2    f½4; 5g f½; g fx1 ; x3 ; x4 g; ; d1 d2    f½4; 4g f½5; 5; ½6; 6g fx2 ; x3 ; x4 g; ; d1 d2    f½4; 4g f½; g fx1 ; x2 ; x3 ; x4 g; ; d1 d2

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J. Li et al. / Information Sciences 189 (2012) 191–207

Table 9 The S discernibility matrix DS of ðU; A; eI; D; eJÞ.

SC0 SC1 SC2 SC3 SC4 SC5 SC6 SC7 SC8 SC9 SC10

SC0

SC1

SC2

SC3

SC4

SC5

SC6

SC7

SC8

SC9

SC10

; ; ; ; ; ; ; ; ; ; ;

; ; A A A A A A A A A

; A ; A A A A A A A A

; A A ; A A A ad A A A

; A A A ; A A bc bc A A

; A A A A ; A A A bc bc

; A A A A A ; A A ad A

; A A ad bc A A ; bc A A

; A A A bc A A bc ; A ad

; A A A A bc ad A A ; bc

; A A A A bc A A ad bc ;

Hence, K has four S reducts: E1 = {a, b}, E2 = {a, c}, E3 = {b, d}, and E4 = {c, d}. It can easily be checked that all the S decision rules derived from K can be implied by the S decision rules derived from each of the reduced real decision formal contexts ðU; Ei ; eI Ei ; D; eJÞ ði ¼ 1; 2; 3; 4Þ. 6. Numerical experiments We conduct in this section some experiments to demonstrate the implementation of the proposed reduction method and furthermore compare our reduction method with the one proposed by Yang et al. [25] in terms of the efficiency. It should be noted that the time complexity of both the methods in this paper and in [25] is exponential. Therefore, the experiments are only feasible for the real decision formal contexts with small or medium sizes. Furthermore, although our method is applicable to any real decision formal contexts, the method in [25] requires that the real decision formal contexts to be reduced are consistent under the given implication mapping. In order to achieve the task of comparing the efficiency of the two reduction methods, we first give two ways of combining small real decision formal contexts into a medium one that can keep the consistency defined in [25]. 6.1. Two ways of constructing a real decision formal context with medium size Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context with U = {x1, . . . , xr},A = {a1, . . . , as} and D = {d1, . . . , dt}. We denote by ðe IÞ ðe JÞ Me ¼ ðmij Þrs and Me ¼ ðmik Þrt the matrices of the conditional real relation values and the decision real relation values, I J respectively. That is, ðe IÞ mij ¼ eIðxi ; aj Þ; ðe JÞ mik ¼ eJðxi ; dk Þ;

i 2 f1; . . . ; rg; j 2 f1; . . . ; sg; i 2 f1; . . . ; rg; k 2 f1; . . . ; tg:

For convenience, we call for short Me and Me the conditional real relation matrix and the decision real relation matrix of K, I J respectively. 6.1.1. Symmetrical mergence For two small real decision formal contexts Kk ¼ ðU k ; Ak ; eI k ; Dk ; eJ k Þ ðk ¼ 1; 2Þ with jU1j = jU2j, jA1j = jA2j and jD1j = jD2j, let Me and Me (k = 1, 2) be their respective conditional and decision real relation matrices. We construct two matrices Ik

Jk

0

Me I B 1 B Me B I2 ðnÞ M ¼B B .. eI B . @ Me I2

Me I2 Me I1

.. . Me

I2

1 . . . Me I2 C

Me C I2 C C .. .. C; . . C A

Me I1

0

Me J B 1 B Me B J2 ðnÞ M ¼B B .. eJ B . @ Me J2

Me J2 Me J1

.. . Me

J2

1

Me J2 C

Me C J2 C C .. .. C; . . C A

Me J1

ðnÞ

ðnÞ

where n is the number of Me (or Me ) in the diagonal of M (or M ). Then we obtain a real decision formal context eI eJ I1 J1 ðnÞ ðnÞ K ¼ ðU; A; eI; D; eJÞ with its conditional and decision real relation matrices being M and M , respectively. We say that K eI eJ is n times of symmetrical mergence of K2 with respect to K1 . 6.1.2. Vertical concatenation Let K ¼ ðU; A; eI; D; eJÞ be a real decision formal context, and Me and Me be its conditional and decision real relation matriI J ces, respectively. We construct two matrices

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J. Li et al. / Information Sciences 189 (2012) 191–207

 T ðmÞ M ¼ MeT ; MeT ; . . . ; MeT ; eI I I I

 T ðmÞ M ¼ MeT ; MeT ; . . . ; MeT ; eJ J J J ðmÞ

ðmÞ

where ‘‘T’’ denotes the transpose of a matrix, and m is the number of Me (or Me) in M (or M ). Then we have a real decieI eJ I J ðmÞ ðmÞ sion formal context K0 ¼ ðU 0 ; A0 ; eI 0 ; D0 ; eJ 0 Þ with its conditional and decision real relation matrices being M and M , respeceI eJ 0 tively. We say that K is m times of vertical concatenation of K. 6.2. Experiments on the large real concept lattices We choose five real decision formal contexts, denoted by Data sets 1, 2, 3, 4 and 5, respectively, to conduct the experiments. Data set 1 is obtained by replacing the value {[4, 5]} in the fourth row and fifth column of Table 2 with {[4, 6]}. Data set 2 is taken from Table 4 in [25]. An auxiliary data set KL is constructed by replacing the values in each row of Data set 2 with

f½4; 11g; f½7; 13g; f½6; 17g; f½1; 9g; f½1; 7g; f½4; 15g: Data sets 3, 4 and 5 are generated by ten times of vertical concatenation of such three data sets that are two, three and four times of symmetrical mergence of Data set 2 with respect to KL , respectively. It can easily be checked that Data sets 1, 2, 3, 4 and 5 are all consistent under the knowledge reduction framework in [25] for the large real concept lattice. The running time on a common personal computer for these five data sets is reported in Table 10. It can be seen that, for both the reduction methods, computing all the reducts based on the large real concept lattices can be completed in a reasonable time when the data sets are of small or medium sizes. But our method takes less time than the method in [25] especially for the medium data sets because our method does not need to examine the consistency of the real decision formal contexts. 6.3. Experiments on the strict real concept lattices In addition to Data sets 1 and 2, we construct other three data sets, denoted by Data sets 30 , 40 and 50 , respectively, to perform the experiments on the strict real concept lattices. We first construct an auxiliary data set KS by replacing the values in each row of Data set 2 with

f½3; 4g; f½5; 6g; f½5; 6g; f½9; 10g; f½7; 8g; f½2; 3g: Data sets 30 , 40 and 50 are then obtained by ten times of vertical concatenation of such three data sets that are two, four and five times of symmetrical mergence of Data set 2 with respect to KS , respectively. Data sets 1, 2, 30 , 40 and 50 are all consistent under the knowledge reduction framework in [25] for the strict real concept lattice. The experimental results in terms of the running time are reported in Table 11. It seems that, for both the reduction methods, computing all the reducts based on the strict real concept lattices can be completed faster than that based on the large real concept lattices. Our method still takes less time than the method in [25].

Table 10 Running time for computing all the reducts based on the large real concept lattices. Data set

Data Data Data Data Data

set set set set set

jUj

1 2 3 4 5

4 4 80 120 160

jAj

4 4 8 12 16

jDj

2 2 4 6 8

Running time (s) The method in [25]

Our method

0.076 0.077 5.835 39.955 2329.807

0.069 0.072 5.119 36.343 2245.241

Table 11 Running time for computing all the reducts based on the strict real concept lattices. Data set

Data Data Data Data Data

set set set set set

1 2 30 40 50

jUj

jAj

jDj

4 4 80 160 200

4 4 8 16 20

2 2 4 8 10

Running time (s) The method in [25]

Our method

0.065 0.081 4.648 75.573 859.308

0.058 0.074 4.379 69.567 702.978

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7. Final remarks In formal concept analysis, rule acquisition is always one of the main purposes in the analysis of all kinds of decision formal contexts. Generally speaking, the decision rules derived directly from a real decision formal context are not concise or compact. Therefore, although there have been some reduction methods for consistent real decision formal contexts under a given implication mapping between the conditional and the decision real concept lattices, how to develop some reduction approaches that can derive more compact decision rules from the reduced real decision formal context and are applicable to any real decision formal contexts is still a very important topic. In this paper, we have built a framework of knowledge reduction in real decision formal contexts from the perspective of rule acquisition and have proposed a corresponding reduction method by constructing a discernibility matrix and its associated Boolean function. The proposed reduction method is applicable to any real decision formal contexts and more compact L (or S) decision rules can be derived from the reduced real decision formal context. Some numerical experiments have also been conducted to assess the efficiency of the proposed method. It should be pointed out that, like most of the existing reduction approaches for formal contexts and decision formal contexts, the implementation of the proposed reduction method is still based on the discernibility matrix and Boolean function. Such Boolean reasoning based reduction methods are computationally expensive and they are even impossibly implemented for large databases. Therefore, some efficient reduction algorithms under the proposed theoretical reduction framework are still needed and deserve to be further studied. In this aspect, developing some heuristic algorithms may be one of the promising directions for the future research although the reduction results obtained by this kind of algorithms are in general approximate.

Acknowledgements The authors are grateful to the anonymous reviewers and the editor for their valuable comments and suggestions which lead to a significant improvement on the manuscript. This work was supported by the National Natural Science Foundations of China (Nos. 10971161 and 70861001).

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