On rule acquisition in decision formal contexts

ORIGINAL ARTICLE

On rule acquisition in decision formal contexts

Jinhai Li • Changlin Mei • Cherukuri Aswani Kumar •

Xiao Zhang

Received: 6 August 2012 / Accepted: 16 January 2013 / Published online: 6 February 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract Rule acquisition is one of the main purposes in

the analysis of decision formal contexts. Up to now, there

have existed several types of rules (e.g., the decision rules

and the granular rules) in decision formal contexts. This

study firstly proposes a new algorithm with less time

complexity for deriving the non-redundant decision rules

from a decision formal context. Then, we invesigate deci-

sion rules and the granular rules in the consistent decision

formal contexts and make a contrast between the decision

rule oriented knowledge reduction and the granular rule

oriented knowledge reduction. Finally, some experiments

are conducted to assess the efficiency of the proposed rule

acquisition algorithm.

Keywords Formal concept analysis � Formal context �Decision formal context � Concept lattice � Rule acquisition

1 Introduction

Formal concept analysis (FCA), proposed by Wille [1], is

oriented towards the discovery of formal concepts and the

construction of concept hierarchies. Nowadays, this theory

has shown a trend of multidisciplinary intersection and

fusion and hence has become an effective tool for con-

ceptual data analysis and knowledge processing.

FCA starts with the notion of a formal context (U, A, I)

which consists of the object set U, the attribute set A and

the binary relation I � U � A indicating that each object of

U has what attributes in A. In FCA, the basic tool used to

analyze databases is the concept lattice which is constituted

by all the formal concepts of a formal context together

with the subconcept–superconcept-relation. According to

Wille’s definition [1], a formal concept is an ordered pair

(X, B) in which X � U contains exactly those objects

shared by all the attributes in B and B � A contains exactly

those attributes that all the objects in X have in common. In

recent years, many studies [2–9] have been devoted to the

issue of reducing the size of the concept lattice of a formal

context in order to improve the understandability of the

resulting concept lattice. In FCA, a useful way of charac-

terizing attribute dependencies in a formal context is via

attribute implication rules or association rules. How to

derive these rules efficiently has drawn much attention in

the literature [10–18].

Decision formal contexts [19], a useful extension of the

formal contexts, were proposed to implement decision

analysis using the concept lattice. Rule acquisition is one of

the main purposes in the analysis of decision formal con-

texts. A few studies [20–25] have recently been devoted to

the rule acquisition in decision formal contexts. To the best

of our knowledge, there have existed several types of rules

(e.g., the decision rules [23–25] and the granular rules [22])

in decision formal contexts. Although a rule acquisition

method for deriving all the non-redundant decision rules

from a decision formal context has been proposed in [24],

this method depends heavily on both the conditional

J. Li (&) � C. Mei � X. Zhang

School of Mathematics and Statistics, Xi’an Jiaotong University,

Xi’an 710049, Shaanxi, People’s Republic of China

e-mail: [email protected]

C. Mei


X. Zhang


C. A. Kumar

School of Information Technology and Engineering,

VIT University, Vellore, India


123

Int. J. Mach. Learn. & Cyber. (2013) 4:721–731

DOI 10.1007/s13042-013-0150-z

concept lattice and the decision concept lattice. Since the

construction of the concept lattice for a large database is

very time-consuming, the method proposed in [24] may not

be efficient enough in handling a large database. If some

efficient methods for extracting the decision rules from a

decision formal context are developed, it is of interest to

investigate the relationship between the decision rules and

the granular rules. Furthermore, knowledge reduction is

always one of the important issues in FCA. Therefore, it is

also essential to clarify the relation and the difference

between the decision rule oriented knowledge reduction

[24] and the granular rule oriented knowledge reduction

[22].

In this paper, we first propose an efficient algorithm to

derive the non-redundant decision rules from a decision

formal context. Then we investigate the relationship

between the non-redundant decision rules and the granular

rules in the consistent decision formal contexts and further

make a contrast between the decision rule oriented

knowledge reduction and the granular rule oriented

knowledge reduction. Finally, we conduct some experi-

ments to assess the efficiency of the proposed rule acqui-

sition algorithm.

2 Preliminaries

In this section, we briefly review some basic notions related

to FCA in order to make the paper self-contained.

Definition 1 ([1]) A formal context is a triple (U, A, I)

consisting of the object set U (called the universe of dis-

course), the attribute set A and the binary relation I �U � A in which (x, a) [ I indicates that the object x has the

attribute a and ðx; aÞ 62 I means the opposite.

A formal context (U, A, I) is said to be regular [19] if for

any (x, a) [ U 9 A, the following conditions hold:

(i) there exist a1, a2 [ A such that (x, a1) [ I and

ðx; a2Þ 62 I;(ii) there exist x1, x2 [ U such that (x1, a) [ I and

ðx2; aÞ 62 I:

It should be noted that an irregular formal context

(U, A, I) can be regularized by removing the rows with

their objects having all the attributes or having no attribute

in A and the columns with their attributes being shared by

all the objects in U or not being shared by any object of U.

Such way of the regularization causes no effect on the

analysis results of the formal context. Thus, without loss of

generality, the formal contexts discussed hereinafter are all

assumed to be regular.

Wille [1] introduced a pair of concept forming operators

on a formal context (U, A, I):

X" ¼ fa 2 A j 8x 2 X; ðx; aÞ 2 IgðX � UÞ;B# ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IgðB � AÞ: ð1Þ

That is, X" is the maximal set of the attributes that all the

objects in X have in common, and B; is the maximal set of

the objects shared by all the attributes in B.

Definition 2 ([1]) Let (U, A, I) be a formal context. The

ordered pair (X, B) with X � U and B � A is called a

formal concept of (U, A, I) if X" ¼ B and B; = X. Here,

the sets X and B are called the extent and the intent of the

formal concept (X, B), respectively.

For two formal concepts (X1, B1) and (X2, B2), if X1 �X2 (or B2 � B1), then (X1, B1) is called a subconcept of

(X2, B2), or (X2, B2) is called a superconcept of (X1, B1).

The subconcept–superconcept-relation between the formal

concepts is denoted by B. Then the set of all the formal

concepts of a formal context (U, A, I) together with the

partial order relation B forms a complete lattice which is

denoted by BðU;A; IÞ and is called the concept lattice of

the formal context (U, A, I). In BðU;A; IÞ; the infimum

and the supremum of {(X1, B1), (X2, B2)} are respectively

defined by

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

ð2Þ

In [26], the authors discussed the axiomatic charac-

terization of the concept lattice. It should be pointed out

that except Wille’s concept lattice, there have existed

several other kinds of concept lattices in the classical

FCA, e.g., the rough concept lattice [27], the object-

oriented concept lattice [28] and the property-oriented

concept lattice [29]. The relationship among Wille’s,

object-oriented and property-oriented concept lattices was

discussed in [5]. In this paper, we only focus on Wille’s

concept lattice to discuss the rule acquisition in decision

formal contexts.

Definition 3 ([22]) Let (U, A, I) be a formal context and

E � A: The restriction of I on U 9 E, denoted by IE, is

defined as IE(x, a) = I(x, a) for any (x, a) [ U 9 E. The

formal context (U, E, IE) is called a subcontext of

(U, A, I).

Similar to the formal context (U, A, I), a pair of concept

forming operators can also be defined on the subcontext

(U, E, IE) as follows:

X"E ¼ fa 2 E j 8x 2 X; ðx; aÞ 2 IEgðX � UÞ;B#E ¼ fx 2 U j 8a 2 B; ðx; aÞ 2 IEgðB � EÞ: ð3Þ

In fact, the operators "E and ;E are the restrictions of the

concept forming operators " and ; on the subcontext

(U, E, IE). Similar to the discussion in Definition 2, we can

give the notion of a formal concept with its extent and

722 Int. J. Mach. Learn. & Cyber. (2013) 4:721–731

123

intent in (U, E, IE). Also, the set of all the formal concepts

of the subcontext (U, E, IE) together with the subconcept–

superconcept-relation B forms a complete lattice denoted

by BðU;E; IEÞ; in line with the notation of the concept

lattice of (U, A, I). We denote by BUðU;E; IEÞ the set of

the extents of all the formal concepts of (U, E, IE).

Proposition 1 ([1]) Let (U, A, I) be a formal context and

E � A: For X;X1;X2 � U and B;B1;B2 � E; the following

properties hold:

(i) X1 � X2 ) X"E

2 � X"E

1 ;

(ii) B1 � B2 ) B#E

2 � B#E

1 ;

(iii) X � X"E#E ;B � B#E"E ;

(iv) ðX"E#E ;X"EÞ; ðB#E ;B#E"EÞ 2 BðU;E; IEÞ:

It should be noted that ðfxg"E#E ; fxg"EÞðx 2 UÞ and

ðfag#E ; fag#E"EÞða 2 AÞ are referred to as the object con-

cepts and the attribute concepts [10], respectively. For

brevity, in the rest of the paper, we write them as

ðx"E#E ; x"EÞ and ða#E ; a#E"EÞ; respectively.

Proposition 2 ([1]) Let (U, A, I) be a formal context,

E � A and T be an index set. For Xt � U;Bt � E (t [ T),

we have

[

t2T

Xt

!"E

¼\

t2T

Xt"E and

[

t2T

Bt

!#E

¼\

t2T

Bt#E :

3 An efficient rule acquisition algorithm for decision

formal contexts

Definition 4 ([19]) A decision formal context is a quin-

tuple (U, A, I, D, J), where (U, A, I) and (U, D, J) with

A \ D ¼ ; are two formal contexts. Here, A and D are

called the conditional attribute set and the decision attri-

bute set of (U, A, I, D, J), respectively.

Like the formal context, a decision formal context

P ¼ ðU;A; I;D; JÞ is also said to be regular [24] if both

(U, A, I) and (U, D, J) are regular. The decision formal

context PE ¼ ðU;E; IE;D; JÞ is called a subcontext of P if

(U, E, IE) is a subcontext of (U, A, I). Without loss of

generality, the decision formal contexts discussed herein-

after are all assumed to be regular. The concept lattice of

(U, D, J) is denoted by BðU;D; JÞ and the set of the

extents of all the formal concepts of (U, D, J) is denoted by

BUðU;D; JÞ:

Definition 5 ([24]) Let P ¼ ðU;A; I;D; JÞ be a decision

formal context and E � A: For any ðX;BÞ 2 BðU;E; IEÞ

and ðY ;CÞ 2 BðU;D; JÞ; if X, B, Y and C are all nonempty

and X � Y ; then the expression B! C is called a decision

rule generated between the formal concepts (X, B) and

(Y, C). Here, B and C are called the premise and the con-

clusion of the decision rule B! C; respectively. The set of

all the decision rules generated between the formal con-

cepts in BðU;E; IEÞ and those in BðU;D; JÞ is denoted by

RðPEÞ; where PE ¼ ðU;E; IE;D; JÞ:

Thus, for any B! C 2 RðPEÞ with ðX;BÞ 2 B

ðU;E; IEÞ and ðY;CÞ 2 BðU;D; JÞ; we have that each x 2U having all the attributes in B also has all the attributes in

C. So, the decision rule is in fact an ‘‘If–then’’ conjunctive

rule. That is, B! C means ‘‘If ^B, then ^C’’. Moreover, it

is easy to verify that B! C is supported by and only by

the objects in X.

It should be pointed out that the decision rules have

something to do with both the attribute implication rules

and the association rules (see, e.g. [10, 11, 20] for the

detailed discussion of the attribute implication rules and

e.g. [16, 17] for the association rules). Concretely, a

decision rule is a special attribute implication rule. How-

ever, an attribute implication rule may not be a decision

rule since the premise or the conclusion of an attribute

implication rule is not required to be the intent of a formal

concept but an attribute set only. Also, a decision rule is a

special association rule. But an association rule may not be

a decision rule because the antecedent of an association

rule may not be the intent of a formal concept and the

confidence is often less than one.


formal context and E � A: For B1 ! C1;B2 ! C2 2 R

ðPEÞ; if B1 � B2 and C2 � C1; we say that B2 ! C2 can be

implied by B1 ! C1: We denote this implication relation-

ship by B1 ! C1 ) B2 ! C2: For any B! C 2 RðPEÞ;if there exists B0 ! C0 2 RðPEÞnfB! Cg such that

B0 ! C0 ) B! C; then B! C is said to be redundant in

RðPEÞ; otherwise, B! C is said to be non-redundant in

RðPEÞ: We denote by R�ðPEÞ the set of all the non-

redundant decision rules in RðPEÞ:

It can be known from Definition 6 that for a given

decision formal context, it is more appealing to extract

the non-redundant decision rules since its redundant

decision rules can be implied by the non-redundant

ones.

Let P ¼ ðU;A; I;D; JÞ be a decision formal context. For

any ðX; YÞ 2 BUðU;A; IÞ �BUðU;D; JÞ; define

aðX; YÞ ¼1; if X � Y and there does not exist X0

2 BUðU;A; IÞ such that X � X0 � Y ;0; otherwise;

8<

:

ð4Þ

Int. J. Mach. Learn. & Cyber. (2013) 4:721–731 723

123

bðX; YÞ ¼1; if X � Yand there does not exist Y0

2 BUðU;D; JÞ such that X � Y0 � Y ;0; otherwise.

8<

:

ð5Þ

In [24], the authors put forward a method to derive the

non-redundant decision rules from a decision formal

context. The method can briefly be described as follows:

It is easy to prove that the time complexity of Algorithm

1 is

O ðjUj þ jAjÞjAjjLAj þ jUjjLAj2jLDj þ jUjjLAjjLDj2� �

;

where |LA| and |LD| denote the cardinalities of the concept

lattices BðU;A; IÞ and BðU;D; JÞ; respectively.

In order to enhance the efficiency of the rule acquisition

method in decision formal contexts, we shall propose in the

following a new algorithm of deriving the non-redundant

decision rules which is of less time complexity than

Algorithm 1.

Definition 7 Let P ¼ ðU;A; I;D; JÞ be a decision formal

context and E � A: For any B! C 2 RðPEÞ; if there

exists B0 ! C0 2 RðPEÞ such that B0 � B and C � C0;

we say that B! C is a feedforward redundant decision

rule in RðPEÞ; otherwise, we say that B! C is a feed-

forward non-redundant decision rule in RðPEÞ:


context and E � A: For any B! C 2 RðPEÞ; if there

exists B0 ! C0 2 RðPEÞ such that B0 � B and C � C0;

we say that B! C is a feedback redundant decision rule in

RðPEÞ; otherwise, we say that B! C is a feedback non-

redundant decision rule in RðPEÞ:

Theorem 1 Let P ¼ ðU;A; I;D; JÞ be a decision formal

context, E � A and B! C 2 RðPEÞ: Then the following

statements hold:

(i) B! C is redundant in RðPEÞ iff B! C is a

feedforward or feedback redundant decision rule in

RðPEÞ:(ii) B! C is non-redundant in RðPEÞ iff B! C is both

a feedforward and a feedback non-redundant deci-

sion rule in RðPEÞ:

Proof It is immediate from Definitions 6, 7 and 8. h


context and E � A: For ðX;BÞ 2 BðU;E; IEÞ with X 6¼ ;and B 6¼ ;; if C ¼

Tx2X x"D 6¼ ;; then B! C is a feed-

forward non-redundant decision rule in RðPEÞ; where "D

denotes the operator " on (U, D, J).

Proof Since the ordered pairs ðx"D#D ; x"DÞðx 2 XÞ are

object concepts of (U, D, J) where ;D denotes the operator

; on (U, D, J), the supremum of fðx"D#D ; x"DÞjx 2 Xg;denoted by (Y, C), satisfies

ðY;CÞ ¼W

x2X

ðx"D#D ; x"DÞ ¼S

x2X

x"D#D

� �"D#D

;T

x2X

x"D

!2 BðU;D; JÞ:

Then

X �[

x2X

x"D#D �[

x2X

x"D#D

!"D#D

¼ Y :

Thus, B! C 2 RðPEÞ according to the assumption.

Furthermore, we can prove that B! C is a feedforward

non-redundant decision rule in RðPEÞ: In fact, if B! C is a

feedforward redundant decision rule in RðPEÞ; there exists

B0 ! C0 2 RðPEÞ such that B0 � B and C � C0: Suppose

ðY0;C0Þ 2 BðU;D; JÞ: Then we obtain Y0 � Y: However,

Y ¼[

x2X

x"D#D

!"D#D

�[

x2Y0

x"D#D

!"D#D

¼ Y0;

which is in contradiction with Y0 � Y : h


context and E � A: For ðX;BÞ 2 BðU;E; IEÞ with X 6¼;;B 6¼ ; and C ¼

Tx2X x"D 6¼ ;; then B ? C is non-

redundant in RðPEÞ iff there does not exist ðX0;B0Þ 2BðU;E; IEÞ with X0 6¼ ; and B0 6¼ ; such that B0 ! C 2RðPEÞ and B0 , B.

Proof Necessity. If there exists ðX0;B0Þ 2 BðU;E; IEÞwith X0 6¼ ; and B0 6¼ ; such that B0 ! C 2 RðPEÞ and

B0 , B, it follows from Definition 8 that B ? C is a

feedback redundant decision rule in RðPEÞ: According to

Theorem 1, B ? C is redundant in RðPEÞ; which is in

contradiction with the assumption that B ? C is non-

redundant in RðPEÞ:

Sufficiency. If B ? C is redundant in RðPEÞ; by The-

orem 1 we have that B ? C is a feedforward or feedback

redundant decision rule in RðPEÞ:

(i) If B ? C is a feedforward redundant decision rule in

RðPEÞ; then there exists B0 ! C0 2 RðPEÞ such that

B0 � B and C , C0. Suppose ðY0;C0Þ 2 BðU;D; JÞand let Y ¼ C#D : Then we obtain Y0 , Y. However, it

follows from Propositions 1 and 2 that


123

Y ¼ C#D ¼\

x2X

x"D

!#D

¼\

x2X

x"D#D"D

!#D

¼[

x2X

x"D#D

!"D#D

�[

x2Y0

x"D#D

!"D#D

¼ Y0;

which is in contradiction with Y0 � Y :(ii) If B! C is a feedback redundant decision rule in

RðPEÞ; there exists B0 ! C0 2 RðPEÞ such that

B0 � B and C � C0: Since C � C0 does not hold, we

obtain C = C0. Thus, B0 ! C 2 RðPEÞ and B0 � B;

which is in contradiction with the assumption. h


context, E � A and X � RðPEÞ:B! C 2 X is called a

premise-minimal decision rule in X if B0 6� B for any

B0 ! C 2 X:

Definition 10 Let P ¼ ðU;A; I;D; JÞ be a decision for-

mal context and E � A: Define MðPEÞ ¼ fB! C jðX;BÞ 2 BðU;E; IEÞ;X 6¼ ;;B 6¼ ;;C ¼

Tx2X x"D 6¼ ;g and

denote by M�ðPEÞ the set of all the premise-minimal

decision rules in MðPEÞ:


context and E � A: Then R�ðPEÞ ¼ M�ðPEÞ:

Proof On one hand, for any B! C 2 M�ðPEÞ; we have

by Definitions 9 and 10 that B! C 2 RðPEÞ and B0 6� B

for any B0 ! C 2 MðPEÞ: It can be known from Theorem

2 that B ? C is a feedforward non-redundant decision rule

in RðPEÞ: If B ? C is a feedback redundant decision rule

in RðPEÞ; there exists B0 ! C0 2 RðPEÞ such that B0 ,

B and C � C0: Since C , C0 is not true, we obtain C = C0.

Thus, B0 ! C 2 MðPEÞ and B0 , B, which is in contra-

diction with B0 6� B for any B0 ! C 2 MðPEÞ: As a result,

B ? C is a feedback non-redundant decision rule in

RðPEÞ: By Theorem 1, B! C 2 R�ðPEÞ is at hand.

On the other hand, for any B! C 2 R�ðPEÞ; there

exist X 2 BUðU;E; IEÞ and Y 2 BUðU;D; JÞ such that

ðX;BÞ 2 BðU;E; IEÞ and ðY;CÞ 2 BðU;D; JÞ: Noting that

[

x2X

x"D#D

!"D#D

;\

x2X

x"D

0@

1A 2 BðU;D; JÞ

and

X �[

x2X

x"D#D

!"D#D

�[

x2Y

x"D#D

!"D#D

¼ Y;

we obtain B!T

x2X x"D 2 RðPEÞ and C �T

x2X x"D :

Then, we can conclude C ¼T

x2X x"D since B! C is a

feedforward non-redundant decision rule in RðPEÞ: By

Definition 10, B! C 2 MðPEÞ follows. Furthermore, we

can prove B! C 2 M�ðPEÞ: In fact, if there exists B0 !C 2 MðPEÞ such that B0 � B; then by Definition 8 B! C

is a feedback redundant decision rule in RðPEÞ; which is in

contradiction with B! C 2 R�ðPEÞ: Thus, based on

Definition 9, we obtain that B! C is a premise-minimal

decision rule in MðPEÞ; i.e. B! C 2 M�ðPEÞ:By Theorem 4, we can obtain all the non-redundant

decision rules from a decision formal context P ¼ðU;A; I;D; JÞ via M�ðPÞ: By Definitions 9 and 10, it is

natural to first compute MðPÞ and then obtain M�ðPÞ:However, here we directly compute M�ðPÞ with-

out generating MðPÞ in advance in order to reduce

the computational complexity, which is described as

follows:

It is easy to prove that the time complexity of Algorithm 2 is

O ðjUj þ jAjÞjAjjLAj þ jDjjLAj2� �

;

where |LA| denotes the cardinality of BðU;A; IÞ: Note that

|U||LD| � |D| generally holds for a given decision formal

context. So, Algorithm 2 is of less time complexity than

Algorithm 1. We have the following example to illustrate

Algorithm 2. h

Example 1 Table 1 shows a decision formal context P ¼ðU;A; I;D; JÞ; where U = {1, 2, 3, 4, 5}, A = {a, b, c,

d, e, f} and D = {d1, d2, d3}. In the table, the number 1 in

the ith row and jth column represents that the object in the

ith row has the attribute in the jth column, and the number

0 in the ith row and jth column means the opposite. Fig-

ure 1 depicts the Hasse diagram of the concept lattice of

the formal context (U, A, I). By Theorem 2, all the feed-

forward non-redundant decision rules generated by the

formal concepts of (U, A, I) are as follows:

r1: a! d1; which is generated by ({1, 3, 5}, {a}) due to

1"D \ 3"D \ 5"D ¼ fd1g and is supported by objects 1,

3, 5;


123

r2: ac! d1d2; which is generated by ({3, 5}, {a, c})

due to 3"D \ 5"D ¼ fd1; d2g and is supported by

objects 3, 5;

r3: acf ! d1d2; which is generated by ({3}, {a, c, f})

due to 3"D ¼ fd1; d2g and is supported by object 3;

r4: b! d2; which is generated by ({2, 4, 5}, {b}) due to

2"D \ 4"D \ 5"D ¼ fd2g and is supported by objects 2,

4, 5;

r5: be! d1d2; which is generated by ({4, 5}, {b, e})

due to 4"D \ 5"D ¼ fd1; d2g and is supported by

objects 4, 5;

r6: abce! d1d2; which is generated by ({5}, {a, b,

c, e}) due to 5"D ¼ fd1; d2g and is supported by

object 5;

r7: bd ! d2d3; which is generated by ({2}, {b, d}) due

to 2"D ¼ fd2; d3g and is supported by object 2.

Thus, by applying Algorithm 2 to the decision formal

context P; the rules r1, r2, r4, r5 and r7 are added into M

one by one. That is, P has five non-redundant decision

rules: r1, r2, r4, r5 and r7.

4 Decision rules and granular rules in the consistent

decision formal contexts

In Sect. 3, we have proposed an efficient algorithm to

extract all the non-redundant decision rules from a decision

formal context. Since the decision rules and the granular

rules are two useful ways of discovering knowledge from a

decision formal context, it is of interest to investigate the

relationship between the non-redundant decision rules and

the granular rules. Furthermore, considering that knowl-

edge reduction is one of the key issues in FCA, it is also

essential to clarify the relation and the difference between

the decision rule oriented knowledge reduction [24] and the

granular rule oriented knowledge reduction [22].

4.1 The relationship between the non-redundant

decision rules and the granular rules


formal context. If x"# � x"D#D for any x 2 U; then P is said

to be consistent; otherwise, P is said to be inconsistent.

In FCA, there are several other kinds of consistent

decision formal contexts (e.g., those in [25, 30–32]) except

the one introduced above. However, for our purpose, we

here only focus on the kind of consistent decision formal

contexts in [22]. That is to say, the consistent decision

formal contexts discussed in the following represents the

ones whose consistency is defined by Definition 11.

Proposition 3 Let P ¼ ðU;A; I;D; JÞ be a decision for-

mal context. Then P is consistent iff x" ! x"D 2 RðPÞ for

any x 2 U:

Proof Necessity. Since P is consistent, it follows from

Definition 11 that x"# � x"D#D for any x 2 U: Noting that

ðx"#; x"Þ 2 BðU;A; IÞ; ðx"D#D ; x"DÞ 2 BðU;D; JÞ and P is

regular, we know that x"#; x"; x"D#D and x"D are all non-

empty, which leads to x" ! x"D 2 RðPÞ:

Sufficiency. It is immediate from Definitions 5 and 11. h

Definition 12 ([22]) Let P ¼ ðU;A; I;D; JÞ be a consis-

tent decision formal context. For any x 2 U; x" ! x"D is

called a granular rule of P:

It can be known from Definition 12 that a granular rule

is a special decision rule. However, a decision rule may not

be a granular rule because the premise or the conclusion of

a decision rule is not required to be the intent of an object

concept but that of a formal concept only.

Let P ¼ ðU;A; I;D; JÞ be a consistent decision formal

context. For any nonempty set X 2 BUðU;A; IÞ withTx2X x" 6¼ ; and

Tx2X x"D 6¼ ;; the mergence of the gran-

ular rules x" ! x"Dðx 2 XÞ is defined as

(U,∅)

({1,3,5},{a})

({3,5},{a,c})

({3},{a,c,f})

({2,4,5},{b})

({2},{b,d})

({4,5},{b,e})

({5},{a,b,c,e})

(∅,A)

Fig. 1 Hasse diagram of the concept lattice of (U, A, I)

Table 1 A decision formal context P 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 0

2 0 1 0 1 0 0 0 1 1

3 1 0 1 0 0 1 1 1 0

4 0 1 0 0 1 0 1 1 0

5 1 1 1 0 1 0 1 1 0


123

^

x2X

x" ! x"D� �

¼\

x2X

x" !\

x2X

x"D : ð6Þ

It is easy to verify that the mergence of x" ! x"D (x 2 X) is

also a decision rule.

Theorem 5 Let P ¼ ðU;A; I;D; JÞ be a consistent

decision formal context and B! C 2 RðPÞ: Then the

decision rule B! C is non-redundant in RðPÞ iff B!C ¼

Vx2B# x" ! x"D

� �and ðx0" \ BÞ# 6� C#D for any

x0 2 C#DnB#:

Proof Since B! C 2 RðPÞ; there exist X 2 BUðU;A; I)

and Y 2 BUðU;D; JÞ such that ðX;BÞ 2 BðU;A; IÞ and

ðY;CÞ 2 BðU;D; JÞ:

If B ? C is non-redundant in RðPÞ; it follows from

Theorem 1 that B ? C is both a feedforward and a feed-

back non-redundant decision rule in RðPÞ: So, B!Tx2X x"D 2 RðPÞ; which leads to C ¼

Tx2X x"D because

B!T

x2X x"D can imply B ? C. Furthermore, noting that

ðX;BÞ ¼_

x2X

ðx"#; x"Þ ¼[

x2X

x"#

!"#;\

x2X

x"

0

@

1

A;

we have B!C¼V

x2B# x"!x"D

� �: To prove ðx0"\BÞ# 6�

C#D for any x0 2 C#DnB#; it is sufficient to show ðx0"\BÞ# 6� Y for any x0 2 YnX: If there exists x0 2 YnX such

that ðx0" \ BÞ# � Y ; then ðx0

" \ BÞ# ¼ ðx0" \ X"Þ# ¼

ðfx0g [ XÞ"# � Y yielding ðfx0g [ XÞ" 6¼ ;: Since ððfx0g[XÞ"#; ðfx0g [ XÞ"Þ 2 BðU;A; IÞ; we obtain ðfx0g[XÞ"!C 2 RðPÞ; i.e. x0

" \ B! C 2 RðPÞ: Since B! C is a

feedback non-redundant decision rule in RðPÞ; we con-

clude x0: \ B = B, which is in contradiction with

x0 2 YnX:If B! C ¼

Vx2B# x" ! x"D

� �and ðx0" \ BÞ# 6� C#D for

any x0 2 C#DnB#; then by Theorem 2, B! C is a feed-

forward non-redundant decision rule in RðPÞ: Thus, to

prove that B! C is non-redundant in RðPÞ; it is sufficient

to show that B! C is a feedback non-redundant decision

rule in RðPÞ: In fact, if B! C is a feedback redundant

decision rule in RðPÞ; there exists B0 ! C0 2 RðPÞ such

that B0 � B and C � C0: Suppose ðX0;B0Þ 2 BðU;A; IÞand ðY0;C0Þ 2 BðU;D; JÞ: Then X � X0 � Y0 � Y:

Therefore, there exists x 2 X0nX such that ðx" \ BÞ# ¼ðfxg [ XÞ"# � X

"#0 ¼ X0 � Y ¼ C#D ; which is in contra-

diction with ðx0" \ BÞ# 6� C#D for any x0 2 C#DnB#:Theorem 5 clarifies the relationship between the non-

redundant decision rules and the granular rules in a con-

sistent decision formal context. h

4.2 A comparison of the decision rule oriented

knowledge reduction and the granular rule oriented

knowledge reduction

In general, both the decision rules and the granular rules

derived directly from a consistent decision formal context

are not concise or compact. In order to derive more com-

pact decision rules and/or granular rules, the issue of rule

acquisition oriented knowledge reduction was discussed in

[22, 24]. Since knowledge reduction is one of the important

issues in FCA, it is of interest to clarify the relation and the

difference between the decision rule oriented knowledge

reduction and the granular rule oriented knowledge

reduction. This issue will be discussed in the following.


formal context and E � A: For X � RðPÞ and

X0 � RðPEÞ; if each decision rule of X can be implied by a

decision rule of X0; we say that X can be implied by X0: We

denote this implication relationship by X0 ) X:


formal context. E � A is called a consistent set of P if

RðPEÞ ) RðPÞ: Furthermore, if E is a consistent set of Pand any F � E is not a consistent set of P; then E is called

a reduct of P:

It can easily be observed from Definitions 13 and 14 that

this kind of knowledge reduction can preserve the decision

rule information of a decision formal context and allows us

to obtain more compact decision rules from a decision

formal context.

Proposition 4 Let P ¼ ðU;A; I;D; JÞ be a decision

formal context. Then E � A is a consistent set of P iff

R�ðPEÞ ) R�ðPÞ:

Proof Necessity. Since E � A is a consistent set of P; it

follows from Definition 14 that RðPEÞ ) RðPÞ:According to Definition 13, it is easy to prove

R�ðPEÞ ) RðPEÞ and RðPÞ ) R�ðPÞ: As a result,

R�ðPEÞ ) RðPEÞ ) RðPÞ ) R�ðPÞ:

Sufficiency. Based on Definition 13, it is also easy to

prove RðPEÞ ) R�ðPEÞ and R�ðPÞ ) RðPÞ: Thus,

according to the assumption R�ðPEÞ ) R�ðPÞ; we can

obtain RðPEÞ ) R�ðPEÞ ) R�ðPÞ ) RðPÞ: By Defi-

nition 14, E is a consistent set of P: h

Definition 15 ([22]) Let P ¼ ðU;A; I;D; JÞ be a consis-

tent decision formal context. E � A is called a granular

consistent set of P if x"E#E � x"D#D for any x 2 U: Fur-

thermore, if E is a granular consistent set of P and any

F � E is not a granular consistent set of P; then E is called

a granular reduct of P:


123

Proposition 5 Let P ¼ ðU;A; I;D; JÞ be a consistent

decision formal context. Then E � A is a granular con-

sistent set of P iff x"E ! x"D 2 RðPEÞ for any x 2 U:

Proof Necessity. If E � A is a granular consistent set of

P; then by Definition 15, x"E#E � x"D#D holds for any x 2U: Since P is regular, we have x"D 6¼ ;: Furthermore, we

can prove x"E 6¼ ;: In fact, if x"E ¼ ;; then x"E#E ¼ U:

Thus, x"D#D ¼ U yielding x"D ¼ ;; which is in contradiction

with x"D 6¼ ;: Consequently, x"E ! x"D 2 RðPEÞ:

Sufficiency. It is immediate from Definitions 5 and 15.

Let P ¼ ðU;A; I;D; JÞ be a consistent decision formal

context and E � A be a granular consistent set of P: We

denote by R#ðPEÞ the set of all the granular rules of the

subcontext PE ¼ ðU;E; IE;D; JÞ: h


decision formal context. If E � A is a granular consistent

set of P; then R#ðPEÞ ) R#ðPÞ:

Proof It is immediate from Definition 13 and Proposi-

tions 3 and 5.

It can easily be seen from Definition 15 and Theorem 6

that the granular rule oriented knowledge reduction can

preserve the granular rule information of a consistent

decision formal context and allows us to derive more

compact granular rules from a consistent decision formal

context.

The following theorem illustrates the relation between

the decision rule oriented knowledge reduction and the

granular rule oriented knowledge reduction. h


decision formal context. If E � A is a consistent set of P;then E is a granular consistent set of P:

Proof Since P is consistent, it follows from Definition 11

that x"# � x"D#D for any x 2 U: Noting that ðx"#; x"Þ 2BðU;A; IÞ; ðx"D#D ; x"DÞ 2 BðU;D; JÞ and P is regular, we

know that x"#; x"; x"D#D and x"D are all nonempty, which leads

to x" ! x"D 2 RðPÞ: If E � A is a consistent set of P; then

there exists B! C 2 RðPEÞ such that B � x" and x"D � C:

Since there exist X 2 BUðU;E; IEÞ and Y 2 BUðU;D; JÞsuch that ðX;BÞ 2 BðU;E; IEÞ and ðY ;CÞ 2 BðU;D; JÞ;we

have by Proposition 1 that

B � x" ) x 2 B#E

) x"# � X

) X"E � x"#"E

) B � x"E

) x"E#E � B#E ¼ X

and x"D�C)C#D�x"D#D)Y�x"D#D : Therefore, x"E#E�X�Y�x"D#D : By Definition 15, we conclude that E is a

granular consistent set of P:

It can be known from Theorem 7 that the decision rule

oriented knowledge reduction of a consistent decision

formal context can preserve the granular rule information.

However, the granular rule oriented knowledge reduction

may not preserve the decision rule information. That is to

say, it may happen that a granular consistent set is not a

consistent set. In what follows, we use a counterexample to

confirm this assertion. h

Example 2 Let P ¼ ðU;A; I;D; JÞ be the decision formal

context in Example 1. Then the following statements hold:

1"# ¼ f1; 3; 5g � f1; 3; 4; 5g ¼ 1"D#D ;

2"# ¼ f2g ¼ 2"D#D ;

3"# ¼ f3g � f3; 4; 5g ¼ 3"D#D ;

4"# ¼ f4; 5g � f3; 4; 5g ¼ 4"D#D ;

5"# ¼ f5g � f3; 4; 5g ¼ 5"D#D :

Thus, it follows from Definition 11 that the decision formal

context P is consistent. According to the results in

Example 1 and the discussion in Sect. 4.1, we obtain the

following seven feedforward non-redundant decision rules

via the mergence of granular rules:

r1: a! d1; which is generated by the extent {1, 3, 5}

due to 1: \ 3: \ 5: = {a} and 1"D \ 3"D \ 5"D ¼fd1g; is supported by objects 1, 3, 5;

r2: ac! d1d2; which is generated by the extent {3,5}

due to 3: \ 5: = {a,c} and 3"D \ 5"D ¼ fd1; d2g; is

supported by objects 3, 5;

r3: acf ! d1d2; which is generated by the extent {3} due

to 3" ¼ fa; c; fg and 3"D ¼ fd1; d2g; is supported by

object 3;

r4: b! d2; which is generated by the extent {2, 4, 5}

due to 2" \ 4" \ 5" ¼ fbg and 2"D \ 4"D \ 5"D ¼fd2g; is supported by objects 2, 4, 5;

r5: be! d1d2; which is generated by the extent {4,5}

due to 4" \ 5" ¼ fb; eg and 4"D \ 5"D ¼ fd1; d2g; is

supported by objects 4, 5;

r6: abce! d1d2; which is generated by the extent {5}

due to 5: = {a, b, c, e} and 5"D ¼ fd1; d2g; is

supported by object 5;

r7: bd ! d2d3; which is generated by the extent {2} due

to 2: = {b,d} and 2"D ¼ fd2; d3g; is supported by

object 2.

For r3 : acf ! d1d2; there is 5 2 fd1; d2g#Dnfa; c; fg#such that


123

ð5" \ fa; c; fgÞ# ¼ f3; 5g � f3; 4; 5g ¼ fd1; d2g#D ;

for r6 : abce! d1d2; there is 4 2 fd1; d2g#Dnfa; b; c; eg#

such that ð4"\fa;b;c;egÞ#¼f4;5g�f3;4;5g¼fd1;d2g#D :

Thus, by Theorem 5, we know that the rules r3 and r6 are

redundant in RðPÞ: Similarly, it can be verified that

r1, r2, r4, r5 and r7 are all non-redundant in RðPÞ:Let E = {a, d, e, f}. Then we can obtain a subcontext

PE ¼ ðU;E; IE;D; JÞ which satisfies

1"E#E ¼ f1; 3; 5g � f1; 3; 4; 5g ¼ 1"D#D ;

2"E#E ¼ f2g ¼ 2"D#D ;

3"E#E ¼ f3g � f3; 4; 5g ¼ 3"D#D ;

4"E#E ¼ f4; 5g � f3; 4; 5g ¼ 4"D#D ; and

5"E#E ¼ f5g � f3; 4; 5g ¼ 5"D#D :

Thus, E is a granular consistent set of P: Figure 2 depicts

the Hasse diagram of the concept lattice of the formal

context (U, E, IE). According to Theorem 5, the non-

redundant decision rules derived from PE are as follows:

r01 : a! d1;

r02 : af ! d1d2;

r03 : e! d1d2;

r04 : d ! d2d3:

It is easy to check that the rules r2 and r4 derived from Pcannot be implied by the non-redundant decision rules

derived from the subcontext PE: By Proposition 4, E is not

a consistent set of P although it is a granular consistent set

of P: That is to say, to remove the attributes b and c from

the decision formal context P will lose the decision rule

information although it can preserve the granular rule

information.

Example 2 illustrates the difference between the deci-

sion rule oriented knowledge reduction and the granular

rule oriented knowledge reduction. Furthermore, by com-

bining the difference with the relation between these two

kinds of knowledge reduction, we understand that the

decision rule oriented knowledge reduction can preserve

more decision rule information than the granular rule ori-

ented knowledge reduction.

5 Experiments

In this section, we conduct some experiments to compare the

proposed algorithm with the existing one in [24] in terms of

the efficiency of extracting the non-redundant decision rules

from a decision formal context. In the experiments, five real-

life databases including Bacteria [33], Zoo [34], Balance

Scale [34], Wine [34], and Car Evaluation [34] are analyzed

to achieve the task of comparing the efficiency. The detailed

information on the five real-life databases is shown in

Table 2. For each of the chosen databases, we took the

classification attribute as the decision attribute and the other

attributes (variables) as the conditional attributes. Then,

using the scaling approach [10] to convert the discrete (but

not Boolean) attributes of Bacteria, Zoo and Balance Scale

into Boolean ones, we obtained three decision formal con-

texts which are denoted by Date sets 1, 2 and 3, respectively.

In the Wine database, there are 178 instances (each of them

denotes a wine) characterized by 13 variables (each of them

denotes a constituent found in each of the wines) whose

values are all continuous. Here, we classified, from small to

large, the values of each variable into three pairwise disjoint

intervals with their length being the same. For example, for

the seventh variable (Flavanoids in the Wine database),

since its minimum value is 0.34 and its maximum value is

5.08, we obtained three pairwise disjoint intervals: [0.34,

1.92), [1.92, 3.50), [3.50, 5.08] according to the above

classification approach. Then, by using the scaling approach,

a decision formal context was obtained and is denoted by

Data set 4. In the Car Evaluation database, its 1728 instances

Table 2 The main characters of the five chosen real-life databases

Database Instances Classes Input attributes excluding the

classification attribute

Boolean Discrete

but not

Boolean

Continuous

Bacteria 17 6 16 0 0

Zoo 101 7 15 1 0

Balance

Scale

625 3 0 4 0

Wine 178 3 0 0 13

Car

Evaluation

1,728 4 0 6 0

(U,∅)

({1,3,5},{a})

({3},{a,f})

({4,5},{e})

({5},{a,e})

({2},{d})

(∅,E)

Fig. 2 Hasse diagram of the concept lattice of (U, E, IE)


123

are characterized by six attributes which are Buying price,

Price of the maintenance, Number of doors, Capacity in

terms of persons to carry, The size of luggage boot, Estimated

safety of the car. For our purpose, the values of the third

attribute are divided into two hierarchies: I (2 or 3) and II (4

or 5). Similarly, using the scaling approach, we obtained

another decision formal context denoted by Data set 5.

Then Algorithms 1 and 2 were applied to Data sets 1, 2,

3, 4 and 5. The corresponding running time is reported in

Table 3, in which |U|, |A| and |D| denote the cardinalities of

the object set, the conditional attribute set and the decision

attribute set of the concerned decision formal context,

respectively, and NR is the abbreviation of the term ‘non-

redundant’. It can be seen from Table 3 that by the running

time of extracting the non-redundant decision rules,

Algorithm 2 is much more efficient than Algorithm 1 for

each of the chosen data sets.

6 Conclusion

Rule acquisition is one of the main purposes in the analysis

of decision formal contexts. In this paper, we have pro-

posed a new algorithm of deriving the non-redundant

decision rules from a decision formal context, proved that

the proposed rule acquisition algorithm is of less time

complexity than the existing one in [24], and conducted

some experiments to compare their efficiency. Further-

more, the relationship between the non-redundant decision

rules and the granular rules has been investigated in the

consistent decision formal contexts. Also, the relation and

the difference between the decision rule oriented knowl-

edge reduction and the granular rule oriented knowledge

reduction have been clarified.

From the point of view of applications, the results

obtained in this paper need to be further extended to the

case of fuzzy decision formal contexts [35], incomplete

decision formal contexts [36] or even real decision formal

contexts [37–39] since in the real world the relationship

between some objects and attributes of a decision formal

context may be fuzzy-valued, interval-valued or even real-

valued. This issue will be discussed in our future work.

Acknowledgments The authors would like to thank the anonymous

reviewers for their valuable comments and helpful suggestions which

lead to a significant improvement on the manuscript. This work was

supported by the National Natural Science Foundation of China (Nos.

10971161, 61005042, 11071281 and 61202018).

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Table 3 A contrast between Algorithm 1 and Algorithm 2 in terms of

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Data

set

|U| |A| |D| Number of

NR-rules

Running time (s)

Algorithm 1 Algorithm 2

Data

set 1

17 16 6 6 0.036 0.018

Data

set 2

101 21 7 9 0.799 0.428

Data

set 3

625 20 3 303 109.347 78.011

Data

set 4

178 39 3 256 2,826.048 680.908

Data

set 5

1,728 19 4 141 4,121.354 2,810.434


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