A general approach to attribute reduction in rough ... - Semantic Scholar
Science in China Series F: Information Sciences © 2007
Science in China Press Springer-Verlag
A general approach to attribute reduction in rough set theory ZHANG WenXiu1 †, QIU GuoFang2 & WU WeiZhi3 1
Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China; 2 School of Management, Xi’an University of Architecture and Technology, Xi’an 710055, China; 3 Information College, Zhejiang Ocean University, Zhoushan 316004, China
The concept of a consistent approximation representation space is introduced. Many types of information systems can be treated and unified as consistent approximation representation spaces. At the same time, under the framework of this space, the judgment theorem for determining consistent attribute set is established, from which we can obtain the approach to attribute reductions in information systems. Also, the characterizations of three important types of attribute sets (the core attribute set, the relative necessary attribute set and the unnecessary attribute set) are examined. rough sets, attribute reduction, information systems, approximation representation spaces
Rough set theory, proposed by Pawlak[1,2] in the 1980s, is a theory for the study of intelligent systems characterized by inexact, uncertain or vague information. Now rough set theory has found successful applications in such fields of artificial intelligence as machine learning, knowledge discovery, decision analysis, process control, pattern recognition, etc. It has become one of flash points in the research area of information science. One fundamental aspect of rough set theory involves a search for particular subsets of condition attributes. Such subsets are called attribute reductions. Many types of attribute reductions ― have been proposed, each of the reductions aimed at some basic requirements[3 11]. In ref. [12], Skowron introduced the notion of discernibility matrix which became a major tool for searching for reductions in information systems. Using the similar idea, Zhang et al.[4,6,8] discussed approaches to attribute reduction in inconsistent and incomplete information systems. In ref. [9], Zhang and Wu gave judgment theorems of consistent attribute sets and approaches to attribute reductions in information systems with fuzzy decision. In this paper, by analyzing problems of attribute reductions in some systems such as informaReceived October 8, 2005; accepted January 8, 2007 doi: 10.1007/s11432-007-0017-6 † Corresponding author (email: [email protected]) Supported by the Major State Basic Research Development Program of China (973 Program) (Grant No. 2002CB312200), and the National Natural Science Foundation of China (Grant Nos. 60673096 and 60373078)
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Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
tion systems, consistent or inconsistent decision information systems, information systems with fuzzy decision and variable precision rough set models, we introduce the concept of a consistent approximation representation space and obtain the corresponding judgment theorem determining consistent attribute sets. Moreover, we examine the characterizations of three important types of attribute sets, especially, provide the conditions to determine the relative necessary and unnecessary attribute sets.
1 Concepts and properties of rough sets Let U = { x1 , x2 ,
, xn } be a finite set of objects called the universe of discourse, P (U ) a
power set of U, and R an equivalence relation on U. Denote
[ xi ]R = { x j ∈ U : ( xi , x j ) ∈ R} .
{
Then U R = [ xi ]R : xi ∈ U
}
forms a partition of U, and ⎧⎪
⎫⎪
⎩⎪ xi ∈X
⎭⎪
σ (U R ) = ⎨ ∪ [ xi ]R : X ⊆ U ⎬ is a σ -algebra on U. Definition 1[1]. Let R be an equivalence relation on U. The pair (U , R ) refers to an approximation space. For any X ⊆ U , denote
{ } { } R ( X ) = { xi ∈ U : [ xi ]R ∩ X ≠ φ } = ∪ {[ xi ]R : [ xi ]R ∩ X ≠ φ }, R ( X ) = xi ∈ U : [ xi ]R ⊆ X = ∪ [ xi ]R : [ xi ]R ⊆ X ,
(1) (2)
R( X ) and R( X ) respectively refer to the lower and upper approximations of X with respect to (U , R) . If R( X ) = R( X ) , then X is called a definable set, otherwise it is called a rough set. Obviously, R , R : P (U ) → σ (U R), standing for the lower and upper approximation operators respectively. Theorem 1[2]. The lower and upper approximation operators satisfy the following properties: (i) R (φ ) = R (φ ) = φ , R (U ) = R (U ) = U , (ii) R ( X ) ⊆ X ⊆ R ( X )
(X ⊆U), (iii) R ( X ∩ Y ) = R ( X ) ∩ R (Y ) , R ( X ∪ Y ) = R ( X ) ∪ R (Y ) (iv) ~ R ( X ) = R ( ~ X ) , ~ R ( X ) = R ( ~ X ) (X ⊆U), where ~ X is the complement of X. Theorem 2[4]. If R is an equivalence relation on U, then
{
( X ,Y ⊆ U ) ,
}
σ (U R ) = X ⊆ U : R ( X ) = R ( X ) . Definition 2.
Let R and R′ be equivalence relations on U, and denote D = U R ′ = { D1 , D2 , , Dr } , D = ( D1 , D2 ,
, Dr ) ,
R ( D ) = ( R ( D1 ) , R ( D2 ) ,
, R ( Dr ) ) ,
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
189
R′ is said to be represented by R if R ( D ) = D. Let R and R′ be equivalence relations on U, and denote
Definition 3.
posR ( R′ ) = ∪ { R ( D ) : D ∈ U R′} =
∪ R(Dj ) ,
j≤r
posR ( R′) represents the positive region of R′ with respect to R. Theorem 3. Let R and R′ be equivalence relations on U. Then the following statements are equivalent: (i) R′ is represented by R, (ii) R ⊆ R′,
(iii) posR ( R ′ ) = U . Proof.
( )
If R′ is represented by R, then R D j = D j ( j ≤ r ), that is, D j ∈ σ (U R ) ( j ≤ r ).
If D j = [ xi ]R′ , then
[ xi ]R ⊆ [ xi ]R′ ,
so R ⊆ R′ . Vice versa, we can conclude that R ' ⊆ R . Thus
( )
we have proved that (i) ⇔ (ii). On the other hand, since R D j ⊆ D j ( j ≤ r ), we observe that
( )
posR ( R ′ ) = U if and only if R D j = D j ( j ≤ r ); thus we have proved that (i) ⇔ (iii).
2 Judgment theorem of attribute reduction and characterization of attributes in consistent approximation representation spaces Let U = { x1 , x2 ,
Definition 4.
, xn } be a finite universe of discourse, A = {a1 , a2 ,
, am }
a finite set of attributes, R = { Ra ⊆ U × U : a ∈ A} a class of equivalence relations on U , and R′ an equivalence relation on U. Then the quadruple (U , A, R , R′ ) is referred to as an approximation representation space. Definition 5. Let S = (U , A, R , R′ ) be an approximation representation space, denote RB =
∩ Ra ( B ⊆ A) . S denotes a consistent approximation representation space if
a∈B
RA ⊆ R′ . An
attribute subset B ⊆ A is referred to as a consistent attribute set of S if RB ⊆ R′. If B is a consistent attribute set of S and B − {b} is not a consistent attribute set of S for all b ∈ B , then B is referred to as a reduction of S. Definition 6. Let S = (U , A, R , R′ ) be an approximation representation space, and denote ⎧⎪{a ∈ A : ( xi , x j ) ∉ Ra }, [ xi ]R ' ∩ [ x j ]R ' = φ , DR ' [ xi ] A ,[ x j ] A = ⎨ [ xi ]R ' ∩ [ x j ]R ' ≠ φ . ⎪⎩ A,
(
(
DR ' [ xi ] A ,[ x j ] A
)
)
is referred to as the discernibility attribute set of xi and xj, and DR′ =
{DR ' ([ xi ]A ,[ x j ]A ) : xi , x j ∈U } Remark.
is called the discernibility matrix of S.
If S = (U , A, R , R′ ) is a consistent approximation representation space, then we
(
)
can conclude that DR ' [ xi ] A ,[ x j ] A ≠ φ
190
for all xi , x j ∈ U . In fact, since RA ⊆ R′, if
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
( xi , x j ) ∉ R′ , we have ( xi , x j ) ∉ RA = ∩ Ra . Then there exists an a ∈ A
such that
a∈ A
that is,
( xi , x j ) ∉ R′
(
)
implies DR ' [ xi ] A ,[ x j ] A ≠ φ . On the other hand, if
(
( xi , x j ) ∉ Ra ;
( xi , x j ) ∈ R′ , by the
)
definition we have DR ' [ xi ] A ,[ x j ] A = A ≠ φ . Theorem 4. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then B ⊆ A is a consistent attribute set of S if and only if
(
)
B ∩ DR ' [ xi ] A ,[ x j ] A ≠ φ Proof.
( xi , x j ∈ U ).
(3)
If B is a consistent attribute set of S, that is, RB ⊆ R′, which is equivalent to the say-
( xi , x j ) ∉ R′ implies ( xi , x j ) ∉ RB , i.e. when [ xi ]R ' ∩ [ x j ]R ' = φ , there exists an such that ( xi , x j ) ∉ Ra . Hence eq. (3) holds, and, therefore, by Remark of Definition 6
ing that a∈B
the proof of this theorem is completed. It should be pointed out that for any consistent approximation representation space S , since A is finite, reductions of S must exist, but may not be unique. Definition 7. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Let
{Bk : k ≤ l}
be all the reductions of S , and denote C=
∩ Bk ,
K=
k ≤l
∪ Bk − C ,
I = A − (K ∪ C) ,
k ≤l
C is referred to as the core attribute set of S, K is referred to as the relative necessary attribute set of S and I is referred to as the unnecessary attribute set of S. Theorem 5. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then the following statements are equivalent: (i) a is an element of core attribute set of S;
(
)
(ii) there exists xi , x j ∈ U such that DR ' [ xi ] A ,[ x j ] A = {a} ; (iii) RA−{a} ⊆/ R′. Proof. (i) ⇒ (ii). Assume that a is an element of core attribute set of S. If any discernibilty attribute set containing attribute a has at least two elements. Let
{ (
)
(
)}
B = ∪ DR ' [ xi ] A ,[ x j ] A − {a}: a ∈ DR ' [ xi ] A ,[ x j ] A . i, j
(
)
(
)
It is easy to see that B ∩ DR ' [ xi ] A ,[ x j ] A ≠ φ xi , x j ∈ U . By Theorem 4, B is a consistent attribute set of S, and a ∉ B . Then there exists a reduction B′ of S such that a ∉ B′, which contradicts the fact that a is an element of core attribute set of S.
(
)
(ii) ⇒ (iii). Assume DR ' [ xi ] A ,[ x j ] A = {a}. It means that
( xi , x j ) ∈ Rb
( xi , x j ) ∉ R′, ( xi , x j ) ∉ Ra ,
and
(b ≠ a ). Then [ xi ]R ' ∩ [ x j ]R ' = φ , but [ xi ] A−{a} = [ x j ] A−{a} . Hence [ xi ] A−{a} ⊆/ [ xi ]R′ ,
which implies that RA−{a} ⊆/ R′. (iii) ⇒ (i). If a is not an element of core attribute set of S , then there exists a reduction B of ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
191
S such that a ∉ B . It is clear that B ⊆ A − {a} , and consequently, RA−{a} ⊆ RB ⊆ R′ , which contradicts the assumption RA−{a} ⊆/ R′ . Theorem 6. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then
a is an element of the unnecessary attribute set of S if and only if R ( a ) ⊆ R′ ∪ Ra , where
{
}
R ( a ) = ∪ RB −{a} : RB ⊆ R′, B ⊆ A . Proof. If a is an element of the unnecessary attribute set of S, then a is not in any attribute reductions; that is, for any B ⊆ A with RB ⊆ R′ , we have RB −{a} ⊆ R′, consequently, R ( a ) ⊆ R′ ⊆ R′ ∪ Ra .
Conversely, if R ( a ) ⊆ R′ ∪ Ra , then for any B ⊆ A with RB ⊆ R′ , we have RB −{a} ⊆ R′ ∪ Ra , i.e. RB −{a} ∩ Rac ⊆ R′ ; thus RB −{a} = RB ∪ ( RB −{a} ∩ Rac ) ⊆ R′ . Consequently, a is not in any attribute reductions, i.e. a is an element of the unnecessary attribute set. For the core attribute set C of S, since RC ⊆ R′ ∪ Ra implies R ( a ) ⊆ R′ ∪ Ra , we see that
RC ⊆ R′ ∪ Ra is the sufficient condition of a being an element of the unnecessary attribute set of S. Theorem 7. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then (i) a ∈ C iff RA−{a} ⊆/ R′ ; (ii) a ∈ I iff R(a) ⊆ R′ ∪ Ra ; (iii) a ∈ K iff RA−{a} ⊆ R′ and R(a ) ⊆/ R′ ∪ Ra . Proof. It follows immediately from Theorems 5 and 6. Example 1. Table 1 illustrates an exemplary information system (U , A, F ) , where U =
{ x1 , x2 ,
, x8 } , A = {a1 , a2 , a3 , a4 } . Let
Ra = {( xi , x j ) ∈ U × U : f a ( xi ) = f a ( x j )}, a ∈ A , R = { Ra : a ∈ A} ,
RB = {( xi , x j ) ∈ U × U : f a ( xi ) = f a ( x j ), a ∈ B}, B ⊆ A . It is easy to see that S = (U , A, R , RA ) is a consistent approximation representation space. It can also be calculated that
DRA = {{a3 } , {a1 , a2 } , {a1 , a2 , a3 } , {a1 , a3 , a4 } , {a2 , a3 , a4 } , A}.
Table 1
192
An information system of Example 1 a1 x1 1 x2 2 x3 1 x4 1 x5 1 x6 1 x7 1 x8 1
a2 1 1 1 2 2 2 2 2
a3 3 2 3 2 2 1 2 1
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
a4 2 1 2 1 1 1 1 1
The consistent approximation representation space has two reductions: B1 = {a1 , a3 } , B2 = {a2 , a3 } . Then we have Since
C = {a3 } , K = {a1 , a2 } , I = {a4 } .
( x6 , x7 ) ∈ RA−{a } 3
and RA− a3 ⊆/ RA , a3 is an element of core C. Since
U Ra1 = {{ x2 } , { x1 , x3 , x4 , x5 , x6 , x7 , x8 }} , U Ra2 = {{ x1 , x2 , x3 } , { x4 , x5 , x6 , x7 , x8 }} , U Ra3 = {{ x1 , x3 } , { x2 , x4 , x5 , x7 } , { x6 , x8 }} , U Ra4 = {{ x1 , x3 } , { x2 , x4 , x5 , x6 , x7 , x8 }} , we can see that Ra3 ⊆ Ra4 , Ra3 ⊆/ Ra1 , Ra3 ⊆/ Ra2 , and RA−{ak } ⊆ RA for all k ≠ 3 . Thus a4 is
an element of the unnecessary attribute set I , and a1 , a2 are elements of the relative necessary attribute set K.
3 Constructive theorems of consistent approximation representation spaces in inconsistent information systems Theorems 4 and 7 respectively give the judgment theorem of consistent attribute set and the characterizations of three types of attribute sets in a consistent approximation representation space. In this section, we will show that various inconsistent information systems can be changed into consistent approximation representation spaces. Thus judgment theorems for determining consistent attribute sets in different senses in these systems will be obtained and the characterization of different types of attribute sets will also be examined. Definition 8. Let I = (U , A, F ) be an information system, d : U → Vd , where Vd is a finite set, then (U , A, F , d ) is referred to as a decision information system, and (U , A, F , d ) is said to be consistent if RA ⊆ Rd , otherwise it is said to be inconsistent, where Rd =
It is easy to see that if
{( x , x ) ∈U × U : d ( x ) = d ( x )} . i
j
i
(U , A, F , d )
j
is a consistent decision information system, then
(U , A, R , Rd ) is clearly a consistent approximation representation space. Letting (U , A, F , d ) be an inconsistent decision information system, we define
{
(
}
U Rd = D j : j ≤ r , D D j
[ xi ]B ) =
D j ∩ [ xi ]B
(
B ⊆ A, xi ∈ U , j ≤ r ,
[ xi ]B ,
)
μ B ( xi ) = D ( D1 [ xi ]B ) , D ( D2 [ xi ]B ) , , D ( Dr [ xi ]B ) ,
{
η B ( xi ) = D j : D ( D j
},
D ( Dk [ xi ]B ) [ xi ]B ) = max k ≤r
(4)
where | X | is the cardinality of the set X. Theorem 8. Let (U , A, F , d ) be an inconsistent decision information system. Then S =
(U , A,R , Rμ )
is a consistent approximation representation space, and B ⊆ A is a consistent
attribute set of S if and only if μ B ( xi ) = μ A ( xi ) ( xi ∈ U ), where
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
193
{( x , x ) ∈U × U : μ
Rμ =
Proof.
i
j
A
( xi ) = μ A ( x j )}.
It is easy to see that Rμ is an equivalence relation on U and RA ⊆ Rμ ; thus
(
S = U , A, R , Rμ
)
is a consistent approximation representation space. If B is a consistent attrib-
( xi , x j ) ∈ RB
ute set of S, then RB ⊆ Rμ ; that is,
(
)
implies xi , x j ∈ Rμ , i.e. x j ∈ [ xi ]B implies
μ A ( x j ) = μ A ( xi ) . Consequently,
(
( k ≤ r , x j ∈ [ xi ]B ) .
)
D Dk /[ x j ] A = D ( Dk /[ xi ] A ) Hence D ( Dk /[ xi ]B ) = Dk ∩ [ xi ]B / [ xi ]B
{ (
)
= ∑ D Dk /[ x j ] A [ x j ] A / [ xi ]B :[ x j ] A ⊆ [ xi ]B
}
= D ( Dk /[ xi ] A )
( k ≤ r , xi ∈ U ) , from which it follows that μ B ( xi ) = μ A ( xi ) ( xi ∈ U ) . Conversely, assume that μ B ( xi ) = μ A ( xi ) ( xi ∈ U ) . If ( xi , x j ) ∈ RB , that is, [ xi ]B = [ x j ]B , then μ B ( xi ) = μ B ( x j ) , and hence μ A ( xi ) = μ A ( x j ) , i.e.,
( xi , x j ) ∈ Rμ . Therefore,
Theorem 9.
(
S = U , A, R , Rη
)
RB ⊆ Rμ .
Let (U, A, F, d) be an inconsistent decision information system. Then is a consistent approximation representation space, and B ⊆ A is a consistent
attribute set of S if and only if η B ( xi ) = η A ( xi ) Rη =
Proof.
( xi ∈ U ) ,
{( x , x ) ∈U × U :η i
j
A
where
( xi ) = η A ( x j )}.
It is easy to see that Rη is an equivalence relation on U and RA ⊆ Rη . Thus S is a
consistent approximation representation space. Similar to the proof of Theorem 8, we can prove that B is a consistent attribute set of S if and only if η B ( xi ) = η A ( xi ) ( xi ∈ U ) . Let (U , A, F , d ) be an inconsistent decision information system, B ⊆ A , and β >0.5. Denote
( ) { ( [ xi ]B ) ≥ β } , β R B ( D j ) = { xi ∈ U : D ( D j [ xi ]B ) > 1 − β } , R βB D j = xi ∈ U : D D j
(
R βB = R βB ( D1 ) , R βB ( D2 ) ,
(
β
β
β
R B = R B ( D1 ) , R B ( D2 ) , Theorem 10.
Then:
(
)
, R βB ( Dr ) , β
)
, R B ( Dr ) .
Let (U , A, F , d ) be an inconsistent decision information system and β >0.5.
(i) S1 = U , A, R , R β
)
is a consistent approximation representation space, and B ⊆ A is a β
β
consistent attribute set of S1 if and only if R B = R A , where Rβ = 194
{( x , x ) : x ∈ R i
j
i
β A
( Dk ) ⇔ x j ∈ R βA ( Dk ) ( k ≤ r )}.
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
(
(ii) S 2 = U , A, R , R
β
)
is a consistent approximation representation space, and B ⊆ A is a β
β
consistent attribute set of S2 if and only if R B = R A , where β
R = Proof.
{( x , x ) : x ∈ R i
j
i
(i) It is easy to see that R
β
β A
}.
β
( Dk ) ⇔ x j ∈ R A ( Dk ) ( k ≤ r )
β
is an equivalence relation on U and RA ⊆ R ; thus S1
is a consistent approximation representation space. Now we are only to prove that RB ⊆ R β
β
iff
β
RB = R A. Assume that R B = R A . By the definition we have R B ( Dk ) = R A ( Dk ) β
( xi , x j ) ∈ RB ,
β
that is,
β
[ xi ]B = [ x j ]B ,
then
( k ≤ r ).
β
xi ∈ R B ( Dk ) ⇔ x j ∈ R B ( Dk ) . β
β
If
Consequently,
xi ∈ R βA ( Dk ) ⇔ x j ∈ R βA ( Dk ) . Hence ( xi , x j ) ∈ R β . Thus we have proved that RB ⊆ R . β
Conversely, suppose that RB ⊆ R . If [ xi ]B = [ x j ]B , then we observe that xi ∈ R A ( Dk ) ⇔ β
β
x j ∈ R βA ( Dk ) . Similar to the proof of Theorem 8, we can conclude that xi ∈ R A ( Dk ) ⇔ β
xi ∈ R B ( Dk ) . Thus R B ( Dk ) = R A ( Dk ) . It follows that R B = R A . β
β
β
β
β
(ii) It is similar to the proof of (i).
4 Constructive theorems of consistent approximation representation spaces in information systems with fuzzy decision In this section we will change an information system with fuzzy decision into a consistent approximation representation space, and then we can obtain results similar to Theorems 4 and 7. Definition 9. A quadruple (U, A, F, D) is referred to as an information system with a fuzzy
{
decision, where (U, A, F ) is an information system and D = D j : U → [ 0,1] , j ≤ r
}
is a fuzzy
decision. Let (U, A, F, D) be an information system with fuzzy decision, B ⊆ A, x ∈ U , j ≤ r . Denote
( ) ( x ) = min {D j ( y ) : y ∈ [ x]B } , R B ( D j ) ( x ) = max {D j ( y ) : y ∈ [ x ]B } , RB ( D j ) ( x ) = ∑ { D j ( y ) : y ∈ [ x ]B } [ x ]B , RB Dj
R B ( x ) = ( R B ( D1 )( x ) , R B ( D2 )( x ) ,
(
Theorem 11.
, R B ( Dr )( x ) ) ,
)
R B ( x ) = R B ( D1 )( x ) , R B ( D2 )( x ) ,
, R B ( Dr )( x ) ,
RB ( x ) = ( RB ( D1 )( x ) , RB ( D2 )( x ) ,
, RB ( Dr )( x ) ) .
Let (U, A, F, D) be an information system with fuzzy decision, then
(i) S1 = (U , A, R , R ) is a consistent approximation representation space and B ⊆ A is a conZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
195
sistent attribute set of S1 if and only if R B ( xi ) = R A ( xi ) R=
(
(ii) S 2 = U , A, R , R
)
{( x , x ) ∈U × U : R i
j
A
( xi ∈ U ) , where
( xi ) = R A ( x j )} .
is a consistent approximation representation space and B ⊆ A is a
consistent attribute set of S2 if and only if R B ( xi ) = R A ( xi ) R=
{( x , x ) ∈U × U : R i
j
A
( xi ∈ U ) ,
where
( xi ) = R A ( x j )} .
(iii) S3 = (U , A, R , R ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S3 if and only if RB ( xi ) = RA ( xi ) R=
Proof.
{( x , x ) ∈U × U : R i
j
A
( xi ) = RA ( x j )}.
( xi , x j ) ∈ RA , that is, R A ( xi ) = R A ( x j ) , and equiva-
(i) It is easy to see that R B is an equivalence relation on U. If
( ) (k ≤ r ) ,
[ xi ] A = [ x j ] A , then R A ( Dk )( xi ) = R A ( Dk ) x j lently,
( xi ∈ U ) , where
( xi , x j ) ∈ R.
so
Thus RA ⊆ R , which means that S1 is a consistent approximation represen-
tation space. Now we are only to prove that RB ⊆ R if and only if R B ( xi ) = R A ( xi ) that R B ( xi ) = R A ( xi )
( )
that RB ⊆ R . Conversely, if
Assume
( xi , x j ) ∈ RB , we have [ xi ]B = [ x j ]B . Then R B ( xi ) = R A ( xi ) = R A ( x j ) , that is, ( xi , x j ) ∈ R . Thus we have concluded RB ⊆ R, then [ xi ]B = [ x j ]B . We have R A ( x j ) = R A ( xi ) , so
( xi ∈ U ) .
R B x j , and consequently,
( xi ∈ U ) .
If
R B ( Dk )( xi ) = min { Dk ( y ) : y ∈ [ xi ]B }
} { { = min { R ( D ) ( x ) :[ x ] ⊆ [ x ] }
= min min Dk ( y ) : y ∈ [ x j ] A :[ x j ] A ⊆ [ xi ]B k
A
j
j A
}
i B
= R A ( Dk )( xi ) ,
from which it follows that R B ( xi ) = R A ( xi )
( xi ∈ U ) .
(ii) and (iii) can be proved similarly to the proof of (i). Let (U, A, F, D) be an information system with fuzzy decision, B ⊆ A, xi ∈ U . Denote
{ ( ) ( x ) = max R ( D )( x )}, M ( x ) = { D : R ( D ) ( x ) = max R ( D )( x )} , M ( x ) = { D : R ( D ) ( x ) = max R ( D )( x )} . M B ( xi ) = D j : R B D j
Theorem 12.
i
B
i
j
B
j
i
B
i
j
B
j
i
k ≤r
k ≤r
k ≤r
B
k
i
B
k
i
B
k
i
Let (U, A, F, D) be an information system with fuzzy decision. Then
(i) S1 = (U , A, R , M ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S1 if and only if M B ( xi ) = M A ( xi ) 196
( xi ∈ U ) , where
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
M=
(
{( x , x ) ∈U × U : M i
j
A
( xi ) = M A ( x j )}.
)
(ii) S 2 = U , A, R , M is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S2 if and only if M B ( xi ) = M A ( xi ) M=
{( x , x ) ∈U × U : M i
j
A
( xi ∈ U ) ,
where
( xi ) = M A ( x j )}.
(iii) S3 = (U , A, R , M ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S3 if and only if M B ( xi ) = M A ( xi ) M=
Proof.
{( x , x ) ∈U × U : M i
j
A
( xi ∈ U ) ,
where
( xi ) = M A ( x j )} .
It is similar to the proof of Theorem 11.
5 Conclusions We have introduced in this paper the concept of a consistent approximation representation space and established attribute reduction theory in this space. We have obtained the judgment theorem of consistent attribute sets and the approach of attribute reductions. We have also examined the characterizations of three important types of attribute sets: the core attribute set, the relative necessary attribute set and the unnecessary attribute set. Many well-known information systems such as consistent or inconsistent decision information systems, information systems with fuzzy decision and variable precision rough set models, etc. and the associating attribute reduction theory can be unified under the framework of consistent approximation representation space theory. However, all the information systems discussed in this paper are complete and the binary relations are all equivalent. Research of the theory of approximation presentation space based on non-equivalent relations will be discussed elsewhere. The authors would like to thank the anonymous referees for their valuable comments and suggestions. 1 2 3 4 5 6 7 8 9 10 11 12
Pawlak Z. Rough sets. Int J Comp Inf Sci, 1982, 11: 341―356 Pawlak Z. Rough Sets ――Theoretical Aspects of Reasoning about Data. Dordrecht: Kluwer Academic Publishers, 1991 Kryszkiewicz M. Comparative study of alternative types of knowledge reduction in insistent systems. Int J Intel Syst, 2001, 16: 105―120 Zhang W -X, Leung Y, Wu W -Z. Information Systems and Knowledge Discovery. Beijing: Science Press, 2003 Beynon M. Reducts within the variable precision rough sets model: A further investigation. Eur J Oper Res, 2001, 134: 592―605 Zhang W -X, Mi J -S, Wu W -Z. Approaches to knowledge reductions in inconsistent systems. Int J Intel Syst, 2003, 18: 989―1000 Qiu G -F, Li H -Z, Xu L -D, et al. A knowledge processing method for intelligent systems based on inclusion degree. Expert Syst, 2003, 20(4): 187―195 Mi J -S, Wu W -Z, Zhang W -X. Approaches to knowledge reduction based on variable precision rough set model. Inf Sci, 2004, 159: 255―272 Zhang M, Wu W -Z. Knowledge reduction in information systems with fuzzy decisions. J Eng Math, 2003, 20(2): 53―58 Leung Y, Wu W -Z, Zhang W -X. Knowledge acquisition in incomplete information systems: a rough set approach. Eur J Oper Res, 2006, 168(1): 164―180 Wu W -Z, Zhang M, Li H -Z, et al. Knowledge reduction in random information systems via Dempster-Shafer theory of evidence. Inf Sci, 2005, 174 (3-4): 143―164 Skowron A, Rauszwer C. The discernibility matrices and functions in information systems. In: Slowinski R, ed. Intelligent Decision Support: Handbook of Applications and Advances of the Rough Set Theory. Dordrecht: Kluwer Academic Publishers, 1992. 331―362 ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
197
Science in China Press Springer-Verlag
A general approach to attribute reduction in rough set theory ZHANG WenXiu1 †, QIU GuoFang2 & WU WeiZhi3 1
Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, China; 2 School of Management, Xi’an University of Architecture and Technology, Xi’an 710055, China; 3 Information College, Zhejiang Ocean University, Zhoushan 316004, China
The concept of a consistent approximation representation space is introduced. Many types of information systems can be treated and unified as consistent approximation representation spaces. At the same time, under the framework of this space, the judgment theorem for determining consistent attribute set is established, from which we can obtain the approach to attribute reductions in information systems. Also, the characterizations of three important types of attribute sets (the core attribute set, the relative necessary attribute set and the unnecessary attribute set) are examined. rough sets, attribute reduction, information systems, approximation representation spaces
Rough set theory, proposed by Pawlak[1,2] in the 1980s, is a theory for the study of intelligent systems characterized by inexact, uncertain or vague information. Now rough set theory has found successful applications in such fields of artificial intelligence as machine learning, knowledge discovery, decision analysis, process control, pattern recognition, etc. It has become one of flash points in the research area of information science. One fundamental aspect of rough set theory involves a search for particular subsets of condition attributes. Such subsets are called attribute reductions. Many types of attribute reductions ― have been proposed, each of the reductions aimed at some basic requirements[3 11]. In ref. [12], Skowron introduced the notion of discernibility matrix which became a major tool for searching for reductions in information systems. Using the similar idea, Zhang et al.[4,6,8] discussed approaches to attribute reduction in inconsistent and incomplete information systems. In ref. [9], Zhang and Wu gave judgment theorems of consistent attribute sets and approaches to attribute reductions in information systems with fuzzy decision. In this paper, by analyzing problems of attribute reductions in some systems such as informaReceived October 8, 2005; accepted January 8, 2007 doi: 10.1007/s11432-007-0017-6 † Corresponding author (email: [email protected]) Supported by the Major State Basic Research Development Program of China (973 Program) (Grant No. 2002CB312200), and the National Natural Science Foundation of China (Grant Nos. 60673096 and 60373078)
www.scichina.com
www.springerlink.com
Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
tion systems, consistent or inconsistent decision information systems, information systems with fuzzy decision and variable precision rough set models, we introduce the concept of a consistent approximation representation space and obtain the corresponding judgment theorem determining consistent attribute sets. Moreover, we examine the characterizations of three important types of attribute sets, especially, provide the conditions to determine the relative necessary and unnecessary attribute sets.
1 Concepts and properties of rough sets Let U = { x1 , x2 ,
, xn } be a finite set of objects called the universe of discourse, P (U ) a
power set of U, and R an equivalence relation on U. Denote
[ xi ]R = { x j ∈ U : ( xi , x j ) ∈ R} .
{
Then U R = [ xi ]R : xi ∈ U
}
forms a partition of U, and ⎧⎪
⎫⎪
⎩⎪ xi ∈X
⎭⎪
σ (U R ) = ⎨ ∪ [ xi ]R : X ⊆ U ⎬ is a σ -algebra on U. Definition 1[1]. Let R be an equivalence relation on U. The pair (U , R ) refers to an approximation space. For any X ⊆ U , denote
{ } { } R ( X ) = { xi ∈ U : [ xi ]R ∩ X ≠ φ } = ∪ {[ xi ]R : [ xi ]R ∩ X ≠ φ }, R ( X ) = xi ∈ U : [ xi ]R ⊆ X = ∪ [ xi ]R : [ xi ]R ⊆ X ,
(1) (2)
R( X ) and R( X ) respectively refer to the lower and upper approximations of X with respect to (U , R) . If R( X ) = R( X ) , then X is called a definable set, otherwise it is called a rough set. Obviously, R , R : P (U ) → σ (U R), standing for the lower and upper approximation operators respectively. Theorem 1[2]. The lower and upper approximation operators satisfy the following properties: (i) R (φ ) = R (φ ) = φ , R (U ) = R (U ) = U , (ii) R ( X ) ⊆ X ⊆ R ( X )
(X ⊆U), (iii) R ( X ∩ Y ) = R ( X ) ∩ R (Y ) , R ( X ∪ Y ) = R ( X ) ∪ R (Y ) (iv) ~ R ( X ) = R ( ~ X ) , ~ R ( X ) = R ( ~ X ) (X ⊆U), where ~ X is the complement of X. Theorem 2[4]. If R is an equivalence relation on U, then
{
( X ,Y ⊆ U ) ,
}
σ (U R ) = X ⊆ U : R ( X ) = R ( X ) . Definition 2.
Let R and R′ be equivalence relations on U, and denote D = U R ′ = { D1 , D2 , , Dr } , D = ( D1 , D2 ,
, Dr ) ,
R ( D ) = ( R ( D1 ) , R ( D2 ) ,
, R ( Dr ) ) ,
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
189
R′ is said to be represented by R if R ( D ) = D. Let R and R′ be equivalence relations on U, and denote
Definition 3.
posR ( R′ ) = ∪ { R ( D ) : D ∈ U R′} =
∪ R(Dj ) ,
j≤r
posR ( R′) represents the positive region of R′ with respect to R. Theorem 3. Let R and R′ be equivalence relations on U. Then the following statements are equivalent: (i) R′ is represented by R, (ii) R ⊆ R′,
(iii) posR ( R ′ ) = U . Proof.
( )
If R′ is represented by R, then R D j = D j ( j ≤ r ), that is, D j ∈ σ (U R ) ( j ≤ r ).
If D j = [ xi ]R′ , then
[ xi ]R ⊆ [ xi ]R′ ,
so R ⊆ R′ . Vice versa, we can conclude that R ' ⊆ R . Thus
( )
we have proved that (i) ⇔ (ii). On the other hand, since R D j ⊆ D j ( j ≤ r ), we observe that
( )
posR ( R ′ ) = U if and only if R D j = D j ( j ≤ r ); thus we have proved that (i) ⇔ (iii).
2 Judgment theorem of attribute reduction and characterization of attributes in consistent approximation representation spaces Let U = { x1 , x2 ,
Definition 4.
, xn } be a finite universe of discourse, A = {a1 , a2 ,
, am }
a finite set of attributes, R = { Ra ⊆ U × U : a ∈ A} a class of equivalence relations on U , and R′ an equivalence relation on U. Then the quadruple (U , A, R , R′ ) is referred to as an approximation representation space. Definition 5. Let S = (U , A, R , R′ ) be an approximation representation space, denote RB =
∩ Ra ( B ⊆ A) . S denotes a consistent approximation representation space if
a∈B
RA ⊆ R′ . An
attribute subset B ⊆ A is referred to as a consistent attribute set of S if RB ⊆ R′. If B is a consistent attribute set of S and B − {b} is not a consistent attribute set of S for all b ∈ B , then B is referred to as a reduction of S. Definition 6. Let S = (U , A, R , R′ ) be an approximation representation space, and denote ⎧⎪{a ∈ A : ( xi , x j ) ∉ Ra }, [ xi ]R ' ∩ [ x j ]R ' = φ , DR ' [ xi ] A ,[ x j ] A = ⎨ [ xi ]R ' ∩ [ x j ]R ' ≠ φ . ⎪⎩ A,
(
(
DR ' [ xi ] A ,[ x j ] A
)
)
is referred to as the discernibility attribute set of xi and xj, and DR′ =
{DR ' ([ xi ]A ,[ x j ]A ) : xi , x j ∈U } Remark.
is called the discernibility matrix of S.
If S = (U , A, R , R′ ) is a consistent approximation representation space, then we
(
)
can conclude that DR ' [ xi ] A ,[ x j ] A ≠ φ
190
for all xi , x j ∈ U . In fact, since RA ⊆ R′, if
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
( xi , x j ) ∉ R′ , we have ( xi , x j ) ∉ RA = ∩ Ra . Then there exists an a ∈ A
such that
a∈ A
that is,
( xi , x j ) ∉ R′
(
)
implies DR ' [ xi ] A ,[ x j ] A ≠ φ . On the other hand, if
(
( xi , x j ) ∉ Ra ;
( xi , x j ) ∈ R′ , by the
)
definition we have DR ' [ xi ] A ,[ x j ] A = A ≠ φ . Theorem 4. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then B ⊆ A is a consistent attribute set of S if and only if
(
)
B ∩ DR ' [ xi ] A ,[ x j ] A ≠ φ Proof.
( xi , x j ∈ U ).
(3)
If B is a consistent attribute set of S, that is, RB ⊆ R′, which is equivalent to the say-
( xi , x j ) ∉ R′ implies ( xi , x j ) ∉ RB , i.e. when [ xi ]R ' ∩ [ x j ]R ' = φ , there exists an such that ( xi , x j ) ∉ Ra . Hence eq. (3) holds, and, therefore, by Remark of Definition 6
ing that a∈B
the proof of this theorem is completed. It should be pointed out that for any consistent approximation representation space S , since A is finite, reductions of S must exist, but may not be unique. Definition 7. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Let
{Bk : k ≤ l}
be all the reductions of S , and denote C=
∩ Bk ,
K=
k ≤l
∪ Bk − C ,
I = A − (K ∪ C) ,
k ≤l
C is referred to as the core attribute set of S, K is referred to as the relative necessary attribute set of S and I is referred to as the unnecessary attribute set of S. Theorem 5. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then the following statements are equivalent: (i) a is an element of core attribute set of S;
(
)
(ii) there exists xi , x j ∈ U such that DR ' [ xi ] A ,[ x j ] A = {a} ; (iii) RA−{a} ⊆/ R′. Proof. (i) ⇒ (ii). Assume that a is an element of core attribute set of S. If any discernibilty attribute set containing attribute a has at least two elements. Let
{ (
)
(
)}
B = ∪ DR ' [ xi ] A ,[ x j ] A − {a}: a ∈ DR ' [ xi ] A ,[ x j ] A . i, j
(
)
(
)
It is easy to see that B ∩ DR ' [ xi ] A ,[ x j ] A ≠ φ xi , x j ∈ U . By Theorem 4, B is a consistent attribute set of S, and a ∉ B . Then there exists a reduction B′ of S such that a ∉ B′, which contradicts the fact that a is an element of core attribute set of S.
(
)
(ii) ⇒ (iii). Assume DR ' [ xi ] A ,[ x j ] A = {a}. It means that
( xi , x j ) ∈ Rb
( xi , x j ) ∉ R′, ( xi , x j ) ∉ Ra ,
and
(b ≠ a ). Then [ xi ]R ' ∩ [ x j ]R ' = φ , but [ xi ] A−{a} = [ x j ] A−{a} . Hence [ xi ] A−{a} ⊆/ [ xi ]R′ ,
which implies that RA−{a} ⊆/ R′. (iii) ⇒ (i). If a is not an element of core attribute set of S , then there exists a reduction B of ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
191
S such that a ∉ B . It is clear that B ⊆ A − {a} , and consequently, RA−{a} ⊆ RB ⊆ R′ , which contradicts the assumption RA−{a} ⊆/ R′ . Theorem 6. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then
a is an element of the unnecessary attribute set of S if and only if R ( a ) ⊆ R′ ∪ Ra , where
{
}
R ( a ) = ∪ RB −{a} : RB ⊆ R′, B ⊆ A . Proof. If a is an element of the unnecessary attribute set of S, then a is not in any attribute reductions; that is, for any B ⊆ A with RB ⊆ R′ , we have RB −{a} ⊆ R′, consequently, R ( a ) ⊆ R′ ⊆ R′ ∪ Ra .
Conversely, if R ( a ) ⊆ R′ ∪ Ra , then for any B ⊆ A with RB ⊆ R′ , we have RB −{a} ⊆ R′ ∪ Ra , i.e. RB −{a} ∩ Rac ⊆ R′ ; thus RB −{a} = RB ∪ ( RB −{a} ∩ Rac ) ⊆ R′ . Consequently, a is not in any attribute reductions, i.e. a is an element of the unnecessary attribute set. For the core attribute set C of S, since RC ⊆ R′ ∪ Ra implies R ( a ) ⊆ R′ ∪ Ra , we see that
RC ⊆ R′ ∪ Ra is the sufficient condition of a being an element of the unnecessary attribute set of S. Theorem 7. Let S = (U , A, R , R′ ) be a consistent approximation representation space. Then (i) a ∈ C iff RA−{a} ⊆/ R′ ; (ii) a ∈ I iff R(a) ⊆ R′ ∪ Ra ; (iii) a ∈ K iff RA−{a} ⊆ R′ and R(a ) ⊆/ R′ ∪ Ra . Proof. It follows immediately from Theorems 5 and 6. Example 1. Table 1 illustrates an exemplary information system (U , A, F ) , where U =
{ x1 , x2 ,
, x8 } , A = {a1 , a2 , a3 , a4 } . Let
Ra = {( xi , x j ) ∈ U × U : f a ( xi ) = f a ( x j )}, a ∈ A , R = { Ra : a ∈ A} ,
RB = {( xi , x j ) ∈ U × U : f a ( xi ) = f a ( x j ), a ∈ B}, B ⊆ A . It is easy to see that S = (U , A, R , RA ) is a consistent approximation representation space. It can also be calculated that
DRA = {{a3 } , {a1 , a2 } , {a1 , a2 , a3 } , {a1 , a3 , a4 } , {a2 , a3 , a4 } , A}.
Table 1
192
An information system of Example 1 a1 x1 1 x2 2 x3 1 x4 1 x5 1 x6 1 x7 1 x8 1
a2 1 1 1 2 2 2 2 2
a3 3 2 3 2 2 1 2 1
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
a4 2 1 2 1 1 1 1 1
The consistent approximation representation space has two reductions: B1 = {a1 , a3 } , B2 = {a2 , a3 } . Then we have Since
C = {a3 } , K = {a1 , a2 } , I = {a4 } .
( x6 , x7 ) ∈ RA−{a } 3
and RA− a3 ⊆/ RA , a3 is an element of core C. Since
U Ra1 = {{ x2 } , { x1 , x3 , x4 , x5 , x6 , x7 , x8 }} , U Ra2 = {{ x1 , x2 , x3 } , { x4 , x5 , x6 , x7 , x8 }} , U Ra3 = {{ x1 , x3 } , { x2 , x4 , x5 , x7 } , { x6 , x8 }} , U Ra4 = {{ x1 , x3 } , { x2 , x4 , x5 , x6 , x7 , x8 }} , we can see that Ra3 ⊆ Ra4 , Ra3 ⊆/ Ra1 , Ra3 ⊆/ Ra2 , and RA−{ak } ⊆ RA for all k ≠ 3 . Thus a4 is
an element of the unnecessary attribute set I , and a1 , a2 are elements of the relative necessary attribute set K.
3 Constructive theorems of consistent approximation representation spaces in inconsistent information systems Theorems 4 and 7 respectively give the judgment theorem of consistent attribute set and the characterizations of three types of attribute sets in a consistent approximation representation space. In this section, we will show that various inconsistent information systems can be changed into consistent approximation representation spaces. Thus judgment theorems for determining consistent attribute sets in different senses in these systems will be obtained and the characterization of different types of attribute sets will also be examined. Definition 8. Let I = (U , A, F ) be an information system, d : U → Vd , where Vd is a finite set, then (U , A, F , d ) is referred to as a decision information system, and (U , A, F , d ) is said to be consistent if RA ⊆ Rd , otherwise it is said to be inconsistent, where Rd =
It is easy to see that if
{( x , x ) ∈U × U : d ( x ) = d ( x )} . i
j
i
(U , A, F , d )
j
is a consistent decision information system, then
(U , A, R , Rd ) is clearly a consistent approximation representation space. Letting (U , A, F , d ) be an inconsistent decision information system, we define
{
(
}
U Rd = D j : j ≤ r , D D j
[ xi ]B ) =
D j ∩ [ xi ]B
(
B ⊆ A, xi ∈ U , j ≤ r ,
[ xi ]B ,
)
μ B ( xi ) = D ( D1 [ xi ]B ) , D ( D2 [ xi ]B ) , , D ( Dr [ xi ]B ) ,
{
η B ( xi ) = D j : D ( D j
},
D ( Dk [ xi ]B ) [ xi ]B ) = max k ≤r
(4)
where | X | is the cardinality of the set X. Theorem 8. Let (U , A, F , d ) be an inconsistent decision information system. Then S =
(U , A,R , Rμ )
is a consistent approximation representation space, and B ⊆ A is a consistent
attribute set of S if and only if μ B ( xi ) = μ A ( xi ) ( xi ∈ U ), where
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
193
{( x , x ) ∈U × U : μ
Rμ =
Proof.
i
j
A
( xi ) = μ A ( x j )}.
It is easy to see that Rμ is an equivalence relation on U and RA ⊆ Rμ ; thus
(
S = U , A, R , Rμ
)
is a consistent approximation representation space. If B is a consistent attrib-
( xi , x j ) ∈ RB
ute set of S, then RB ⊆ Rμ ; that is,
(
)
implies xi , x j ∈ Rμ , i.e. x j ∈ [ xi ]B implies
μ A ( x j ) = μ A ( xi ) . Consequently,
(
( k ≤ r , x j ∈ [ xi ]B ) .
)
D Dk /[ x j ] A = D ( Dk /[ xi ] A ) Hence D ( Dk /[ xi ]B ) = Dk ∩ [ xi ]B / [ xi ]B
{ (
)
= ∑ D Dk /[ x j ] A [ x j ] A / [ xi ]B :[ x j ] A ⊆ [ xi ]B
}
= D ( Dk /[ xi ] A )
( k ≤ r , xi ∈ U ) , from which it follows that μ B ( xi ) = μ A ( xi ) ( xi ∈ U ) . Conversely, assume that μ B ( xi ) = μ A ( xi ) ( xi ∈ U ) . If ( xi , x j ) ∈ RB , that is, [ xi ]B = [ x j ]B , then μ B ( xi ) = μ B ( x j ) , and hence μ A ( xi ) = μ A ( x j ) , i.e.,
( xi , x j ) ∈ Rμ . Therefore,
Theorem 9.
(
S = U , A, R , Rη
)
RB ⊆ Rμ .
Let (U, A, F, d) be an inconsistent decision information system. Then is a consistent approximation representation space, and B ⊆ A is a consistent
attribute set of S if and only if η B ( xi ) = η A ( xi ) Rη =
Proof.
( xi ∈ U ) ,
{( x , x ) ∈U × U :η i
j
A
where
( xi ) = η A ( x j )}.
It is easy to see that Rη is an equivalence relation on U and RA ⊆ Rη . Thus S is a
consistent approximation representation space. Similar to the proof of Theorem 8, we can prove that B is a consistent attribute set of S if and only if η B ( xi ) = η A ( xi ) ( xi ∈ U ) . Let (U , A, F , d ) be an inconsistent decision information system, B ⊆ A , and β >0.5. Denote
( ) { ( [ xi ]B ) ≥ β } , β R B ( D j ) = { xi ∈ U : D ( D j [ xi ]B ) > 1 − β } , R βB D j = xi ∈ U : D D j
(
R βB = R βB ( D1 ) , R βB ( D2 ) ,
(
β
β
β
R B = R B ( D1 ) , R B ( D2 ) , Theorem 10.
Then:
(
)
, R βB ( Dr ) , β
)
, R B ( Dr ) .
Let (U , A, F , d ) be an inconsistent decision information system and β >0.5.
(i) S1 = U , A, R , R β
)
is a consistent approximation representation space, and B ⊆ A is a β
β
consistent attribute set of S1 if and only if R B = R A , where Rβ = 194
{( x , x ) : x ∈ R i
j
i
β A
( Dk ) ⇔ x j ∈ R βA ( Dk ) ( k ≤ r )}.
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
(
(ii) S 2 = U , A, R , R
β
)
is a consistent approximation representation space, and B ⊆ A is a β
β
consistent attribute set of S2 if and only if R B = R A , where β
R = Proof.
{( x , x ) : x ∈ R i
j
i
(i) It is easy to see that R
β
β A
}.
β
( Dk ) ⇔ x j ∈ R A ( Dk ) ( k ≤ r )
β
is an equivalence relation on U and RA ⊆ R ; thus S1
is a consistent approximation representation space. Now we are only to prove that RB ⊆ R β
β
iff
β
RB = R A. Assume that R B = R A . By the definition we have R B ( Dk ) = R A ( Dk ) β
( xi , x j ) ∈ RB ,
β
that is,
β
[ xi ]B = [ x j ]B ,
then
( k ≤ r ).
β
xi ∈ R B ( Dk ) ⇔ x j ∈ R B ( Dk ) . β
β
If
Consequently,
xi ∈ R βA ( Dk ) ⇔ x j ∈ R βA ( Dk ) . Hence ( xi , x j ) ∈ R β . Thus we have proved that RB ⊆ R . β
Conversely, suppose that RB ⊆ R . If [ xi ]B = [ x j ]B , then we observe that xi ∈ R A ( Dk ) ⇔ β
β
x j ∈ R βA ( Dk ) . Similar to the proof of Theorem 8, we can conclude that xi ∈ R A ( Dk ) ⇔ β
xi ∈ R B ( Dk ) . Thus R B ( Dk ) = R A ( Dk ) . It follows that R B = R A . β
β
β
β
β
(ii) It is similar to the proof of (i).
4 Constructive theorems of consistent approximation representation spaces in information systems with fuzzy decision In this section we will change an information system with fuzzy decision into a consistent approximation representation space, and then we can obtain results similar to Theorems 4 and 7. Definition 9. A quadruple (U, A, F, D) is referred to as an information system with a fuzzy
{
decision, where (U, A, F ) is an information system and D = D j : U → [ 0,1] , j ≤ r
}
is a fuzzy
decision. Let (U, A, F, D) be an information system with fuzzy decision, B ⊆ A, x ∈ U , j ≤ r . Denote
( ) ( x ) = min {D j ( y ) : y ∈ [ x]B } , R B ( D j ) ( x ) = max {D j ( y ) : y ∈ [ x ]B } , RB ( D j ) ( x ) = ∑ { D j ( y ) : y ∈ [ x ]B } [ x ]B , RB Dj
R B ( x ) = ( R B ( D1 )( x ) , R B ( D2 )( x ) ,
(
Theorem 11.
, R B ( Dr )( x ) ) ,
)
R B ( x ) = R B ( D1 )( x ) , R B ( D2 )( x ) ,
, R B ( Dr )( x ) ,
RB ( x ) = ( RB ( D1 )( x ) , RB ( D2 )( x ) ,
, RB ( Dr )( x ) ) .
Let (U, A, F, D) be an information system with fuzzy decision, then
(i) S1 = (U , A, R , R ) is a consistent approximation representation space and B ⊆ A is a conZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
195
sistent attribute set of S1 if and only if R B ( xi ) = R A ( xi ) R=
(
(ii) S 2 = U , A, R , R
)
{( x , x ) ∈U × U : R i
j
A
( xi ∈ U ) , where
( xi ) = R A ( x j )} .
is a consistent approximation representation space and B ⊆ A is a
consistent attribute set of S2 if and only if R B ( xi ) = R A ( xi ) R=
{( x , x ) ∈U × U : R i
j
A
( xi ∈ U ) ,
where
( xi ) = R A ( x j )} .
(iii) S3 = (U , A, R , R ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S3 if and only if RB ( xi ) = RA ( xi ) R=
Proof.
{( x , x ) ∈U × U : R i
j
A
( xi ) = RA ( x j )}.
( xi , x j ) ∈ RA , that is, R A ( xi ) = R A ( x j ) , and equiva-
(i) It is easy to see that R B is an equivalence relation on U. If
( ) (k ≤ r ) ,
[ xi ] A = [ x j ] A , then R A ( Dk )( xi ) = R A ( Dk ) x j lently,
( xi ∈ U ) , where
( xi , x j ) ∈ R.
so
Thus RA ⊆ R , which means that S1 is a consistent approximation represen-
tation space. Now we are only to prove that RB ⊆ R if and only if R B ( xi ) = R A ( xi ) that R B ( xi ) = R A ( xi )
( )
that RB ⊆ R . Conversely, if
Assume
( xi , x j ) ∈ RB , we have [ xi ]B = [ x j ]B . Then R B ( xi ) = R A ( xi ) = R A ( x j ) , that is, ( xi , x j ) ∈ R . Thus we have concluded RB ⊆ R, then [ xi ]B = [ x j ]B . We have R A ( x j ) = R A ( xi ) , so
( xi ∈ U ) .
R B x j , and consequently,
( xi ∈ U ) .
If
R B ( Dk )( xi ) = min { Dk ( y ) : y ∈ [ xi ]B }
} { { = min { R ( D ) ( x ) :[ x ] ⊆ [ x ] }
= min min Dk ( y ) : y ∈ [ x j ] A :[ x j ] A ⊆ [ xi ]B k
A
j
j A
}
i B
= R A ( Dk )( xi ) ,
from which it follows that R B ( xi ) = R A ( xi )
( xi ∈ U ) .
(ii) and (iii) can be proved similarly to the proof of (i). Let (U, A, F, D) be an information system with fuzzy decision, B ⊆ A, xi ∈ U . Denote
{ ( ) ( x ) = max R ( D )( x )}, M ( x ) = { D : R ( D ) ( x ) = max R ( D )( x )} , M ( x ) = { D : R ( D ) ( x ) = max R ( D )( x )} . M B ( xi ) = D j : R B D j
Theorem 12.
i
B
i
j
B
j
i
B
i
j
B
j
i
k ≤r
k ≤r
k ≤r
B
k
i
B
k
i
B
k
i
Let (U, A, F, D) be an information system with fuzzy decision. Then
(i) S1 = (U , A, R , M ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S1 if and only if M B ( xi ) = M A ( xi ) 196
( xi ∈ U ) , where
ZHANG WenXiu et al. Sci China Ser F-Inf Sci | April 2007 | vol. 50 | no. 2 | 188-197
M=
(
{( x , x ) ∈U × U : M i
j
A
( xi ) = M A ( x j )}.
)
(ii) S 2 = U , A, R , M is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S2 if and only if M B ( xi ) = M A ( xi ) M=
{( x , x ) ∈U × U : M i
j
A
( xi ∈ U ) ,
where
( xi ) = M A ( x j )}.
(iii) S3 = (U , A, R , M ) is a consistent approximation representation space and B ⊆ A is a consistent attribute set of S3 if and only if M B ( xi ) = M A ( xi ) M=
Proof.
{( x , x ) ∈U × U : M i
j
A
( xi ∈ U ) ,
where
( xi ) = M A ( x j )} .
It is similar to the proof of Theorem 11.
5 Conclusions We have introduced in this paper the concept of a consistent approximation representation space and established attribute reduction theory in this space. We have obtained the judgment theorem of consistent attribute sets and the approach of attribute reductions. We have also examined the characterizations of three important types of attribute sets: the core attribute set, the relative necessary attribute set and the unnecessary attribute set. Many well-known information systems such as consistent or inconsistent decision information systems, information systems with fuzzy decision and variable precision rough set models, etc. and the associating attribute reduction theory can be unified under the framework of consistent approximation representation space theory. However, all the information systems discussed in this paper are complete and the binary relations are all equivalent. Research of the theory of approximation presentation space based on non-equivalent relations will be discussed elsewhere. The authors would like to thank the anonymous referees for their valuable comments and suggestions. 1 2 3 4 5 6 7 8 9 10 11 12
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