Knowledge Dependency Relationships in Incomplete Information ...

Knowledge Dependency Relationships in Incomplete Information System Based on Tolerance Relations Chen Wu, Xiaohua Hu, Lijuan Wang, Xibei Yang, Yi Pan

Abstract— In this paper we define the concepts of knowledge dependency, partial knowledge dependency and dependency degree in rough set models based on tolerance relations for incomplete information system. These concepts, not properly defined and studied in previous research study for incomplete information system, but are very important from data mining and rough set theory community. We consider two kinds of dependency relationships: one is related to the conditional attribute subset and the decision, and the other is related to two different conditional attribute subsets. We then propose several theorems based on our revised knowledge dependency relationships. We give formal proofs of the theorems and verify their correctness with some examples.

I. INTRODUCTION

R

ough Set Theory (RST), proposed by Palwak Z. in the early 1980s as a mathematical tool to deal with non-determination, fuzziness, and uncertainty, has been widely used in a lot of research fields such as Artificial Intelligence, Data mining, Pattern Recognition, Knowledge Acquisition, Machine Learning and so on [1,2]. The RST is originally developed for complete information systems and is based on the assumption that all objects have definitive values in every attribute. The classification is made by an indiscernibility relation defined by Pawlak Z. But in incomplete information systems (IIS), it is not always possible to establish an indiscernibility relation due to the existence of null values. So the original RST may not be applied in incomplete information systems, which is common in the real world. Therefore some new expanded RST models are necessary in dealing with the incomplete information system, mainly in expanding indiscernibility relation to non-indiscernibility Chen Wu is with School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003,China; and with the College of Information Science and Technology, Drexel University, Philadelphia, PA 19104, USA. (e-mail: [email protected]). Xiaohua Hu is with the College of Information Science and Technology, Drexel University, Philadelphia, PA 19104, USA. (e-mail: [email protected]). Lijuan Wang is with School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003,China . Xibei Yang is School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212003,China . Yi Pan is Dept. of Computer Science, Georgia State University, Atlanta, GA, USA

relation such as tolerance relation, similarity relation, limited tolerance relation, etc. [3, 4, 5, 6, 7] There are many methods of knowledge dependency and the dependency degree far widely applied in the measurement of significance of attributes and the rule extraction. But these methods mainly depend on the assumption of complete information system, and every attribute’s value must be known. But in the real application, it is very often that data set is incomplete because of data measurement’s errors and limits. Thus, the quondam knowledge dependency and dependency degree can hardly reflect the facts very well. A definition of extensive knowledge dependency degree, considering the former as a special example is given in [8]. The functional dependency and multiple valued dependencies in the relational database with null values are discussed in [9], but they are in the relational database model based on relational algebra, not with the rough set theory. So, in this paper, we generalize the equivalence relation into the generic tolerance relation, and give the new definitions of functional dependency and the dependency degree under the incomplete information system. Furthermore, we discuss whether or not a series of properties under the complete information systems is tenable under the incomplete information system. II. ELEMENTARY CONCEPTS Definition 1 An incomplete information system (IIS) is a

IIS =< U , V, A, f > , where U is a non-empty finite set of objects and A is a non-empty finite set of Va a ∈ A U Va quadruple

attributes, such that for any

:

?

.Here

is

V

called the value set of attribute a. Attribute domain a may contain a null value, meaning unknown or uncertain, denoted

V = U a∈ AV a Represents the value set f is called an information function in of all attributes in IIS . by special symbol ‘*’.

IIS , f (x , a) ∈ Va

for any a ∈ A and any x ∈ U . In incomplete information systems, null values are divided into two types, one is existed, and the other is non-existing. In our paper, the assumption of existing type of null values is considered. We take the example in [3, 10, 11] to explain them. Example 1 Suppose U = {1,2,3,4,5,6}; A = {P, M, S, X}, P stand for Price, M for Mileage, S for Size, and X for Max-speed, V = {high, low, full, compact, *}, so Table 1 is an IIS.

Table 1 An incomplete information system Car

P

M

S

X

1

high

high

full

low

2

low

*

compact

low

3

*

*

compact

high

4

high

*

full

high

5

*

*

full

high

6

low

high

compact

*

Table 2 Different values of attributes p and q p(x1 )

IIS =< U , V , AT , f > is an incomplete information system. For any a ∈ A , the tolerance relation SIM ({a}) is defined as follows: SIM ({a})={(x, y) ∈ U× U| a(x)=a(y) ∨ a(x)=* ∨ a(y)=* } Definition 2

Definition 3 We denote

SIM(B) = ∩ SIM({a}) for a∈B

a

a

b

q(x2 ) b

a

a

b

a

a

a

p(x1 )

p(x2 )

*

a

q(x1 ) b

q(x2 ) b

*

*

a

b

*

*

b

*

a

*

b

a

*

*

*

a

*

*

a

*

b

b

*

*

b

b

a

*

b

*

*

*

b

*

a

*

*

b

*

*

*

b

a

*

*

*

*

*

*

*

In the conventional definition of knowledge dependency,

p→q

if

and

only

q ( x1 ) = q( x2 )

if

when

p ( x1 ) = p( x2 ) .But in Definition 6, we have 16 cases shown

B ⊆ A.

Obviously,

q(x1 )

Suppose

.

any

p(x2 )

SIM ({a}) is a consistence relation,

possessing reflexivity and symmetry.

δ B ( x ) denote the object set {y ∈ U|(x, y) ∈ SIM(B)} , δ B ( x ) is the tolerance class Definition 4 Let

in Table 2, considering various appearances of null values in IIS, when a ? *,b ? *. Definition 7 Suppose P, Q ⊆ A are two different attribute subset in IIS, we call

Q totally depends from P

{δ B ( x ) | x ∈U } . Any element of U / SIM ( B ) is called a

P ⇒ Q ) if for any q ∈ Q , there exists an attribute p ∈ P such that p → q . Theorem 1 p→q if and only if SIM ({ p}) ⊆ SIM ({q}) for any p , q ∈ A . Proof (1) For p→q ,

tolerance class.

if

B ⊆ A. U / SIM ( B ) denote the Definition 5 Let

for any

x ∈U

with respect to

corresponding classification, which is the family set

III. FUNCTIONAL DEPENDENCY AND DEPENDENCY DEGREE IN IIS

p , q ∈ A are two any given and different attributes in IIS, we denote p → q if and only if for Definition 6 Suppose

any

x1 , x2 ∈U

,

x1 ≠ x2

,

when

p ( x1 ) = p( x2 ) ∨ p( x1 ) = * ∨ p( x2 ) = * is true, we have q ( x1 ) = q( x2 ) ∨ q( x1 ) = * ∨ q ( x2 ) = * .

(denoted by

p ( x1 ) = p( x2 ) ∨ p( x1 ) = * ∨ p( x2 ) = *

( x1 , x2 ) ∈ SIM ({ p})

.So

it

is

q ( x1 ) = q( x2 ) ∨ q( x1 ) = * ∨ q( x2 ) = *

,

then

certain

that

.

Therefore

( x1 , x2 ) ∈ SIM ({q}) .Thus, when p → q , It is held that SIM ({ p}) ⊆ SIM ({q}) ; (2) If SIM ({ p}) ⊆ SIM ({q}) , then ( x1 , x2 ) ∈ SIM ({ p}) , and furthermore p ( x1 ) = p( x2 ) ∨ p( x1 ) = * ∨ p( x2 ) = * ( x1 , x2 ) ∈ SIM ({q})

.

This

means

.

So

q ( x1 )

= q ( x2 ) ∨ q ( x1 ) = * ∨ q ( x2 ) = * .Therefore p → q . Thus, when SIM ({ p}) ⊆ SIM ({q}) , p → q is true.

⇒ Q if and only if SIM ( P ) ⊆ SIM (Q ) for any P , Q ⊆ A . Proof (1) In P ⇒ Q , for any q ∈ Q , there exists p ∈ P , such that SIM ({ p}) ⊆ SIM ({q}) . According to Definition 3, we can obtain SIM ( P ) ⊆ SIM (Q ) . Theorem 2 P

(2)

According

to

SIM ( P ) ⊆ SIM (Q )

Definition

3, means

But

if

we

still

use

(3.1),

(3.2),

we

can

get

POS P ({ X }) = U and k = γ P ({ X }) = 1 . P totally depends on X. It does not agree with the fact. So we have to relax the strict definitions accordingly. Let us consider two cases when we do define knowledge dependency in incomplete information systems: (1) Assume that attribute subset P is the condition attribute subset, and Q is the decision attribute subset and

I a∈P SIM ({a}) ⊆ I a∈Q SIM ({ a}) . Thus, for any SIM ({q}) , we can find SIM ({ p}) such that SIM ({ p}) ⊆ SIM ({q}) .It means that for any q ∈ Q , there exists p ∈ P such that p → q . Moreover, According to Theorem 1, we have P ⇒ Q .

the decision attribute subset has no null values. (2) Attribute subsets P and Q are both the condition

The dependency relation in Theorem 1 and 2 is based on the tolerance relation. In traditional rough set theory, the definition of partial knowledge dependency is based on the relative positive set:

Q = ∅ , then POS P (Q) = ∅ .

POS P (Q ) = U X∈U / IND (Q ) U C∈U / I N D( P) C( C ⊆ X ) U / IND( P ) , U / IND( Q ) are the equivalence class sets with respect to P and Q respectively. The dependency degree k of Q with respect to P is where

It is written as

(3.2)

k → Q , and | T | is the cardinal P 

number of set T in (3.2). This definition is proper for complete information systems, but it is not applicable in IIS, because we generally may not get equivalence relations. After extending the equivalence relation to the tolerance relation, we have to extend the equivalence classes in the definition of positive region to tolerance classes. It is a very obvious choice to use the following expression:

POS P (Q ) = U X∈U / SIM ( Q) U C∈U / SIM ( P) C (C ⊆ X ) (3.3) However, if we still directly revise the definition of the dependency

degree

for

P

 → k

Q

as

k = γ P (Q) = POS P ( Q) / | U | , we would encounter some problems. We find that in attribute subset Q , for any attribute a ∈ Q and any x ∈ U , if a ( x) = * , then S A ( x) = U . Therefore, there must have POS P (Q ) = U and

P and Q may have null values

In the case of (1), we give a definition as follows.

P , Q ⊆ A , we define the dependency set from P to Q as (3.3). Particularly, if Definition 8

Definition 9

For any

Q partially depends from P in degree k ,

k → Q , where k is defined in formula P 

denoted by

POS P ( Q) is defined in formula (3.3) . Obviously, 0≤ k ≤ 1. We also can have the following expressions. If k =0, then Q is totally independent from P ; If k =1, then Q is totally dependent from P , also denoted by P ⇒ Q . If 0 < k < 1 , then Q partially depends from P . (3.2) in which

( 3.1)

k = γ P (Q) = POS P ( Q) / | U |

attribute subsets. Both

k =1. So Q

is totally dependent on P . This does not agree with the fact obviously. Let us take Table 1 as an example. For attributes P and X, we know that f (P, 1)= f (P, 4)= high, but f (X, 1)=low, f (X, 4)=high, so f (X, 1) ≠ f (X, 4). Obviously, P does not totally depend on X.

In the case of (2), we need to revise the definition of knowledge dependency as follows. Definition 10 Suppose P , Q ⊆ A are any two given conditional

attribute

subsets.

If

for

∀ x ∈U

,

y ∈ δ P ( x ) means y ∈ δ Q ( x) , then we call Q is totally dependent from P , denoted by P ⇒ Q . Definition 11

Suppose P ,

Q ⊆ A are any two given

conditional attribute subsets. If for ∀ x ∈ U , there exists

y ∈ δ P ( x ) and y ∈ δ Q ( x) , then we call Q is partially dependent from

k → Q, P in degree k , denoted by P 

where

k = ∑ δ P ( x ) I δ Q ( x) / ∑ δ P( x ) x∈U

Obviously, 0 ≤

k =1, Q P ⇒ Q.

if

( 3.4)

x∈U

k ≤ 1. If k=0, Q

is not dependent from

is totally dependent from

P;

P , denoted by

Example 2 In Table 1, δ P (1) = δ P (4) ={1,3,4,5}, δ P (2) =

δ P (6)

={2,3,5,6},

δ P (3)

=

δ P (5)

={1,2,3,4,5,6};

δ M (1) = δ M (2) = δ M (3) = δ M (4) = δ M (5) = δ M (6) ={1,2,3,4,5,6}; δ S (1) = δ S (4) = δ S (5) = {1,4,5}, δ S (2) = δ S (3) = δ S (6) = {2,3,6}; δ X (1) = δ X (2) = {1,2,6}, δ X (3) = δ X (4) = δ X (5) ={3,4,5,6}, δ X (6) = {1,2,3,4,5,6}. According to Definition 10, we obtain P ⇒ M , S ⇒ M , X ⇒ M , S ⇒ P. k1 According to Definition 11, we have P → X , k2 k3 k4 X → P , S → X , X → S , where k 1=9/14, k 2=3/4, k 3=2/3, k 4=1/2.

δ S (3) ∩ δ X (3) = {3,6} ≠ ∅ . So at the meanwhile, we can k 1 → X , X  k 2 → P , S → k3 X , obtain P  k 4 X , where k , k , k , k are different real numbers S → 1 2 3 4 between 0 and 1. They validate the correctness of Theorem 4.

P , Q ⊆ A , if P ⊆ Q , then P ⇒ Q . Proof If P ⊆ Q ,then for ∀ x ∈ U , δ P ( x ) ⊆ δ Q ( x ) .So Theorem 5 For ∀

according to Theorem 3, we can get Example

5

P⇒ Q.

N = {P, M } in Table 1. Then

Let

{P} ⊆ N and {M } ⊆ N , δ N (1) = δ N (4) = {1,3,4,5} ,

δ N (2) = δ N (6) = {2,3,5,6} IV. PROPERTIES OF KNOWLEDGE DEPENDENCY UNDER IIS The theorems in this section are mainly related to Definition 10 and Definition 11, we mainly consider the dependency between two condition attributes, and the two attributes may have null values at the same time. The dependency between condition attributes and the decision attributes is a special case of it, so the discussion about it is omitted. Theorem 3 Suppose P , Q ⊆ A are any given condition attribute subsets. P ⇒ Q if and only if δ P ( x ) ⊆ δ Q ( x ) for ∀ x ∈ U . Proof (1) For ∀ x ∈ U , if δ P ( x ) ⊆ δ Q ( x ) , then

δ P ( x ) ∩ δ Q ( x ) = δ P ( x) .So k=1 according to (2.3), and P ⇒ Q. (2) For

∀ P , Q ⊆ A ,if P ⇒ Q , then k=1 according to

(2.3) and

∑ | δ P (x ) ∩ δ Q( x ) | = ∑ | δ P( x ) | . Thus x∈U

x∈U

SP ( x) ⊆ S Q ( x) . Example 3 In Table 1, for

∀ x ∈U ,δ P ( x ) ⊆ δ M ( x ) ,

δ S ( x ) ⊆ δ M ( x) , δ X ( x ) ⊆ δ M ( x ) ), δ X ( x) ⊆ δ P ( x ) , and at the meaning time, P ⇒ M , S ⇒ M , X ⇒ M , S ⇒ P , so they validate the correctness of Theorem 3. Theorem 4 Let P , Q ⊆ A be any given condition k P  → Q if and only if ∃x ∈U such that δ P ( x ) ∩ δ Q ( x ) ≠ ∅ .

attribute subsets.

k P  → Q , then according to Definition 11, ∃x ∈U such that y ∈ δ P ( x ) and y ∈ δQ ( x ) . Therefore

Proof If

δ P ( x) ∩ δQ ( x) ≠ ∅ . Example

4

δ P (2) ∩ δ X (2) = {2,6} ≠ ∅

δ N (3) = δ N (5) = {1,2,3,4, 5,6} . Consequently, we can obtain that for ∀ x ∈ U , δ N ( x ) ⊆ δ P ( x ) , δ N ( x) ⊆ δ M ( x ) are held according to theorem 3. Thus,

N ⇒ P , N ⇒ M . So we validate the correctness of theorem 5. Theorem 6 For (1) if

Table

1, ,

∀ P, Q, S ⊆ A ,we have

k 1→ Q , Q ⇒ S , then P  k 2 → S , and P 

k1 ≤ k2 ;

(2) if P

k 1 → S , then P  k 2 → S , and ⇒ Q, Q 

k1 ≤ k2 .

k 1→ Q means P  δ P ( x ) ∩ δQ ( y ) ≠ ∅ .For

Proof (1) According to Theorem 4, that

∃x, y ∈U such that

∀ x ∈ U , δ P ( x ) ⊆ δ Q ( x ) .According to Theorem 3, Q ⇒ S means δ Q ( x) ⊆ δ S ( x) . Therefore,

δ P ( x ) ⊆ δ S (x )

,

and

then

δ P ( x) ∩ δQ ( x)

k 2 → S , and k ≤ k . ⊆ δ S ( x ) ∩ δ Q ( x ) . Thus P  1 2 (2) We can prove it by the same method as in (1). Example 6

In Table 1,

k 1→ S , S ⇒ X 

P,

k 2 → P , and k =1/2, k =3/4, so k ≤ k ; X  1 2 1 2 k1→ X S ⇒ P , P 

,

k 2 X , and k =9/14, S → 1

k 1 =2/3, so k 1 ≤ k 2 .Thus, we validate the correctness of

Theorem 6.

∀ P, Q, S ⊆ A , it is held that k 1→ Q and P  k 2 → Q ∪ S , then; (1) if P  Theorem 7 For

In

,

k 1→ Q and P ∪ S → k 2 Q , then P 

(2) if k1 ≤ k2 .

Proof

P ⊆ P ∪ S , we can get P ∪ S

⇒ P , according to Theorem 4. Referring to Theorem 3, we obtain that for ∀ x ∈ U , δ P∪S ( x ) ⊆ δ P ( x ) . So δ P∪S ( x ) ∩ δ Q ( x ) ⊆ δ P ( x ) ∩ δQ ( x ) and k1 ≥ k2 . (2) We can prove this situation by the same method as (1). Example 7 In Table 1,we get following results: for have

T = { X , S}

short.

Then

, denoted by

according

δ T (1) = {1}

δ T (3) = {3,6}

,

,

to

δ T (4) =

δ T ( 2 ) ∩ δ P ( x ) = {2,6} ,

we ,

k 1 =9/14. When x=2,

∃x, y ∈U such that k 2 = 3/7. Thus,

k 1 → T , k =3/7; P  k 2 → X , k =9/14, k ≤ k . P  1 2 1 2 So Theorem 8 is right for this case. Theorem 9 Suppose P, Q, S

⊆ A are any given. If k 1 → Q , then P → k 2 Q , S  k 3 → Q , and P ∪ S 

for

According

get

1,

we

δ R (1) = δ R ( 4 ) = δ R ( 5 ) = {1,4,5}

,

δ R ( 2 ) = δ R (3) =

.

δ R (6) =

Proof According to Theorem 4,

P⊆P∪S ,

S ⊆ P ∪ S , so δ P∪S ( x ) ⊆ δ P ( x ) ,

δ P∪S ( x ) ⊆ δ S ( x ) . Furthermore δ P ( x ) ∩ δ Q ( x ) ≠ ∅ , δ S (x ) ∩ δ Q (x ) ≠ ∅ δ P∪S ( x ) ∩ δ Q ( x ) ⊆

,

δP ( x) ∩ δQ ( x)

and ,

δ P∪S ( x ) ∩ δ Q ( x ) ⊆ δ S ( x ) ∩ δ Q ( x ) . Thus, we can get the conclusion that

R = { P , S } , denoted by R = P ∪ S Table

for

k 1 → Q means P ∪ S  that ∃ x, y ∈ U such that δ P ∪S ( x ) ∩ δ Q ( y ) ≠ ∅ . For

1,

we get the conclusion that k1 ≥ k2 .

to

T = S∪ X

Table

δ T (5) = {4,5} ,

so

by

max( k 2 , k 3 ) ≤ k1

k2 X ∪S , δ P ( x ) ∩ δ T ( y ) ≠ ∅ , P →

short.

T = { S , X } ,denoted short. Then X ⊆ T ,

T = X ∪S

δ T (2) = {2,6}

k1→ X , δ T (6) = {2,3,6} . P 

(2) Let

k 1 ≤ k 2 is held..

Example 8 In Table 1, Suppose (1) Because

(1) Let

δ S ( x ) ∩ δ P ( x) ⊆ δ S ( x ) ∩ δ Q ( x ) ,

k 2 Q , S  k 3 → Q , and max P →

( k 2 , k 3 ) ≤ k1 . Example 9 In Table 1, let

R = P∪ S

R = { P , S } , denoted by

When

k 1→ X and P ∪ S  k2 X , k 1 =2/3. From the calculate above, we know P →

x=2,

δ R (2) ∩ δ R (2) = {2,6} , so ∃x, y ∈U such that k2 X , δ R ( x ) ∩ δ X ( y ) ≠ ∅ , and therefore P ∪ S →

k 3→ X , and k =9/14, k =2/3. Thus it illustrates the S  2 3 conclusion max( k 2 , k3 ) ≤ k1 , so we validate the correctness

and k2 = 2/3. Thus, we get the conclusion that k1 ≤ k2 .

of Theorem 9.

k1→ X P 

,

and

k1

{2,3,6} =9/14.

In this way, we check the correctness of Theorem 7 by (1) and (2). Theorem 8

For

∀ P, Q, S ⊆ A , if Q ⊆ P ,

k 1 → P , then S  k 2 → Q , and k ≤ k . S  1 2 Proof According to Theorem 5, from Q ⊆ P , we can get P ⇒ Q . According to Theorem 3, we can get that for ∀ x ∈ U , δ P ( x ) ⊆ δ Q ( x ) . Meanwhile, according to

k 1 → P means that ∃ x ∈ U , for S  y ∈ U , there exists δ S ( x ) ⊆ δ P ( x ) ,

Theorem 4,

∀

δ S ( x ) ⊆ δQ ( x)

.So

k 2→ Q S 

.Because

, we can get

V. CONCLUSIONS In this paper, we discuss two cases for knowledge dependency in expanded rough sets model based on tolerance relations, the knowledge dependency between condition attribute and decision attribute and the one between two condition attributes. We refine the definitions of total dependency, partial dependency, and the degrees under the two cases. Furthermore, we discuss a series of what properties under the complete information systems are still tenable or conservative under the incomplete ones. We find that the properties such as reflexivity, transitivity, extensibility, and decomposability under complete information system are still preserved under IIS, but the degrees have been changed. According to the research of knowledge dependency under IIS, we can get some new understandings about the knowledge dependency under IIS, and analytical results.

Consequently, it is a very useful tool in data mining under IIS. Acknowledgement: Hu’s research work was supported in part from the NSF Career grant IIS 0448023, NSF CCF 0514679 and the research grant from PA Dept of Health. Yi Pan's research was supported in part by the National Science Foundation (NSF) under Grants ECS-0196569, ECS-0334813, and CCF-0514750, the National Institutes of Health (NIH) under Grants R01 GM34766-17S1,and P20 GM065762-01A1. REFERENCES [1] Pawlak Z. Rough set theory and its applications to data analysis. Journal of Cybernetics and Systems, 1998(29):661-688. [2] Pawlak Z. Rough sets and intelligent data analysis . Information Sciences, 2002 (147): 1-12. [3] Kryszkiewicz .M. Rough Set Approach to Incomplete Information Systems. Information Sciences, 1998(112):39-49. [4] Jerzy Stefanowski. Incomplete information tables and rough classification. Journal of Computational Intelligence, 2001(17):545-566. [5] Hao Zhongxiao. On database under null values. Beijing: Mechanic Industry Publishing Company,1996 [6] Stefanowski J. Tsoukias A. On the extension of rough sets under incomplete information. In: N Zhong, A.Tsoukias, S.Ohsugaeds. Proc of the 7 th Int’l Workshop on New Directions in Rough Sets. Data Mining, and Granular-Soft Computing. Berlin:Springer-Verlag,1999,73-81 [7] Wang Guoying. Extension of Rough set under incomplete informationsystem. Journal of computer research and development, 2002, 39(10):1238-1243 [8] Hu Dan, Li Hongxing. The measurement of de velopment degree of knowledge. Journal of Beijing Normal University(Natural Science), 2004,40(3): 320-325 [9] Deng Fang’an, Xu Yang. Rough set approach to data reasoning for incomplete information system. Journal of computer engineering and applications, 2004,30:51-53 [10] Zhang Wenxiu, Liang Jiye. The theory and measures of rough set. Beijing: Science publishing company, 2001:206-212 [11] Huang Sumei, Wu Chen. Study of method of Knowledge reducing and decision reasoning for rough set in incomplete information system. Journal of East China shipbuilding institute (Natural science edition), 2004,18(6):51-54