2020 4th International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2020) ISBN: 978-1-60595-674-9
A New Method of Translating Covering Rough Set into
Classical Rough Set
Ying-ying HE
1, Lin-hai CHENG
1, Yu ZHANG
1and Yue-jin LV
2,3,* 1College of Electrical Engineering, Guangxi University, Nanning, China
2College of Mathematics and Information Science, Guangxi University, Nanning, China
3
Lushan College, Guangxi University of Science and Technology, Liuzhou, China
*Corresponding author
Keywords: Rough sets, Covering approximate spaces, Partition approximate spaces, Transposed class, Symmetric class.
Abstract. Binary relational rough sets enrich the applicable scope of classic rough set, but they lose some excellent properties. Therefore, covering translates to partition has become one of the key points in the study of covering approximate spaces. However, the existing translation methods have some shortcomings, such as the inconsistency between the partitions translated by covering and covering reduction, the inconsistency between the monotonicity of the covering and the translated partition, and the limited application of the translation method, and so on. In view of this situation, the basic requirements for covering translate partition are put forward, and then the concepts of transposed and symmetric classes are defined. On this basis, the translation method is proposed which gets over the shortcomings of the existing methods, and the effectiveness of the way is proved theoretically. At the same time, in consideration of the situation that practical data cannot form a single coverage, the new method is extended to the multi-covering approximate space by the minimum description of multi-covering elements.
Introduction
Rough sets theory is a mathematical tool for dealing with uncertain data which proposed by Pawlak-a Polish scholar [1]. On the basis of classification, knowledge is understood as a kind of classification ability. Formal knowledge is a partition of domain, and each partition is called a knowledge [1]. Compared with other mathematical tools for uncertainties, rough set theory doesn't need any prior information, and it has received widespread attention [2]. Nowadays, rough set theory has been extensive used in expert system, data mining, patter recognition and artificial intelligence [3,4,5,6].
Translating covering rough set into classical rough set can not only expends the applicable range of the classical rough set but also maintains the basic properties and approximate accuracy of classical rough set. So, it has attracted some scholars’ attention [10,11,12,13], however, the existing methods have disadvantages. In [10], the partitioning generated by this relation is the complete set of universes when the covering is a partitioning. What’s more, when C is a reduction of C ', there is discordance between the equivalent relation translated from C and the equivalent relation
translated from C '. In [11,12], the partitioning translated by those methods may become thicker when the covering becomes thinner. And in [13], it is not possible to generate a partitioning approximate space from the covering approximate space whose relationship between objects is unknown. Aiming at these problems, this paper proposes a new method of translate covering approximate spaces into partition approximate spaces by the concepts of transposed class and symmetric class. The new method not only solves the unreasonable place of exiting methods, but also applies to the covering approximate spaces with unknown relationship between objects. Simultaneously, it is of theoretical significance to extend the new method to the similarity space of the covering family in case that the real problem cannot form a single coverage.
Preliminaries
Definition 1. [14] Let U be nonempty finite set, and C is a set class of U . If the subset of C is nonempty and C U , C is a covering of U .
Definition 2.[14] Let U be nonempty finite set, and C is a covering of U , then ( , )
S U C is a covering approximation space.
Definition 3. [14] Supposethere is a covering approximation space, denote as S (U C, ), for
x U , we call F ( )x
G xG G C
is a subordinate group of x. G is a set of C . Definition 4. [14] Suppose there is a covering approximation space, denote as S (U C, ). x is an object in U . For x, his minimum description is defined as
( ) i i ( ) j ( ), j i i j
d
M x G G F x G F x G G G G (1) Definition 5. [15] Suppose C is a covering of U while G is a set of C , if there are
1, 2 i
G G G C G and
1
i
j j
G G , then G is a reducible set of C , otherwise, G is
irreducible. We say C is irreducible, if every set of C is irreducible.
If G is a reducible set of C , we will remove G until C is irreducible, then we get a reduction of C , denoted by Red( )C .
Definition 6. [16] Suppose C is a covering of U , for X U , we define the covering lower and upper approximation operations as :
( ) U i d( ), i
C X x G M x G X (2)
( ) U i d x( ), i
C X x G M G X (3)
Definition 7. [2] SupposeS
U, , ,V f A
is an information system, and an equivalence relation Nind Q( ), the approximation accuracy of the set X is( ) ( )
( )
N
N N
X X
The Exiting Translation Methods
Definition 8. (covering partial order relation) There is a covering approximation space, denote as S ( ,U C). C1 and C2 are coverings of U , if xi U , MdC1( )xi MdC2( )xi and
1 2
C C ,we say C1 is strictly coarser than C2, we record it as C1 C2.
Generally, a translate method should meets the following requirements:(1) if C is a partition, converted partition D should satisfies D C . (2) if C is a reduction of C ', the partition D
and D' translated by C and C ',should satisfy D D '. (3) if
1 2
C C , then the partition D1
and D2 translated by C1 and C2,should satisfy D1 D2. (4) The similarity accuracy of the
translated classic rough set model should not lower than that of the covering rough set before conversion.
The method of translate covering to partition proposed in reference [10]is as follows:
Definition 9. (equivalence relation) Suppose C is a covering of U , for xi U , F ( )x is a
neighborhood of xi, denoted by
xi , we say
xi is covering degree ofxi. If xi and xjsatisfy
xi xj , we say xi is equivalent to xj in covering degree.Definition 10 evidently not satisfies the above four points.
Example 1 Let U
x x x x x1, , , ,2 3 4 5
， C1
G G G1, 2, 3
， C2'
G G G G1, 2, 4, 5
，
2 1, 2, 5
C G G G ，and G1
x x1, 2
，G2
x x3, 4
，G3
x5 ，G4
x x x x1, , ,2 4 5
，G5
x x4, 5
。(1)For the coveringC1, the partition translated by C1 is
x x x x x1, 2, ,3 4, 5
, inconsistent with C1.(2)the partition translated by C2'is
x x x1, ,2 5
, x3 , x4
; the partition translated by C2 is
x x x x1, , ,2 3 5 , x4
.Though coveringC2 is a reduction of covering ' 2C , the partitions aren't consistent.
(3)C1 C2, but the partition translated by C1 is coarser than which by C2.
(4)Suppose X
x x x1, ,2 3
, then the approximation accuracy of C2 is1/ 2, While D2 is0.The method of translate covering to partition proposed in reference [11]is as follows:
Definition 10. Suppose C
G G1, 2, ,Gn
is an covering of U , x U , we denoted that:
( )
C Gj
I x j x (5)
Definition 11. There is a covering of U , denote as C
G G1, 2, ,Gn
, I
1, 2, ,n
and
I , we denoted that: I x i
I x, G i j I x, Gj
. Then we hold that IC( ) :x xU is a partition.Definition 11 not satisfies the point (3).
Example 2 Let U
x x x x x1, , , ,2 3 4 5
, C1
G G G1, 2, 3
, C2
G G G1, 4, 3
, and G1
x x x1, ,2 3
,
2 1, ,3 4
G x x x , G3
x x4, 5
, G4
x x1, 3
.The partition translated by C1 is D1
x x1, 3
, x2 ,
x4 , x5
and the partition translated by2
C is D2
x x1, 3
, x2 , x x4, 5
. From the definition 9we know that C2 C1 , however2 1
D D .
The method of translate covering to partition proposed in reference [12]is as follows:
Definition 12. (equivalence relation) There is a covering approximation space, record as ( ,U )
C
A C B C
A B
x y
x y (6)
Definition 12 not satisfies the point (3) too.
Example 3 (continue from example 2) We have C2 C1, from define 16 we can translate C1 to
a partition D1, D1
x x1, 3
, x2 , x4 , x5
, similarity we can translate C2 to a partition D2,
2 1, 3 , 2 , 4, 5
D x x x x x . Obviously, D2 D1 .
The method of translate covering to partition proposed in reference [13]is as follows:
Definition 13. ( -equivalence relation) There is an information system, record as
( , , , )
S U A V f BA is a subset of attributes, parameter and -similarity relation RB
, for U
xi , B( )
xi is a set contains all -maximally compatible class of xi , we define
-equivalence relation as follows:
( , ) , ( ) ( )
B b b
N x xi j U U b B xj xi (7) The method proposed by reference [13]is effectively when the similarity relation is clear between objects, but when the relationship between objects is uncertain, the method lose its basis.
A New Method to Translate Covering to Partition
It can be seen from the analysis of the third section that the existing method have their shortcomings. On this basis, we present a new method to translate a covering to a partition which overcomes the shortcomings of existing methods, and extend it into the covering family approximate spaces.
Definition 14. (Transposed class) There is a covering approximation space, denote as ( , )
S U C , Md( )x is minimum description ofx, we define the transpose class of x as follows:
( ) ( )
d d
T x y U x M y (8) By transposing, we get a set, for each object in this set, its minimal description containing x, that means the object x has relation with all objects in the set. and the transposed class have the following property.
Theorem 1. Let C1 C2, T Cd 1and T Cd 2 are transpose class of C1 and C2, then we have
1 2
d d
T C T C .
Proof. Let MdC1( )xj MdC2( )xj . For xi U
xj ,we have MdC1( )xi MdC2( )xi . Suppose xt MdC2( )xj butxt MdC1( )xj , then we have xj TdC2( )xt and xj TdC1( )xt . Thus1( ) 2( )
dC dC
T xt T xt , so xr U
xt we have TdC1( )xr TdC2( )xr , that mains T Cd 1 T Cd 2. This completes the proof.Definition 15. (symmetric class) There is a covering approximation space, record as S (U C, ), the minimum description of xis Md( )x , the transposed class of x is Td( )x , the symmetric class is defined as follows:
( ) ( ( )) ( )
d d d
S x M x T x (9) We know that, all objects in Md( )x have relation with x, and in Td( )x , the object x has relation with all objects in the set. Through the above, we can get that in symmetric class Sd(x), all objects have symmetric relation with x, that means, all objects are similar to xsymmetrically.
Definition 16. (equivalence relation) There is a covering approximation space, denote as ( , )
( ) ( )
d d
C
R S S
i j i j
x x x x (10)
Theorem 2. The equivalence relation RC satisfies transitivity, reflexivity and symmetry. Proof. Because the definition of equivalence relation is the equality of symmetric classes, its reflexivity, symmetry and transitivity are obvious. This completes the proof.
Definition 17. (equivalence class) SupposeS (U C, ) is a covering approximation space, the equivalence relation of the covering is RC , for x U , its equivalence class is:
( ) U C
D x y xR y (11) After the three steps above, we have successfully transformed the similarity relationship into the equivalence relationship, this method satisfies the four conditions proposed in the previous section which a translate method should meets, we prove them as follows
Theorem 3. There is a covering approximation space, record as S (U C, ), the equivalence class of all objects constitute
e a partition of U . Denoted by D .
Proof. (1) From the Throrem1 we know that xi D( )xi , thus
1
( )
U
D i U
i
x .
(2) If xi D( )xj D( )xi D( )xj , suppose D( )xi D( )xj xl . According to the definition 15 we have Sd( )xl Sd( )xi Sd( )xl Sd( )xj , thus Sd( )xi Sd( )xj . So xi D( )xj , we obtain
( )
D
i j
x x .
Combine (1) and (2), we have the equivalence class of all objects constitute a partition of U. This completes the proof.
Theorem 4. If C
G G1, 2, ,Gn
is a partition of U, then
D( )x xU
C .Proof. (1) If x xi, j Gr , then Md( )xi Md( )xj Td( )xi Td( )xj Gr . So we have
( ) ( )
d d
S xi S xj Gr. That mains, after translate, xi and xj are belongs to a same equivalence class.
(2) If xi Gr, xj Gt, from (1) we know that Sd( )xi Gr Sd( )xj Gt . Thus, after translation, xi and
j
x are not belongs to a same equivalence class.
Combine (1) and(2) , we have, if C is a partition, converted partition D satisfies D C . This completes the proof.
Theorem 5. Suppose there is a covering of U denote as C 'while the reduction ofC ' is C , we have
DC( )x xU
DC'( )x xU
.Proof. If the reduction of C'is C , from definition 4, the minimum description of C and C ' are the same. Thus,
the partition translated from C and C ' are the same. This completes the proof.
Theorem 6. Suppose C1 and C2 are two coverings of U , if there are C1 C2 ,then we have
1 2
C C
D D .
Proof. Its easily obtained by Definitions 8 and 17.
Theorem 7. Suppose there is a covering of U,denote as C . D is a partition translated by C , for X U , we have C(X )D(X ).
Proof. For y D( )x , according to definition15 and definition 16 we have y( Md( ))x Td( )x . Then
( )
d
M
y x , thus D( )x Md( )x .
(1) For X U and xC X( ), if G Mi d( ),x Gi X. Then Md( )x X , thus
( ) d( )
D x M x X . That means xD X( ), so, D X( )C X( );
(2) for X U , if xC X( ),xD X( ). Because D( )x Md( )x , then ( Md( ))x X , thus
( ),
i d i
G M x G X . That meansxC X( ), therefore we have C X( )D X( ).
In practical applications, the covering of domain is not completely a single covering, but also multiple covering cases. The multi-covering approximation spaces is more complicated than the single-covering approximation spaces. The key problem of the transformation is how to define the minimum description of the universe element. According to the definition of single covering minimum description, we propose the minimum description of multi-covering approximate spaces to extend the method to multi-covering approximate spaces.
Definition 18. [17](covering family) Let CL
C C1, 2, ,Cn
, if C Ci is a covering of U ,we say CL is a covering family of U .
Definition 19. (minimum description of covering family) Let CL
C C1, 2, ,Cn
is a coveringfamily of U , x U , define its minimum description as:
( ) , ( ) ( ),
d d d
ML x Gi l x G Gi l i l M Ci x x Gj t M Cj x Gj t Gi l Gi l Gj t (12) In the covering family, the transposed class, symmetry class, equivalence relation and equivalent class of domain elements are recorded as TLd( )x ,SLd( )x ,DLS andDL( )x .
Theorem 8. If C CL ,then Covering approximate space C and multi-covering approximate space CL are related as follows:
(1) MLd( )x Md( )x (2) TLd( )x Td( )x (3) SLd( )x Sd( )x (4)DL( )x D( )x Proof. From definition 14,15,17,19, it is obviously.
Conclusions
In this paper, we analyze the shortcomings of the existing transformation methods, and proposed a new method to translate covering into partition. Firstly, the basic requirements for translation methods are proposed. Secondly, through the definition of transposed and symmetrical classes, we have successfully transformed the similarity relationship into the equivalence relationship, this method satisfies the four conditions which a translate method should meets, and it’s effectiveness is proved by theories. Finally, the minimum description of the covering family is proposed considering that there may be multiple covering of data in real life. The method of translation single covering into partition is extended to the approximate spaces of covering family. Translating covering into partition not only enriches the applicable range of classic rough set, but also retains the basic properties of classic rough set. It is one of the researches focuses of covering information systems. What’s more, translating covering into partition can improve the classification quality of the original data, it’s more useful for data processing and better for decision making. In the next work, we will consider applying this method to the uncertainty measure of covering rough sets and further consider the reduction of attributes.
Acknowledgement
This reach was financially supported by the National Natural Science Foundation of China (No.71361002).
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