Numerical characterizations
of covering rough sets
based on evidence theory
Chen Degang, Zhang Xiao
Department of Mathematics and Physics, North China Electric Power
University, Beijing, 102206, P. R. China
Outline:
1. Introduction
2. Basic notions related to covering rough sets
3. Belief function and plausibility function of
covering rough sets
4. Numerical characterizations of attribute reduction
of covering information system
5. Numerical characterizations of attribute reduction
of covering decision systems
6. Conclusions
1. Introduction
What is covering rough sets?
Covering rough sets are improvements of traditional rough
sets by considering cover of universe instead of partition.
Why do we need covering rough sets?
Partition and equivalence relation are too restrictive to many
applications. One response to this argument is to extend an
equivalence relation to more general relations such as
similarity relation [21], tolerance relation [4,20] or others
[22,25,26]. Another response is to consider a cover instead
of partition and obtain covering rough sets [1-3,5,9,16,28,2932].
1. Introduction
The existing study on covering rough sets
Zakowski employed coverings of universe to establish the generalized
rough sets [28]. Bonikowski et al. [1] studied the structures of covers.
Mordeson [9] examined the relationship between the approximations of
sets defined with respect to covers and some axioms satisfied by
traditional rough sets. Chen et al. [5] discussed the covering rough set
under the framework of a complete completely distributive lattice. Zhu and
Wang [29-32] compared three kinds of generalized rough sets to deal with
vagueness and granularity in information system.
Chen et al.[6] began to develop definition and methods of attribute
reduction with covering rough sets. In [6] the intersection of coverings was
defined and the discernibility matrix was employed to compute all reducts.
Their study established a theoretical foundation for attribute reduction of
covering decision systems.
1. Introduction
Two facts motivate our idea
Among these work on covering rough sets, less effort has been
concentrated on developing measures for covering rough sets up to
now. As well known, in traditional rough set theory different kinds of
measures are proposed to reveal numerical characterizations of
rough sets and applied to develop algorithms of finding reducts.
This fact motivates our idea in this paper to develop measures to
characterize covering rough sets numerically.
As pointed in [23,24], there is closed connection between rough set
theory and evidence theory. This connection further motivates us to
set up connection between covering rough sets and evidence theory,
i.e., to characterize approximations and attribute reductions in
covering rough sets by employing measures in evidence theory.
2. Basic notions related to covering rough sets
We recall the basic concepts related to covering rough sets [6].
Definition 2.1. Let U be a universe, and C a family of subsets of U .
C is called a covering of U if none elements in C is empty and C  U .
Definition 2.2. Let C  {C1 , C2 , , Cn }be a covering of U. For every
x U , let Cx  {C j : C j  C, x  C j } , then Cov(C)  {Cx : x U }
U
U
is also a covering of . We call it the induced covering of C .
Definition 2.3. Let   {Ci : i  1, , m} be a family of coverings ofU
. For every x U , let  x  {Cix : Cix  Cov(Ci )}, thenCov( )  { x : x U }
is also a covering of U. We. call it the induced covering of  .
For every X  U , the lower and the upper approximations of X
with respect to  are defined as follows
 ( X )  { x :  x  X }  ( X )  { x :  x X  }
The positive region Pos ( X )   ( X ) .
3. Belief function and plausibility function of covering
rough sets
In this section, we first discuss the property of lower approximation
and propose a new upper approximation for covering rough sets.
Lemma 3.1. Let(U ,  ) be a covering information system and
Δ  {Ci : i  1,..., m} be a family of coverings of U . For x U and X  U
, we have
 ( X )  { x :  x  X }  {x :  x  X }
In this paper, we define a new upper approximation of X with respect
*
to the induced cover of  as  * ( X )  {x :  x X  }. Here  is
developed in terms of an induced cover, it certainly can be defined for
arbitrary covering. Furthermore, we have the following conclusions.
3. Belief function and plausibility function of covering
rough sets
Theorem 3.2. Suppose Δ  {Ci : i  1,..., m} is a family of coverings of
U , the covering lower approximation  and upper approximation  *
have the following properties:
(Contraction)
( X )  X
(2)  (~ X ) ~ * ( X ) (Duality)
(3L)  ( )  
(Normality)
(1L)
(4L)  (U )  U
(1U)
X  * ( X )
(Extension)
* (~ X ) ~  ( X ) (Duality)
(3U)  * ( )  
(Co-normality) (4U) * (U )  U
(Normality)
(Co-normality)
(5L)( X Y )  ( X ) (Y ) (Multiplication)(5U) * ( X Y )  * ( X ) * (Y )(Addition)
(6L)X  Y   ( X )  (Y )(Monotone) (6U) X  Y  * ( X )  * (Y ) (Monotone)
(7L)  ( ( X ))   ( X ) (Idempotency)(7U) * (* ( X ))  * ( X ) (Idempotency)
3. Belief function and plausibility function of covering
rough sets
Theorem 3.3. Let (U ,  ) be a covering information system,U  {x1 , x2 , , x,n }
for any X  U , denote
*
Bel ( X )   ( X ) n , Pl ( X )   ( X ) n .
Then Bel and Pl are belief and plausibility functions on U respectively,
and the corresponding mass distribution is
 f 1 ( i )

, X   i , here
mΔ ( X )  
n
f : U  Cov( Δ) defined as f ( xi )  i ,

0, otherwise
and f 1 (i )  {xk : f ( xk )  i } .
4. Numerical characterizations of attribute reduction of
covering information system
The reduct of covering information systems is the minimal subset of 
that preserves the induced coveringCov( ) .
Theorem 4.1. Let S  (U ,  ) be a covering information system and  be
a family of coverings, U  {x , x ,
1
2
, xn } , Cov( )  {1 ,  2 ,
,  n }, P   ,
then P is a reduct of S iff
n
n
1
1
BelP (i )  1, and for any nonempty subset P   P , BelP (i )  1.

i 1  i
i 1  i
4. Numerical characterizations of attribute reduction of
covering information system
Theorem 4.2. Let S  (U ,  ) be a covering information system and  be
a family of coverings, U  {x1 , x2 , , xn },Cov( )  {1 ,  2 ,
n
1
Pl Δ (i )  M , then P is a reduct of S iff

i 1  i
n

i 1
,  n }, P   ,
n
1
1
,
and
for
any
nonempty
subset
,
PlP (i )  M
P   P  PlP' (i )  M.
i
i 1  i
4. Numerical characterizations of attribute reduction of
covering information system
From Theorem 4.1 and 4.2 we conclude that the purpose of attribute
reduction in covering information
systems is to find a minimal subset
n
n
1
Bel
(

)


1
or 
PlP (i )  M . In
i
P   which preserves  P i
i 1
i 1  i

Theorem 4.3 M  1may not hold since Δ (i )  i may not always hold.
Generally we always have Δ (i )  i even i is a basic granule, and
this is one difference between covering rough sets and traditional
rough sets since every basic granule equals to its lower and upper
approximations in traditional rough sets.
Now we define the significance of a covering in  in a covering
information system.
4. Numerical characterizations of attribute reduction of
covering information system
Definition 4.3. Let S  (U ,  )be a covering information system.
U  {x1 , x2 , , xn } , we define the significance of the covering C  Δ by
n
n
i 1
i 1
Sig Δ (C)   Bel Δ (i ) i   Bel Δ{C} (i ) i .
Theorem4.4. Let S  (U ,  ) be a covering information system. For
every C  Δ , C is indispensable in D in  iff Sig Δ (C)  0 .
4. Numerical characterizations of attribute reduction of
covering information system
Theorem 4.5. Core( Δ)  {C  Δ : Sig Δ (C)  0}.
Definition 4.6. LetS  (U ,  ) be a covering information system.
U  {x1 , x2 , , xn } , for every covering C  Core( Δ) , we define the
significance of the covering C relative to Core( )by
n
SigCore ( Δ) (C)   BelCore ( Δ)
i 1
n
{C}
(i ) i   BelCore ( Δ) (i ) i
i 1
.
Algorithm 1. Acquire the core and the reduct for a covering
information system.
(1) let Core( )   ;
n
n
i 1
i 1
(2) for each C  Δ, calculate Sig Δ (C)   BelΔ (i ) i   Bel Δ{C} (i ) i
(3) if for every C  Δ , Sig Δ (C)  0, then Core( )   , go to step (6);
(4) If Sig Δ (C)  0 , then let Core( Δ)  Core( Δ) {C};
n
(5) if
 Bel
i 1
Core (  )
(i ) i  1 then return Core( ) , else go to step (6);
(6) let P  Core( ) ;
(7) for each C  { Δ  P}, calculate Sig P (C) ;
(8) if Sig P (C)  max C { Δ-P } Sig P (Ci ) , then P  P {C};
i
n
(9) if
 Bel
i 1
P
(i ) i  1 then stop and output as a reduct P , else
go back to step (7).
2
2
Let   m and U  n , the time complexity of Algorithm 1 is O(m  n ).
By Algorithm 1, we can acquire not only the core but also a proper
reduct.
5. Numerical characterizations of attribute reduction of
covering decision systems
Similar to attribute reduction of decision systems in traditional rough sets,
attribute reduction of covering decision systems aims to find the minimal set
of conditional attributes to preserve the positive region of decision attribute
[6].
Lemma 5.1. LetS  (U ,  , D  {d }) be a covering decision system,
U / D  {D1 , D2 , , Dr }, then we have
r
 Bel ( D )  Pos ( D)
j 1
j
U.
Theorem 5.2. Let S  (U ,  , D  {d }) be a covering decision system,
U  {x1 , x2 , , xn },U / D  {D1 , D2 , , Dr } , P   , then P is a relative reduct
r
of S iff
r
 Bel
j 1
 Bel
j 1
P
r
P
( D j )   Bel ( D j ), and for any nonempty subset P   P,
r
j 1
( D j )   BelP ( D j )
j 1
.
5. Numerical characterizations of attribute reduction of
covering decision systems
The covering decision systems can be divided into consistent covering
decision systems and inconsistent covering decision systems[6]. Generally
speaking, for covering decision systems, we only consider to find a minimal
subset of to preserve the sum of belief functions of all decision classes.
Definition 5.3. Let S  (U ,  , D  {d }) be a covering decision system.
U / D  {D1 , D2 , , Dr }, for everyC  Δ , we define the significance of the
r
r
covering C relative to D in  by Sig Δ D (C)   Bel Δ ( D j )   Bel Δ{C} ( D j )
j 1
i 1
Theorem 5.4. Let S  (U ,  , D  {d }) be a covering decision system.
For every C  Δ , C is indispensable relative to D in  iff Sig Δ D (C)  0 .
5. Numerical characterizations of attribute reduction of
covering decision systems
By Theorem 5.4 and the definition of CoreD ( ), we have the following
result.
Theorem 5.5. CoreD ( Δ)  {C  Δ : Sig Δ D (C)  0} .
Definition 5.6. Let S  (U ,  , D  {d }) be a covering decision system,
U / D  {D1 , D2 ,
, Dr }. For everyCi  CoreD ( ) but C  Δ , we define the
relative significance of the covering C to CoreD ( )by
r
SigCoreD ( Δ) (C)   BelCoreD ( Δ)
j 1
r
{C}
( D j )   BelCoreD ( Δ) ( D j )
i 1
.
Algorithm 2. Acquire the core and the reduction in a covering
decision system.
(1) let Core( )   ;
r
r
j 1
i 1
(2) for each C  Δ , calculate Sig Δ D (C)   Bel Δ ( D j )   Bel Δ{C} ( D j ) ;
(3) if for every C  Δ, Sig Δ D (C)  0 , then CoreD ( )   , go to step (6);
(4) If Sig Δ
r
D
(C)  0, then CoreD ( Δ)  CoreD ( Δ) {C};
r
(5) if  BelCore (  ) ( D j )   Bel ( D j ) then return CoreD ( ) , else go to
j 1
step (6);
j 1
(6) let P  CoreD ( ) ;
r
(7) for each C  { Δ  P}, calculate Sig P (C)   BelP
j 1
r
{C}
(8) if Sig P (C)  max Ci{ Δ-P } Sig P (Ci ) , then P  P {C};
(9) if
r
 Bel
j 1
( D j )   BelP ( D j ) ;
i 1
r
P
( D j )  Bel ( D j ) then stop and output as a reduct P ,
j 1
else go back to step (7).
Let   m and U  n , the time complexity of Algorithm 2 is O(m2  n2 ).
By Algorithm 2, we can acquire not only the core but also a proper
reduct.
6. Conclusions
The covering rough set theory is a generalization of traditional rough
set theory characterized by covers instead of partitions. Since there is
closed connection between rough set theory and evidence theory, we
try to propose belief and plausibility functions to characterize covering
rough sets. The relationships between attribute reduction and belief
(plausibility) function are analyzed in covering information and
decision systems respectively. We give the concepts of significance
and relative significance of coverings to find reducts in covering
information and decision systems. In addition, the relevant algorithms
are designed. In a word, we develop a numerical method to find
reducts by employing the belief function in covering information
(decision) systems. Our future work will concentrate on developing
other measures for covering rough sets based on belief and
plausibility functions in this paper.
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