Rough sets and decision tables - Semantic Scholar
ROUGH
SETS AND
DECISION
TABLES
Z. P a w l a k I n s t i t u t e of C o m p u t e r Science P o l i s h A c a d e m y of S c i e n c e s ~.0. B o x 22 00-901 W a r s a w , PKiN, P o l a n d
I. I n t r o d u c t i o n
~e
show
in t h i s
Pawlak
(1982 a n d
~heory
(see P o l l a c k ,
in t h i s p a p e r control
In t h i s
that
the concept
can be used
Hicks
have been
applied
of the r o u g h
as a b a s i s
and Harrison
(see M r 6 z e k
to o t h e r
2~ D e c i s i o n
which
1985))
algorithm
advantages
article
to the
(1984))
(see
for the d e c i s i o n
(1971)).
The
ideas
implementation and
set
showed
tables
introduced
of c e m e n t
considerable
kiln
practical
methods°
tables
section
we give
a formal
will be used throughout
A aecision
table S =
definition
of a d e c i s i o n
table
this paper°
is a s y s t e m
(Univ,
Att,
Val,
f)
where: Univ Att
- is a f i n i t e = Con
U Dec
set of s t a t e s ,
- is the
Conditions
called
the u n i v e r s e
set of a t t r i b u t e s ;
attributes
and
Dec
- is the
Con
- is t h e
set of
set of d e c i s i o n s
attrib-
utes. Val
f
I ~ Val , where a6Att%'1 a ( d o m a i n of a).
: Univ
× Att ~ Val
function, A aecision f(x,a)
rule
for every
such
Val
is t h e set of v a l u e s
- is a t o t a l
function,
that
6 V a l a for
f(x,a)
in S is a f u n c t i o n x{Univ
of an a t t r i b u t e
aEAtt
a
and aEAtt.
f
x
called
the d e c i s i o n
every x6Univ
: A t t ~ Val,
such
that
and f
a6Att. x
(a)
=
188
If
fx
is a u e c i s i o n
called c onditi0ns A ~ecision every
rule
y6Univ,
fx
y ~ x
wise the d e c i s i o n
rule
in
and d e c i s i o n s in
S
fx/COn
rule
S
then
is d e t e r m i n i s t i c = fy/COn
f
fx/COn
of the d e c i s i o n
and
rule
fx/Dec fx
(consistent)
implies
fx/Dec
is n o n d e t e r m i n i s t i c
are
respectively. if for
= fy/Dec;
other-
(inconsistent)°
X
A decision sion r u l e s
table
is d e t e r m i n i s t i c
are d e t e r m i n i s t i c ;
~eterministic
otherwise
A decision
table
Val~={v
S'=
table
6 Val
An e x a m p l e
the d e c i s i o n
(X,Att,Vai',f')
if all table
its d e c i S
is n o n -
if
X ~ Univ,
f'= f / X × A t t
: V fx (a) = v ~ x£X
of a d e c i s i o n
table
is s h o w n b e l o w :
a
b
c
d
e
I
I
0
2
2
0
2
0
1
1
I
2
3
2
0
0
I
I
4
I
I
0
2
2
5
I
0
2
0
I
6
2
2
0
I
I
7
2
I
I
I
2
8
0
I
1
0
1
Fig. In the d e c i s i o n
is said to be a X - r e s t r i c t i o n
S : (Univ,Att,Val,f),
Univ
ano d o m a i n s
(consistent)
(inconsistent).
of the e e c i s i o n and
S
table Univ
1
= {1,2,~..,81,
of all a t t r i b u t e s
are e q u a l
Con Val
= {a,b,c],
Dec
= ~d,e~
= { 0 ~ 1 , 2 I.
3. R o u g h s e t £
Let
S = (Univ,Att,Val,f)
be a d e c i s i o n
t a b l e and let
a ~ Att,
x,y 6 Univ. With every relation
A
s u b s e t of a t t r i b u t e s defined
(x,y) If
(x,y)
p e c t to
A
6 A in
lity relation tion
A
~ A
A ~ Att
iff
fx(a)
we say that
x
= fy(a)
for e v e r y
and
are
S (A - i n d i s c e r n i b l e ) in
S.
Equivalence
are c a l l e d A - e l e m e n t a r y classes
of A is d e n o t e d by
y and
classes sets in
FV
alence
we a s s o c i a t e
the e q u i v a l e n c e
thus
.
A
o
A
a 6 A.
indiscernible
with res-
is c a l l e d an i n d i s c e r n i b i -
of the i n d i s c e r n i b i l i t y S
rela-
and the f a m i l y of an e q u i v -
t89
Let
A ~ Att
and
X ~ Univ
in a d e c i s i o n
table
S=(Univ,Att,
in
S
we m e a n
of
X
in
Val, f) . By
A-lower
(A-upper)
olx ~X Let
We s h a l l A -
6 Univ
use
also
the
region
A - doubtful
region
A - negative ' region
X
= A_X
is
we
X
the
sets
N X ~ ~.
be c a l l e d
A
following
definitions:
of
set
X
of
set
X
of
set X
say t h a t
in
sets w i l l
-
~
is the
set
is the
X
S.
AX;
set B n A ( X ) ;
is the
set
A - nondefinable
Nondefinable
of
X9
: Ix] A
- A X will
ositive
If AX set
niv =
= {X
BnA(X ) = AX
a_ploroximation
set N e g A X
is A - d e f i n a b l e
= Univ in
- AX.
S; o t h e r w i s e
S.
be c a l l e d
also
rough
sets
in
S.
The n u m b e r c a r d AX ~A(X)
= card
Will and
be c a l l e d
the a c c u r a c i of the X
with
respect
to
A
in
S,
the n u m b e r ~i(X)
will
be c a l l e d
sion
table
space
S =
TS =
definable terior
that
each
in
S,
closure
following
1)
Ax ~_ x =_~x
set
subset
of a t t r i b u t e s
defines
where the
X
with
topological
the
A
thus
S.
in a d e c i -
of all
approximations
TS,
in
topological
is the f a m i l y
and upper space
to
A c Att
uniquelly
DefA(S)
lower
respect
are
A in-
approximations
properties:
2)
~
3)
A(x u Y) _~A_X U SA
5)
A(X
N Y)
6)
~(x
n Y) c K x
7)
A(-X)
: - ~(X)
6)
~(-x)
= - i(x}
Moreover
= A~
and
in the
the
perties :
of the
(Univ,Att,Val,f)
(Univ,DefA(S)) ,
sets
and
= I - SA(X)
the r o u g h n e s s
L e t us n o t i c e
have
AX
= ~; A U n i v
= A Univ
= Univ
: A_X N A_Y
in this
n ~Y
topological
space
we h a v e
the
following
two pro-
190
9) 10)
AAX
= ~AX
~X
= ~X
F r o m the t o p o l o g i c a l
v i e w the r o u g h
sets can be c l a s s i f i e d
as
follows: a) Set X is r o u g h l y A - d e f i n a b l e
in
S
if
A_X # ~
and
AX ~ Univ,
b)
Set X is i n t e r n a l l y A - n o n d e f i n a b l e
in
S
if
c)
Set X is e x t e r n a l l Z A - n o n d e f i n a b l e
in
S
if A X = U n i v and ~ X #
d) Set X is t p t a l ! [ A - n o n d e f i n a b l e
4. D e p e n d e n c y
S
if
be a d e c i s i o n
X i -c U n i v , a f a m i l y of s u b s e t s
A X = ~ and AX = U n i v .
A-lower
table,
r ~ F=~],X2,-'-,Xn I ' where
of U n i v and A -c Att.
(A-upper)approximation
of
F
the
set
in
S
we m e a n
F
by
the
families AF = { ~ X ] , ~ X 2 ,o.., ~ X n ~ AF = { A X I I A X 2 , . . . , Whe A - p o s i t i v e
r99ion
AXn~.
of a family
F
is
POs~(F) The A - ~ o u b t f u ~
= U _AXi " X .l£ F r e g i o n of a f a m i l y F is the
BnA(F)
= U
set
BnAXi "
X.6F l The A - n_e~at±ve r e g i o n of a f a m i l y
F
NegA(F)
AX..
= Univ - U X.6F l
The
,
of a t t r i b u t e s
Let Z=(bniv,Att,Val,f)
By
in
A_X : @ and A X ~ U n i v ,
is the set
l
nummer c a r d P O S A (F) YA(F)
= card U
will be calle~
the ~ u a l i t y
at the a p p r o x i m a t i o n
a n d the n u m b e r card P O S A (F) ~A(F)
= Z card AX. 1 X. 6F l
of
A in
S,
t91
w i l l be c a l l e d the a c c u r a c y L e t B,C ~ A t t Val,f) t
and
k
subsets
real n u m b e r
We say t h a t if
be two
at a ~ p r o x i m a t i o n
C
F
of a t t r i b u t e s
such t h a t
depends
of
0 4 k {
by
in
A
in
S o
S = (Univ,Att,
I.
in a d e g r e e k on B in S, in s y m b o l s B ~ C~
k = TB ( C ) . If
write
k = I we say t h a t
also
B ~ C
C
totally depends
i n s t e a d of
If
0 < k < I
If
k = 0
we say t h a t
we say t h a t
• he f o l l o w i n g
on
B
in
S
and we
B ~ C. C
C
properties
roughly dePends
on
B
is t o t a ! l [ ! g d e p e n d e n t on
in B
S* in
S.
are valid:
Property~ A Qecision Con ~ Dec
in
table
A aecision ministic
S = (Univ,Att,Val,f)
table
S = (Univ,Att,Val,f)
if con h Dec and
A ~ecision ministic
if
is d e t e r m i n i s t i c
iff
S.
table
is c a l l e d
roughly
deter-
0 < k < I.
S = (Univ,Att,Val,f)
is c a l l e d t o t a l l y
nondeter-
C o n ~ Dec.
P__rroperty The f o l l o w i n g
properties
I) C o n ] Dec
in
S/POScon(Dec* )
2) Con ~ Dec
in
S/Bncon(Dec* ) .
Note.
The a b o v e p r o p e r t y
posed
into two p a r t s
and the s e c o n d
says that e v e r y
(possibly
POScon(Dec*) YCon(Dec table
empty)
decision
t a b l e can be d e c o m -
such t h a t one is d e t e r m i n i s t i c
totally nondeterministic.
It is e a s y to c o m p u t e
*
are true:
) = 0,5,
that
= ~3,4,6,7,~, i.e.
Bncon(Dec*~
0,~
Con
is r o u g h l y d e t e r m i n i s t i c
ing two d e c i s i o n
in the d e c i s i o n
Dec,
c
shown
in Fig.
Ir
= 41,2,5,8],
that is to m e a n that the d e c i s i o n
a n d c a n be d e c o m p o s e d
tables Univ
table
a
b
d
e
3
2
0
0
1
1
4
t
t
0
2
2
6
2
2
0
1
1
7
2
1
1
1
2
Fig°
2
into the f o l l o w -
t92
Univ
The d e c i s i o n ~he
table
shown
table
a
b
c
d
e
I
I
0
2
I
0
2
0
I
I
I
2
5
I
0
2
0
I
8
0
I
I
0
I
Fig.
3
shown
in Fig°
in Fig.
3 is t o t a l l y
2 is t o t a l l y
deterministic
and
nondeterministico W
L e t us a l s o
notice
all d e t e r m i n i s t i c The m e a n i n g
decisions
~Con(Dec*)
S =
Set
A
is i n d e p e n d e n t
Set
A
is d e p e n d e n t
Set
B c A
(Univ,Att,Val,f)
set in
Subset B
) = I/3
which
is the
ratio
decisions
in the
table.
of
is o b v i o u s .
of a t t r i b u t e s
Let
pendent
~Con(Dec
to all p o s s i b l e
of the n u m b e r
5. R e d u c t i o n
if
that
be a d e c i s i o n
in
S
table
if for e v e r y if t h e r e
s
exists
is a r e d u c t
of
A
is a r e d u c t
of
A
with
respect
of
A
such
that
S
if
let
B c A, ~
in
in
and
B c A
B
A ~ Atto
m ~. such
that ~--~.
is the m a x i m a l
inde-
S~
B ~ A
is an i n d e p e n d e n t
subset
to
YB(C
C ~ Att
*)
=
in
S (or
YA(C* )
POSB(C* ) = POSA(C*)). L e t us n o t i c e coincide
with
Property
3
If respect
A h B to
B
in in
In p a r t i c u l a r decision can
table
simplify
that
if
the r e d u c t
S S,
A = C
of
and
C
then
the r e d u c t
is a r e d u c t
A
with
respect
to C
of
A,
or r e d u c t
of c o n d i t i o n s
attributes
of
A with
Con
in a
C h B,
if C is a r e d u c t
S a n d C o n h Dec,
the d e c i s i o n
of
A~
table
then
C h Dec,
by r e d u c i n g
This
the
is to
mean
that
set of c o n d i t i o n s
we at-
tributes. For
example
of c o n d i t i o n s decision
table
in the d e c i s i o n
attributes as shown
is
table
C = 4a,b],
below:
shown thus
in Fig.
I the
only
we can
simplify
the
reduct
193
Univ
a
b
d
1
I
0
2
0
2
0
I
I
2
3
2
0
I
I
4
!
I
2
2
5
I
0
0
I
6
2
2
I
0
7
2
I
I
2
8
0
1
0
1
can
be
d
e
Fig. and
consequently
the
decision
4
table
Univ
a
b
3
2
0
I
I
4
!
I
2
2
6
2
2
I
I
7
2
I
I
2
Fig. Univ
table
with
easily
a
b
d
e
I
I
0
2
0
2
0
I
I
2
5
!
0
0
I
8
0
I
0
I
set
of
seen
that
as
follows
6
the
decision
attributes
in t h e
(or a p p r o x i m a t e
reduct
is i n d e p e n d e n t . We can
also
define
respect
to
a subset
Let sion
be
decomposed
5
Fig. It c a n
e
0 ~
table
S =
Subset in S a n d
e ~
Directly Property If B
A
in t h e
a real
reduct
following
number
way:
and
let
A
in
B ~
Att
in a d e c i -
.
is a ~ - r e d u c t
of
S
if B
is a ~ - r e d u c t
in S a n d
from
these
YB(C*)
of A
in S w i t h
= YA(C*)
definitions
we
respect
£o
have
4 is a
A ~
is i n d e p e n d e n t
= 1-~.
B of A
B is i n d e p e n d e n t
be
C)
approximate
(Univ,Att,Val,f)
B of
YB(A*)
Subset
I
the
&-reduct
of A
in S t h e n
B
~
A.
to C ~ Att
if
194
Property
5
If B is a ~ - r e d u c t then B k~
In p a r t i c u l a r in S, t h e n C k - ~ conditions between
respect
to C c Att,
a n d A ~ C,
Dec.
That
in S a n d C ~
is to m e a n
Con
is a ~ - r e d u c t
that we can reduce
in s u c h a w a y t h a t t h e d e g r e e
and conditions
attributes
the
of Con
set o f
of d e p e n d e n c y
is d e c r e a s e d
b y the
~ .
of d e c i s i o n
With
every
conditions CB,
if C o n ~ D e c
attributes,
decisions
constant
6. C o s t
of A in S w i t h
C.
when
table
decision
attributes S
table
B ~ Con,
is u n d e r s t o o d CB =
X a£B
S =
c 1(a)
(Univ,Att~Val,f)
we associate
defined +
cost
and
CS,B,
subset or
of
in s h o r t
thus:
X , c 2(x) x6BnB(Dec )
,
where c]
: C o n ~ R +,
c2
a n d R + is t h e
set of n o n n e g a t i v e
ing t h e v a l u e
of a t t r i b u t e
sification
of
x,
due to
: Univ ~ R +
reals;
a a n d c2(x)
smaller
c1(a)
- is the c o s t of m e a s u r -
- is t h e c o s t o f
set of a t t r i b u t e s
imprecise
(imprecise
clas-
decision
in s t a t e x). There the t o t a l
is of c o u r s e
trade-off
cost CB by proper
The p r o b l e m
will
between
choosing
be d i s c u s s e d
c I a n d c 2 a n d we c a n m i n i m i z e
of c o n d i t i o n s
attributes.
in s o m e d e t a i l s
in a s u b s e q u e n t
paper.
7. D e c i s i o n
nets
iJiany p r o b l e m s ent d e c i s i o n ditions
are
tables
In o r d e r
each
to do so a s s u m e
tables.
table,
but
in s u c h a w a y
making
process
a set of d i f f e r -
that
if some c o n -
c a n be s w i t c h e d
from
one. that each decision
table,
table contains
of d e c i s i o n
in a n e t
the d e c i s i o n
to a n o t h e r
of the decision
decision
are names
table
not one decision
connected
satisfied,
one d e c i s i o n
a "name"
require
one
and the
specific
Thus
table
set of d e c i s i o n attribute
each condition
is l a b e l l e d
by
attributes
in
- values
of w h i c h
in the d e c i s i o n
table
195
specifies
also next decision
With graph, are
each
nodes
labellea
ponding
are
to b e u s e d ,
decision labelled
by conditions,
which
tables
to m a k e
a decision~
we can associate
by decision determine
tables,
transfers
a directed
and between
corres-
tables.
For Fig.
set of s u c h
of w h i c h
table
example
consider
three
decision
tables A,B,C
as shown
in
7. A
u
a
B
b
c
N
u
C
a
b
c
N
u
a
b
c
N
I
0
1
2
A
1
I
2
2
B
I
0
2
2
C
2
2
1
1
B
2
0
0
0
A
2
I
0
I
C
3
0
0
0
~
3
2
I
I
A
3
2
I
0
A
4
I
1
2
C
4
2
0
2
B
5
1
2
I
C
Fig.
7
N - is t h e
"next table"
The graph
attribute.
associated
with
this
set at t a b l e s
is s h o ~
in Fig~
8.
C.
Fig. For
the
by conditions Several nets,
Sake of but
corresponding
theoretical
for e x a m p l e
in Fig.
simplicity
we
the p r o b l e m
labelled
arise
in c o n n e c t i o n
of c o n s i s t e n c y
but
branches
of the g r a p h
not
states.
problems
8 is i n c o n s i s t e n t ! )
8
we
with
(the d e c i s i o n
shall n o t d i s c u s s
these
decision not
shown
problems
t96
in these paper.
Acknow!edgenlento
Thanks are due to dr~ Ao Skowron for critical
remarks.
References Mr@zeK, A., (]984). Information Polish Acad. Sci. (to appear)
Systems and Control Algorithms,
Pawlak, Zo, (]982). Rough Sets. International and Computer Sciences, I](5), 34]-356
BUllo
Journal of Information
Pawlak, Zo, (]985). Decision tables and decision algorithms. POlish Acad. Sci. (to appear) Pollack, S., Hicks, H., and Harrison, W., (1971)o Decision Theory and Practices Wiley and Sons, Inc. New York.
Bull.
Tables:
SETS AND
DECISION
TABLES
Z. P a w l a k I n s t i t u t e of C o m p u t e r Science P o l i s h A c a d e m y of S c i e n c e s ~.0. B o x 22 00-901 W a r s a w , PKiN, P o l a n d
I. I n t r o d u c t i o n
~e
show
in t h i s
Pawlak
(1982 a n d
~heory
(see P o l l a c k ,
in t h i s p a p e r control
In t h i s
that
the concept
can be used
Hicks
have been
applied
of the r o u g h
as a b a s i s
and Harrison
(see M r 6 z e k
to o t h e r
2~ D e c i s i o n
which
1985))
algorithm
advantages
article
to the
(1984))
(see
for the d e c i s i o n
(1971)).
The
ideas
implementation and
set
showed
tables
introduced
of c e m e n t
considerable
kiln
practical
methods°
tables
section
we give
a formal
will be used throughout
A aecision
table S =
definition
of a d e c i s i o n
table
this paper°
is a s y s t e m
(Univ,
Att,
Val,
f)
where: Univ Att
- is a f i n i t e = Con
U Dec
set of s t a t e s ,
- is the
Conditions
called
the u n i v e r s e
set of a t t r i b u t e s ;
attributes
and
Dec
- is the
Con
- is t h e
set of
set of d e c i s i o n s
attrib-
utes. Val
f
I ~ Val , where a6Att%'1 a ( d o m a i n of a).
: Univ
× Att ~ Val
function, A aecision f(x,a)
rule
for every
such
Val
is t h e set of v a l u e s
- is a t o t a l
function,
that
6 V a l a for
f(x,a)
in S is a f u n c t i o n x{Univ
of an a t t r i b u t e
aEAtt
a
and aEAtt.
f
x
called
the d e c i s i o n
every x6Univ
: A t t ~ Val,
such
that
and f
a6Att. x
(a)
=
188
If
fx
is a u e c i s i o n
called c onditi0ns A ~ecision every
rule
y6Univ,
fx
y ~ x
wise the d e c i s i o n
rule
in
and d e c i s i o n s in
S
fx/COn
rule
S
then
is d e t e r m i n i s t i c = fy/COn
f
fx/COn
of the d e c i s i o n
and
rule
fx/Dec fx
(consistent)
implies
fx/Dec
is n o n d e t e r m i n i s t i c
are
respectively. if for
= fy/Dec;
other-
(inconsistent)°
X
A decision sion r u l e s
table
is d e t e r m i n i s t i c
are d e t e r m i n i s t i c ;
~eterministic
otherwise
A decision
table
Val~={v
S'=
table
6 Val
An e x a m p l e
the d e c i s i o n
(X,Att,Vai',f')
if all table
its d e c i S
is n o n -
if
X ~ Univ,
f'= f / X × A t t
: V fx (a) = v ~ x£X
of a d e c i s i o n
table
is s h o w n b e l o w :
a
b
c
d
e
I
I
0
2
2
0
2
0
1
1
I
2
3
2
0
0
I
I
4
I
I
0
2
2
5
I
0
2
0
I
6
2
2
0
I
I
7
2
I
I
I
2
8
0
I
1
0
1
Fig. In the d e c i s i o n
is said to be a X - r e s t r i c t i o n
S : (Univ,Att,Val,f),
Univ
ano d o m a i n s
(consistent)
(inconsistent).
of the e e c i s i o n and
S
table Univ
1
= {1,2,~..,81,
of all a t t r i b u t e s
are e q u a l
Con Val
= {a,b,c],
Dec
= ~d,e~
= { 0 ~ 1 , 2 I.
3. R o u g h s e t £
Let
S = (Univ,Att,Val,f)
be a d e c i s i o n
t a b l e and let
a ~ Att,
x,y 6 Univ. With every relation
A
s u b s e t of a t t r i b u t e s defined
(x,y) If
(x,y)
p e c t to
A
6 A in
lity relation tion
A
~ A
A ~ Att
iff
fx(a)
we say that
x
= fy(a)
for e v e r y
and
are
S (A - i n d i s c e r n i b l e ) in
S.
Equivalence
are c a l l e d A - e l e m e n t a r y classes
of A is d e n o t e d by
y and
classes sets in
FV
alence
we a s s o c i a t e
the e q u i v a l e n c e
thus
.
A
o
A
a 6 A.
indiscernible
with res-
is c a l l e d an i n d i s c e r n i b i -
of the i n d i s c e r n i b i l i t y S
rela-
and the f a m i l y of an e q u i v -
t89
Let
A ~ Att
and
X ~ Univ
in a d e c i s i o n
table
S=(Univ,Att,
in
S
we m e a n
of
X
in
Val, f) . By
A-lower
(A-upper)
olx ~X Let
We s h a l l A -
6 Univ
use
also
the
region
A - doubtful
region
A - negative ' region
X
= A_X
is
we
X
the
sets
N X ~ ~.
be c a l l e d
A
following
definitions:
of
set
X
of
set
X
of
set X
say t h a t
in
sets w i l l
-
~
is the
set
is the
X
S.
AX;
set B n A ( X ) ;
is the
set
A - nondefinable
Nondefinable
of
X9
: Ix] A
- A X will
ositive
If AX set
niv =
= {X
BnA(X ) = AX
a_ploroximation
set N e g A X
is A - d e f i n a b l e
= Univ in
- AX.
S; o t h e r w i s e
S.
be c a l l e d
also
rough
sets
in
S.
The n u m b e r c a r d AX ~A(X)
= card
Will and
be c a l l e d
the a c c u r a c i of the X
with
respect
to
A
in
S,
the n u m b e r ~i(X)
will
be c a l l e d
sion
table
space
S =
TS =
definable terior
that
each
in
S,
closure
following
1)
Ax ~_ x =_~x
set
subset
of a t t r i b u t e s
defines
where the
X
with
topological
the
A
thus
S.
in a d e c i -
of all
approximations
TS,
in
topological
is the f a m i l y
and upper space
to
A c Att
uniquelly
DefA(S)
lower
respect
are
A in-
approximations
properties:
2)
~
3)
A(x u Y) _~A_X U SA
5)
A(X
N Y)
6)
~(x
n Y) c K x
7)
A(-X)
: - ~(X)
6)
~(-x)
= - i(x}
Moreover
= A~
and
in the
the
perties :
of the
(Univ,Att,Val,f)
(Univ,DefA(S)) ,
sets
and
= I - SA(X)
the r o u g h n e s s
L e t us n o t i c e
have
AX
= ~; A U n i v
= A Univ
= Univ
: A_X N A_Y
in this
n ~Y
topological
space
we h a v e
the
following
two pro-
190
9) 10)
AAX
= ~AX
~X
= ~X
F r o m the t o p o l o g i c a l
v i e w the r o u g h
sets can be c l a s s i f i e d
as
follows: a) Set X is r o u g h l y A - d e f i n a b l e
in
S
if
A_X # ~
and
AX ~ Univ,
b)
Set X is i n t e r n a l l y A - n o n d e f i n a b l e
in
S
if
c)
Set X is e x t e r n a l l Z A - n o n d e f i n a b l e
in
S
if A X = U n i v and ~ X #
d) Set X is t p t a l ! [ A - n o n d e f i n a b l e
4. D e p e n d e n c y
S
if
be a d e c i s i o n
X i -c U n i v , a f a m i l y of s u b s e t s
A X = ~ and AX = U n i v .
A-lower
table,
r ~ F=~],X2,-'-,Xn I ' where
of U n i v and A -c Att.
(A-upper)approximation
of
F
the
set
in
S
we m e a n
F
by
the
families AF = { ~ X ] , ~ X 2 ,o.., ~ X n ~ AF = { A X I I A X 2 , . . . , Whe A - p o s i t i v e
r99ion
AXn~.
of a family
F
is
POs~(F) The A - ~ o u b t f u ~
= U _AXi " X .l£ F r e g i o n of a f a m i l y F is the
BnA(F)
= U
set
BnAXi "
X.6F l The A - n_e~at±ve r e g i o n of a f a m i l y
F
NegA(F)
AX..
= Univ - U X.6F l
The
,
of a t t r i b u t e s
Let Z=(bniv,Att,Val,f)
By
in
A_X : @ and A X ~ U n i v ,
is the set
l
nummer c a r d P O S A (F) YA(F)
= card U
will be calle~
the ~ u a l i t y
at the a p p r o x i m a t i o n
a n d the n u m b e r card P O S A (F) ~A(F)
= Z card AX. 1 X. 6F l
of
A in
S,
t91
w i l l be c a l l e d the a c c u r a c y L e t B,C ~ A t t Val,f) t
and
k
subsets
real n u m b e r
We say t h a t if
be two
at a ~ p r o x i m a t i o n
C
F
of a t t r i b u t e s
such t h a t
depends
of
0 4 k {
by
in
A
in
S o
S = (Univ,Att,
I.
in a d e g r e e k on B in S, in s y m b o l s B ~ C~
k = TB ( C ) . If
write
k = I we say t h a t
also
B ~ C
C
totally depends
i n s t e a d of
If
0 < k < I
If
k = 0
we say t h a t
we say t h a t
• he f o l l o w i n g
on
B
in
S
and we
B ~ C. C
C
properties
roughly dePends
on
B
is t o t a ! l [ ! g d e p e n d e n t on
in B
S* in
S.
are valid:
Property~ A Qecision Con ~ Dec
in
table
A aecision ministic
S = (Univ,Att,Val,f)
table
S = (Univ,Att,Val,f)
if con h Dec and
A ~ecision ministic
if
is d e t e r m i n i s t i c
iff
S.
table
is c a l l e d
roughly
deter-
0 < k < I.
S = (Univ,Att,Val,f)
is c a l l e d t o t a l l y
nondeter-
C o n ~ Dec.
P__rroperty The f o l l o w i n g
properties
I) C o n ] Dec
in
S/POScon(Dec* )
2) Con ~ Dec
in
S/Bncon(Dec* ) .
Note.
The a b o v e p r o p e r t y
posed
into two p a r t s
and the s e c o n d
says that e v e r y
(possibly
POScon(Dec*) YCon(Dec table
empty)
decision
t a b l e can be d e c o m -
such t h a t one is d e t e r m i n i s t i c
totally nondeterministic.
It is e a s y to c o m p u t e
*
are true:
) = 0,5,
that
= ~3,4,6,7,~, i.e.
Bncon(Dec*~
0,~
Con
is r o u g h l y d e t e r m i n i s t i c
ing two d e c i s i o n
in the d e c i s i o n
Dec,
c
shown
in Fig.
Ir
= 41,2,5,8],
that is to m e a n that the d e c i s i o n
a n d c a n be d e c o m p o s e d
tables Univ
table
a
b
d
e
3
2
0
0
1
1
4
t
t
0
2
2
6
2
2
0
1
1
7
2
1
1
1
2
Fig°
2
into the f o l l o w -
t92
Univ
The d e c i s i o n ~he
table
shown
table
a
b
c
d
e
I
I
0
2
I
0
2
0
I
I
I
2
5
I
0
2
0
I
8
0
I
I
0
I
Fig.
3
shown
in Fig°
in Fig.
3 is t o t a l l y
2 is t o t a l l y
deterministic
and
nondeterministico W
L e t us a l s o
notice
all d e t e r m i n i s t i c The m e a n i n g
decisions
~Con(Dec*)
S =
Set
A
is i n d e p e n d e n t
Set
A
is d e p e n d e n t
Set
B c A
(Univ,Att,Val,f)
set in
Subset B
) = I/3
which
is the
ratio
decisions
in the
table.
of
is o b v i o u s .
of a t t r i b u t e s
Let
pendent
~Con(Dec
to all p o s s i b l e
of the n u m b e r
5. R e d u c t i o n
if
that
be a d e c i s i o n
in
S
table
if for e v e r y if t h e r e
s
exists
is a r e d u c t
of
A
is a r e d u c t
of
A
with
respect
of
A
such
that
S
if
let
B c A, ~
in
in
and
B c A
B
A ~ Atto
m ~. such
that ~--~.
is the m a x i m a l
inde-
S~
B ~ A
is an i n d e p e n d e n t
subset
to
YB(C
C ~ Att
*)
=
in
S (or
YA(C* )
POSB(C* ) = POSA(C*)). L e t us n o t i c e coincide
with
Property
3
If respect
A h B to
B
in in
In p a r t i c u l a r decision can
table
simplify
that
if
the r e d u c t
S S,
A = C
of
and
C
then
the r e d u c t
is a r e d u c t
A
with
respect
to C
of
A,
or r e d u c t
of c o n d i t i o n s
attributes
of
A with
Con
in a
C h B,
if C is a r e d u c t
S a n d C o n h Dec,
the d e c i s i o n
of
A~
table
then
C h Dec,
by r e d u c i n g
This
the
is to
mean
that
set of c o n d i t i o n s
we at-
tributes. For
example
of c o n d i t i o n s decision
table
in the d e c i s i o n
attributes as shown
is
table
C = 4a,b],
below:
shown thus
in Fig.
I the
only
we can
simplify
the
reduct
193
Univ
a
b
d
1
I
0
2
0
2
0
I
I
2
3
2
0
I
I
4
!
I
2
2
5
I
0
0
I
6
2
2
I
0
7
2
I
I
2
8
0
1
0
1
can
be
d
e
Fig. and
consequently
the
decision
4
table
Univ
a
b
3
2
0
I
I
4
!
I
2
2
6
2
2
I
I
7
2
I
I
2
Fig. Univ
table
with
easily
a
b
d
e
I
I
0
2
0
2
0
I
I
2
5
!
0
0
I
8
0
I
0
I
set
of
seen
that
as
follows
6
the
decision
attributes
in t h e
(or a p p r o x i m a t e
reduct
is i n d e p e n d e n t . We can
also
define
respect
to
a subset
Let sion
be
decomposed
5
Fig. It c a n
e
0 ~
table
S =
Subset in S a n d
e ~
Directly Property If B
A
in t h e
a real
reduct
following
number
way:
and
let
A
in
B ~
Att
in a d e c i -
.
is a ~ - r e d u c t
of
S
if B
is a ~ - r e d u c t
in S a n d
from
these
YB(C*)
of A
in S w i t h
= YA(C*)
definitions
we
respect
£o
have
4 is a
A ~
is i n d e p e n d e n t
= 1-~.
B of A
B is i n d e p e n d e n t
be
C)
approximate
(Univ,Att,Val,f)
B of
YB(A*)
Subset
I
the
&-reduct
of A
in S t h e n
B
~
A.
to C ~ Att
if
194
Property
5
If B is a ~ - r e d u c t then B k~
In p a r t i c u l a r in S, t h e n C k - ~ conditions between
respect
to C c Att,
a n d A ~ C,
Dec.
That
in S a n d C ~
is to m e a n
Con
is a ~ - r e d u c t
that we can reduce
in s u c h a w a y t h a t t h e d e g r e e
and conditions
attributes
the
of Con
set o f
of d e p e n d e n c y
is d e c r e a s e d
b y the
~ .
of d e c i s i o n
With
every
conditions CB,
if C o n ~ D e c
attributes,
decisions
constant
6. C o s t
of A in S w i t h
C.
when
table
decision
attributes S
table
B ~ Con,
is u n d e r s t o o d CB =
X a£B
S =
c 1(a)
(Univ,Att~Val,f)
we associate
defined +
cost
and
CS,B,
subset or
of
in s h o r t
thus:
X , c 2(x) x6BnB(Dec )
,
where c]
: C o n ~ R +,
c2
a n d R + is t h e
set of n o n n e g a t i v e
ing t h e v a l u e
of a t t r i b u t e
sification
of
x,
due to
: Univ ~ R +
reals;
a a n d c2(x)
smaller
c1(a)
- is the c o s t of m e a s u r -
- is t h e c o s t o f
set of a t t r i b u t e s
imprecise
(imprecise
clas-
decision
in s t a t e x). There the t o t a l
is of c o u r s e
trade-off
cost CB by proper
The p r o b l e m
will
between
choosing
be d i s c u s s e d
c I a n d c 2 a n d we c a n m i n i m i z e
of c o n d i t i o n s
attributes.
in s o m e d e t a i l s
in a s u b s e q u e n t
paper.
7. D e c i s i o n
nets
iJiany p r o b l e m s ent d e c i s i o n ditions
are
tables
In o r d e r
each
to do so a s s u m e
tables.
table,
but
in s u c h a w a y
making
process
a set of d i f f e r -
that
if some c o n -
c a n be s w i t c h e d
from
one. that each decision
table,
table contains
of d e c i s i o n
in a n e t
the d e c i s i o n
to a n o t h e r
of the decision
decision
are names
table
not one decision
connected
satisfied,
one d e c i s i o n
a "name"
require
one
and the
specific
Thus
table
set of d e c i s i o n attribute
each condition
is l a b e l l e d
by
attributes
in
- values
of w h i c h
in the d e c i s i o n
table
195
specifies
also next decision
With graph, are
each
nodes
labellea
ponding
are
to b e u s e d ,
decision labelled
by conditions,
which
tables
to m a k e
a decision~
we can associate
by decision determine
tables,
transfers
a directed
and between
corres-
tables.
For Fig.
set of s u c h
of w h i c h
table
example
consider
three
decision
tables A,B,C
as shown
in
7. A
u
a
B
b
c
N
u
C
a
b
c
N
u
a
b
c
N
I
0
1
2
A
1
I
2
2
B
I
0
2
2
C
2
2
1
1
B
2
0
0
0
A
2
I
0
I
C
3
0
0
0
~
3
2
I
I
A
3
2
I
0
A
4
I
1
2
C
4
2
0
2
B
5
1
2
I
C
Fig.
7
N - is t h e
"next table"
The graph
attribute.
associated
with
this
set at t a b l e s
is s h o ~
in Fig~
8.
C.
Fig. For
the
by conditions Several nets,
Sake of but
corresponding
theoretical
for e x a m p l e
in Fig.
simplicity
we
the p r o b l e m
labelled
arise
in c o n n e c t i o n
of c o n s i s t e n c y
but
branches
of the g r a p h
not
states.
problems
8 is i n c o n s i s t e n t ! )
8
we
with
(the d e c i s i o n
shall n o t d i s c u s s
these
decision not
shown
problems
t96
in these paper.
Acknow!edgenlento
Thanks are due to dr~ Ao Skowron for critical
remarks.
References Mr@zeK, A., (]984). Information Polish Acad. Sci. (to appear)
Systems and Control Algorithms,
Pawlak, Zo, (]982). Rough Sets. International and Computer Sciences, I](5), 34]-356
BUllo
Journal of Information
Pawlak, Zo, (]985). Decision tables and decision algorithms. POlish Acad. Sci. (to appear) Pollack, S., Hicks, H., and Harrison, W., (1971)o Decision Theory and Practices Wiley and Sons, Inc. New York.
Bull.
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