Information Technology Reference
In-Depth Information
2
6. Similarly, we can make the pair of
P
1
and
P
3
and the pair of
P
2
and
P
3
be
indiscernible. However, when we make all of
P
1
,
P
2
, and
P
3
indiscernible, and select
{
/
3
≥
0
.
P
3
falls outside of POS
ʲ
c
1
}
as a reduct,
P
1
∪
P
2
∪
}
(
d
)
, because the distribution
{
c
1
is
6.
To overcome that difficulty, for each type of reducts, we consider two approximate
discernibility functions
F
(
X
1
,
X
2
,
X
3
)
=
(
4
,
2
,
1
)
, and
μ
X
1
(
P
1
∪
P
2
∪
P
3
)
=
4
/
7
≤
0
.
and
F
:
F
characterizes a sufficient condition of the
preservation and
F
ʲ
characterizes a necessary condition.
First, we discuss discernibility functions characterizing sufficient conditions. By
Theorems
3
and
4
, we know that
F
L
ʲ
ʲ
ʲ
ʲ
,
F
U
ʲ
, and
F
LU
F
L
F
U
ʲ
are discernibility
functions of sufficient conditions for B-reducts, P-reducts, and UN-reducts with
ʲ
=
ʲ
∧
ʲ
.
Definition 13
Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate. Discernibility functions
F
B
ʲ
,
F
P
ʲ
, and
F
UN
ʲ
are defined as follows:
F
B
F
L
F
U
ʲ
,
F
P
F
L
ʲ
,
F
UN
F
U
ʲ
=
ʲ
∧
ʲ
=
ʲ
=
ʲ
.
Clearly, we have the following proposition.
Proposition 2
([
20
])
Let A be a subset of C , and
ʲ
∈[
0
,
0
.
5
)
be an admissible
error rate. We have the following implications:
•
If F
B
c
A
ʲ
(
˜
)
=
1
then A satisfies
(
VPB1
)
and
(
VPB2
)
with
ʲ
,
If F
P
c
A
•
ʲ
(
˜
)
=
1
then A satisfies
(
VPP1
)
and
(
VPP2
)
with
ʲ
,
If F
UN
ʲ
c
A
•
(
˜
)
=
1
then A satisfies
(
VPUN1
)
and
(
VPUN2
)
with
ʲ
.
Next, let us discuss a discernibility function characterizing a necessary condition.
Consider necessary discernibility functions for P-reducts. In the sufficient discerni-
bility function
F
P
F
L
ʲ
, pairs of objects included in the positive region are discerned
when they have different values of
ʲ
=
ʻ
C
. However, such pairs are not necessarily dis-
cerned because there may be a P-reduct such that some of pairs become indiscernible.
On the other hand, for each pair
u
i
and
u
j
, if they are excluded from the positive
region of the common condition attributes, i.e.,
C
\
m
ij
, they should be discerned
because no subset
A
m
ij
satisfies (
VPP1
) and (
VPP2
). From this consider-
ation, discernibility functions characterizing necessary conditions for preservation
of the boundaries, the positive region, and the unpredictable region are obtained as
follows.
ↆ
C
\
Definition 14
Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate. Moreover, let
c
i
be a
˜
Boolean variable pertaining to a condition attribute
c
i
∈
C
. Discernibility functions
F
B
ʲ
,
F
P
ʲ
, and
F
UN
ʲ
are defined as follows:
F
B
ʲ
(
˜
c
1
,
˜
c
2
,...,
˜
c
m
)
=
m
ij
˜
c
,
c
∈
B
ʲ
(
i
,
j
)
∈
ʔ
F
P
ʲ
(
˜
c
1
,
˜
c
2
,...,
˜
c
m
)
=
m
ij
˜
c
,
c
∈
P
ʲ
(
i
,
j
)
∈
ʔ
Search WWH ::
Custom Search