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The conjunction of ( VPL ) and ( VPB1 ) is equivalent to that of ( VPL ) and ( VPU ) ,
The conjunction of ( VPU ) and ( VPP1 ) is equivalent to that of ( VPL ) and ( VPU ) .
We define 4 different types of reducts.
Definition 11 ([ 20 ]) Let
ʲ ∈[
0
,
0
.
5
)
be an admissible error rate.
An LU-reduct with
ʲ
in VPRSM is a minimal condition attribute subset A
C
satisfying the following condition:
LA A (
LA C (
and UA A (
UA C (
X i ) =
X i )
X i ) =
X i )
V d .
for all i
(VPLU)
An LUN-reduct with
ʲ
in VPRSM is a minimal condition attribute subset A
C
satisfying the following conditions:
LA A (
LA C (
and UNP A (
UNP C (
X i ) =
X i )
for all i
V d ,
d
) =
),
(VPLUN1)
d
B satisfies (VPLUN1)
for all B
A
.
(VPLUN2)
A BUN-reduct with
ʲ
in VPRSM is a minimal condition attribute subset A
C
satisfying the following conditions:
BN A (
BN C (
and UNP A (
UNP C (
X i ) =
X i )
for all i
V d ,
d
) =
),
(VPBUN1)
d
B satisfies (VPBUN1)
for all B
A
.
(VPBUN2)
A PUN-reduct with
ʲ
in VPRSM is a minimal condition attribute subset A
C
satisfying the following conditions:
POS A (
POS C (
and UNP A (
UNP C (
d
) =
d
)
d
) =
d
),
(VPPUN1)
.
B satisfies (VPPUN1)
for all B
A
(VPPUN2)
In Fig. 7.2 , we show the relationships among 9 types of reducts. Names of reducts
are abbreviated to their first characters. Reducts located in the upper part of Fig. 7.2
preserve regions much more. Therefore, such reducts are larger in the sense of set
inclusion than the other reducts located in the lower part. A line segment connecting
two types of reducts implies that, for each reduct of the upper type say A satisfies
the preserving condition of a reduct of the lower one. From Fig. 7.2 , we know that
LU-reducts preserve regions most. On the other hand, UN-reducts and P-reducts do
not preserve many regions.
The next proposition says that composite reducts such as LUN- or BUN-reducts
can be constructed from their base reducts such as L- and UN-reducts or B- and
UN-reducts. The proposition is useful to enumerate the composite reducts.
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