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•
A U-reduct with
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following condition:
UA
A
(
UA
C
(
X
i
)
=
X
i
)
for all
i
∈
V
d
.
(VPU)
Mi et al. [
33
] independently proposed L-reducts and U-reducts under the names
of lower and upper distribution reducts.
Additionally, we can define a reduct preserving the unpredictable region.
Definition 10
([
20
]) Let
ʲ
∈[
0
,
0
.
5
)
be an admissible error rate. A UN-reduct with
ʲ
in VPRSM is a minimal condition attribute subset
A
ↆ
C
satisfying the following
conditions:
UNP
A
(
UNP
C
(
d
)
=
d
),
(VPUN1)
ↇ
.
B
satisfies (VPUN1) for all
B
A
(VPUN2)
Remark 7
We modify the definitions of B- and UN-reducts from our paper [
20
],
because there aremistakes inBoolean functions for B- andUN-reducts. By adding the
second condition, the preserving conditions of B- and UN-reducts become monotone
with respect to the set-inclusion of condition attribute sets.
By definitions, (
VPL
) and (
VPU
) obviously imply (
VPP1
,2) and (
VPUN1
,2),
respectively. Moreover, (
VPP1
,2) also implies (
VPQ1
,2). Hence, we have the fol-
lowing relations among different types of reducts.
Theorem 3
([
20
])
Let A be a subset of C . We have the following statements in
VPRSM with a fixed parameter
ʲ
∈[
0
,
0
.
5
)
,
(a)
A is an L-reduct then A satisfies (
VPP1
,2),
(b)
A is a U-reduct then A satisfies (
VPUN1
,2),
(c)
A is a P-reduct then A satisfies (
VPQ1
,2).
Contrary to the classical RSM, (
VPB1
,2) is not equivalent to (
VPU
). In RSM, pre-
serving boundaries implies preventing ambiguity expansion, namely upper approx-
imations. However, in VPRSM, the ambiguity expansion can be prevented not
only by preserving boundaries but by preserving them with the unpredictable
region. Furthermore, we can define other compositions of different
types of
reducts.
Simply combining 5 types of structure-based reducts, we obtain 2
5
31
types of reducts (ignoring (a) and (b) of Theorem
3
). To reduce the number, we first
investigate relationships of preserving conditions of reducts.
−
1
=
Theorem 4
([
20
])
Let A be a subset of C . We have the following statements in
VPRSM with a fixed parameter
ʲ
∈[
0
,
0
.
5
)
,
•
The conjunction of
(
VPB1
)
and
(
VPP1
)
is equivalent to that of
(
VPB1
)
and
(
VPUN1
)
,
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