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reducts (e.g. different authors would give different definitions), we can easily quote
the equivalent of them from here. The other is that equivalent reducts (e.g. U-reducts
and B-reducts) in RSM could become different in an extended RSM (e.g. variable
precision RSM).
Remark 4
From the discussion above, we know that a U-reduct preserves more
information than an L-reduct. However, when
p
=
2, we have the following relation:
UA
A
(
X
1
)
=
U
\
LA
A
(
X
2
),
UA
A
(
X
2
)
=
U
\
LA
A
(
X
1
).
Namely, we obtain upper approximations from lower approximations. Hence, in that
case, an L-reduct is a U-reduct.
Remark 5
From Theorem
1
, we see that preserving the measure
is equivalent to
preserving the lower approximations. Contrary, we can define a measure preserving
which is equivalent to preserving the upper approximations. For example [
23
], we
define,
ʳ
i
∈
V
d
|
\
UA
A
(
X
i
)
|
U
˃
A
(
d
)
=
,
(
p
−
1
)
|
U
|
then
˃
A
(
d
)
=
˃
C
(
d
)
is same as condition (
U
).
7.2.4 Boolean Functions Representing Reducts
Boolean reasoning [
37
] is a methodology where solutions of a given problem is
associated with those of Boolean equations. In this section, we develop positive
(monotone) Boolean functions whose solutions are given by condition attribute sub-
sets satisfying the preserving conditions (
L
)or(
U
). Moreover, prime implicants of
the Boolean functions exactly correspond to L-reducts or U-reducts. The Boolean
functions are useful for enumerating reducts.
The results of this section are well-known and appeared in many papers e.g. [
1
,
43
,
45
,
50
], but in slightly different expressions from ours. A unified formulation
of Boolean functions of different types of reducts is provided using the generalized
decision function.
Here, we briefly introduce Boolean functions and Boolean formulas [
9
,
14
]. Let
q
be a natural number. A Boolean function is a mapping
f
q
:{
0
,
1
}
ₒ{
0
,
1
}
, where
q
w
∈{
0
,
1
}
is called a Boolean vector whose
i
th component is
w
i
.Let
x
1
,
x
2
,...,
x
q
be Boolean variables. A Boolean formula in the Boolean variables
x
1
,
x
2
,...,
x
q
is
a composition of
0
, 1, the variables an
d
operators
o
f conjunction
∧
, disjunction
∨
,
complementation
x
3
, and so on (for complete
definition, see e.g. [
9
]). The Boolean formula is a Boolean function of the variables
x
1
,
·
, such as
x
1
∧
(
x
2
∨
x
3
)
,
(
x
1
∧
x
2
)
∨
x
q
. Conversely, any Boolean function can be expressed by a Boolean
formula. For two Boolean functions
f
and
g
,
g
x
2
,...
≤
f
means that
f
and
g
satisfy
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