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formulation of several Boolean functions corresponding to several types of reducts
in the classical and extended rough set models.
In this chapter, we show our theoretical results of structure-based reducts in the
classical and extended rough set models, including definitions of reducts and their
strong-weak hierarchy. The results consist of our papers [
20
,
23
,
25
,
31
]. Our main
contributions are to propose structure-based reducts, investigate strong-weak rela-
tions of reducts, and connect reducts with prime implicants of Boolean functions in a
unified formulation in the variable precision and dominance-based rough set models.
For the structure-based reducts in the variable precision rough set model, we revise
their definitions from our previous work [
20
]. Parts of the results were independently
developed by other authors [
33
,
45
,
49
,
52
].
This chapter is organized as follows. In Sect.
7.2
, we study structure-based reducts
in the classical rough set models. Firstly, we define a decision table and the rough set
model of the decision table. Then, we introduce several types of reducts including
structure-based reducts and others. We show that all types of reducts are reduced to
two different types. Finally, we connect all reducts of each type with the prime
implicants of a specific Boolean function. Sections
7.3
and
7.4
are devoted to
structure-based reducts in the variable precision rough set model and those in the
dominance-based rough set model, respectively. Those sections have almost the same
organization as that of Sect.
7.2
, namely, defining a rough set model and reducts,
investigating strong-weak relations of reducts, and connecting reducts with prime
implicants of Boolean functions. Concluding remarks are given in Sect.
7.5
.
7.2 Structure-Based Attribute Reduction in Rough Set Models
7.2.1 Decision Tables
In rough set theory, analysed data sets form decision tables [
36
,
39
]. A decision
table is defined by
.
1
U
is a finite set of objects.
AT
is a finite set of attributes.
V
is a set of attribute values. Each attribute
a
D =
(
U
,
AT
=
C
∪{
d
}
,
{
V
a
}
a
∈
AT
)
∈
AT
is a function
a
:
U
ₒ
V
a
, where
V
a
ↆ
V
is a set of values for
a
. For an object
u
∈
U
and an attribute
a
∈
AT
,
a
(
u
)
is the value of
u
with respect to
a
.For
A
=
{
a
i
1
,
a
i
2
,...,
a
i
k
}ↆ
AT
,
V
A
is the Cartesian product of
{
V
a
i
l
}
l
=
1
,
2
,...,
k
, namely,
V
A
=
ʠ
a
i
l
∈
A
V
a
i
l
={
(v
i
1
,v
i
2
,...,v
i
k
)
|
v
i
l
∈
V
a
i
l
,
l
=
1
,
2
,...,
k
}
.
A
(
u
)
is the tuple of
the values of
u
with respect to
A
, namely,
A
.The
attribute set
AT
is divided into a condition attribute set
C
and a decision attribute
d
to investigate the dependency of the decision attribute on condition attributes or
the causal effect of condition attributes on the decision attribute. Throughout this
chapter, we consider that the objects and the condition attributes are indexed by
(
u
)
=
(
a
i
1
(
u
),
a
i
2
(
u
),...,
a
i
k
(
u
))
1
A decision table is often defined by the finite set of objects
U
and the finite set of attributes
AT
,
i.e.,
(
U
,
AT
)
, however we use that definition to clarify the sets of values for the attributes.
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