Databases Reference
In-Depth Information
b
is a flexible attribute and
d
is a decision attribute. Also, we assume that
H
denotes a
high
profit and
L
denotes a
low
one.
In order to induce rules in which the THEN part consists of the decision
attribute
d
and the IF part consists of attributes belonging to
A
St
∪
A
Fl
,for
instance
LERS
[5] can be used for rules extraction.
In order to e
ciently extract rules when the number of attributes is large,
we can use sub-tables (
U,B
)of
S
where
B
is a
d
-reduct (see [7]) in
S
.
The set B is called
d
-reduct in S if there is no proper subset
C
of
B
such
that
d
depends on
C
. The concept of
d
-reduct in
S
was introduced with a
purpose to induce rules from
S
describing values of the attribute
d
depending
on minimal subsets of
A
St
A
Fl
.
By
L
(
r
) we mean all attributes listed in the IF part of a rule
r
. For example,
if
r
1
=[(
a
1
,
2)
∪{
d
}
.
By
d
(
r
1
) we denote the decision value of that rule. In our example
d
(
r
1
)=8.If
r
1
,
r
2
are rules and
B
∧
(
a
2
,
1)
∧
(
a
3
,
4)
−→
(
d,
8)] is a rule then
L
(
r
1
)=
{
a
1
,a
2
,a
3
}
A
Fl
is a set of attributes, then
r
1
/B
=
r
2
/B
means that the conditional parts of rules
r
1
,
r
2
restricted to
attributes
B
are the same. For example if
r
2
=[(
a
2
,
1)
⊆
A
St
∪
∗
(
a
3
,
4)
−→
(
d,
1)],
then
r
1
/
.
In our example, we get the following optimal rules:
{
a
2
,a
3
}
=
r
2
/
{
a
2
,a
3
}
1. (
a,
0)
−→
(
d,L
), (
c,
0)
−→
(
d,L
)
2. (
b,R
)
−→
(
d,L
), (
c,
1)
−→
(
d,L
)
3. (
b,P
)
−→
(
d,L
), (
a,
2)
∗
(
b,S
)
−→
(
d,H
)
4. (
b,S
)
∗
(
c,
2)
−→
(
d,H
)
w
) denotes the fact that the value of
attribute
a
has been changed from
v
to
w
. Similarly, the term (
a,v
Now, let us assume that (
a,v
−→
w
)(
x
)
means that
a
(
x
)=
v
has been changed to
a
(
x
)=
w
. Saying another words,
the property (
a,v
) of object
x
has been changed to property (
a,w
).
Let
S
=(
U,A
St
∪
−→
) is a decision table and rules
r
1
,
r
2
have
been extracted from
S
. The notion of action rule was introduced in [10]. Its
definition is given below. We assume here that:
A
Fl
∪{
d
}
•
B
St
is a maximal subset of
A
St
such that
r
1
/B
St
=
r
2
/B
St
•
d
(
r
1
)=
k
1
,
d
(
r
2
)=
k
2
and
k
1
≤ k
2
•
(
∀a ∈
[
A
St
∩ L
(
r
1
)
∩ L
(
r
2
)])[
a
(
r
1
)=
a
(
r
2
)]
•
(
∀
i
∈
p
)(
∀
b
i
∈
[
A
Fl
∩
L
(
r
1
)
∩
L
(
r
2
)])[[
b
i
(
r
1
)=
v
i
]&[
b
i
(
r
2
)=
w
i
]]
By (
r
1
,r
2
)-action rule on
x
∈
U
we mean the expression
r
:
[(
b
1
,v
1
−→
w
1
)
∧
(
b
2
,v
2
−→
w
2
)
∧
...
∧
(
b
p
,v
p
−→
w
p
)](
x
)
=
k
2
)](
x
).
where (
b
j
,v
j
→
⇒
[(
d,k
1
−→
w
j
) means that the value of the
j
t
h
flexible attribute
b
has been changed from
v
j
to
w
j
.
The notion of an extended action rule was given in [11]. The following two
conditions have been added to the original definition of the action rule:
