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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
∪
) be a decision table and rules
r
1
,
r
2
are extracted
from
S
. The notion of an extended action rule (E-action rule) was given in [21].
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
≤
q
)(
∀
e
i
∈
[
A
St
∩
[
L
(
r
2
)
−
L
(
r
1
)]])[
e
i
(
r
2
)=
u
i
],
•
(
∀
i
≤
r
)(
∀
c
i
∈
[
A
Fl
∩
[
L
(
r
2
)
−
L
(
r
1
)]])[
c
i
(
r
2
)=
t
i
],
•
(
∀
i
∈
p
)(
∀
b
i
∈
[
A
Fl
∩
L
(
r
1
)
∩
L
(
r
2
)])[[
b
i
(
r
1
)=
v
i
]&[
b
i
(
r
2
)=
w
i
]].
Let
A
St
∩
L
(
r
1
)
∩
L
(
r
2
)=
B
.By(
r
1
,r
2
) -E-action rule on
x
∈
U
we mean the
expression
r
:
[
{
a
=
a
(
r
1
):
a
∈
B
}∧
(
e
1
=
u
1
)
∧
(
e
2
=
u
2
)
∧
...
∧
(
e
q
=
u
q
)
∧
(
b
1
,v
1
−→
w
1
)
∧
(
b
2
,v
2
−→
w
2
)
∧
...
∧
(
b
p
,v
p
−→
w
p
)
∧
(
c
1
,
−→
t
1
)
∧
(
c
2
,
−→
t
2
)
∧
...
∧
(
c
r
,
−→
t
r
)](
x
)=
⇒
[(
d, k
1
−→
k
2
)](
x
)
Object
x
∈
U
supports (
r
1
,r
2
)-E-action rule
r
in
S
=(
U, A
St
∪
A
Fl
∪{
d
}
), if
the following conditions are satisfied:
•
∀
i
≤
p
)[
b
i
∈
L
(
r
)][
b
i
(
x
)=
v
i
]
∧
d
(
x
)=
k
1
(
•
∀
i
≤
p
)[
b
i
∈
L
(
r
)][
b
i
(
y
)=
w
i
]
∧
d
(
y
)=
k
2
(
•
∀
j
≤
p
)[
a
j
∈
(
A
St
∩
L
(
r
2
))][
a
j
(
x
)=
u
j
]
(
•
(
∀j ≤ p
)[
a
j
∈
(
A
St
∩ L
(
r
2
))][
a
j
(
y
)=
u
j
]
•
object
x
supports rule
r
1
•
object
y
supports rule
r
2
By the support of E-action rule
r
in
S
, denoted by
Sup
S
(
r
), we mean the set
of all objects in
S
supporting
r
. Saying another words, this set is defined as:
{
x
:[
a
1
(
x
)=
u
1
]
∧
[
a
2
(
x
)=
u
2
]
∧
...
∧
[
a
q
(
x
)=
u
q
]
∧
[
b
1
(
x
)=
v
1
]
∧
[
b
2
(
x
)=
v
2
]
∧
...
∧
[
b
p
(
x
)=
v
p
]
∧
[
d
(
x
)=
k
1
]
}
.
By the confidence of
r
in
S
, denoted by
Conf
S
(
r
), we mean
[
Sup
S
(
r
)
/Sup
S
(
L
(
r
))]
[
Conf
(
r
2
)]
To find the confidence of (
r
1
,r
2
)-E-action rule in
S
, we divide the number of
objects supporting (
r
1
,r
2
)-action rule in
S
by the number of objects supporting
left hand side of (
r
1
,r
2
)-E-action rule times the confidence of the classification
rule
r
2
in
S
.
×
9.3
Discovering E-Action Rules
In this section we present a new algorithm, called Action-Tree algorithm, for dis-
covering E-action rules. Basically, we partition the set of classification rules
R
discovered from a decision system
S
=(
U, A
St
∪
), where
A
St
is the set
of stable attributes,
A
Fl
is the set of flexible attributes and,
V
d
=
A
Fl
∪{
d
}
{
d
1
,d
2
, ..., d
k
}
