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meaning with symbol |S| (the number of the elements in S) in mathematical
language. If rule θ→Ψ is totally consistent, the rough operator can be omitted.
Table 11.11 The reducts of all attribute values in Table 11.9
U
a b d e
1 0 2 0
0 × 1 2
2 0 1 1
1 1 2 2
1 0 0 1
× 2 1 1
2 1 1 2
0 × 0 1
Thus, the decision algorithm generated from Table 11.11 is:
1
2
3
4
5
6
7
8
a
1
b
0
→
0.5
d
2
e
0
,
a
0
→
0.5
d
1
e
2
,
a
2
b
1
→
d
1
e
2
,
a
2
b
0
∨
b
2
→
d
1
e
1
,
a
0
→
0.5
d
0
e
1
,
a
1
b
1
→
d
2
e
2
,
a
1
b
0
→
0.5
d
0
e
1
.
2
. The Method Based on The Decomposition of Inconsistent Tables
Each decision table T=(
U, A, C, D
) can be split into two decision tables
T
1
=(
U
1
,
A
,
C
,
D
) and
T
2
=(
U
2
,
A
,
C
,
D
). Thus, we have
C
¼
1
D
in table
T
1
and
C
¼
0
D
in
table
).
From this proposition we know, if the dependency degrees of condition
attributes are known and a decision table is inconsistent (its dependency degree is
less than 1), the table can be decomposed to two sub-tables. One table is totally
inconsistent whose dependency degree is 0, and the other table is totally
consistent whose dependency degree is 1. Of course, only when the dependency
degree is more than 0 and not equal to 1, the decomposition can be processed.
The totally consistent table can be reduced with the method introduced in
Chapter 4. The totally inconsistent table needs not to be processed, and the
decision rules with rough operators can be generated directly from it.
T
2
. Here,
U
1
=
POS
C
(
D
),
U
2
=∪
BN
C
(
X
), and
X
∈
U
|
IND
(
D
