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indiscernibility. Apparently, if identical premises have different consequences,
the rule is inconsistent.
11.4 Reduction of Decision Tables
Before decision making, we usually face a problem: whether are all condition
attributed necessary? Or, whether can decision table be reduced? The decision
tables after reduction has the same functions with the tables before reduction, but
the tables after reduction has less attributes. Therefore, the reduction of attributes
is very important in real applications. With the reduction, we can make decisions
based on fewer conditions, that is, same results can be acquired by simpler
methods. Strictly speaking, although decision algorithm and decision table are
different concepts, the decision algorithms expressed by decision tables are more
compact, more understandable and simpler than those expressed by decision
logic language. Decision algorithms can describe decision tables with the
methods of logics, thus some properties of them can be used mutually.
11.4.1
Dependency of Attributes
To process data and make decisions, the internal relations of data and the
dependency of attributes must be analyzed. The dependency of attributes is
relative to the dependency of knowledge we introduced before. If there exists a
consistent
PQ
-decision algorithm in
S
, we say attribute set
Q
is totally dependent
(or dependent for short ) on attribute set
P
, denoted as
P
¼
Q
; if there exists some
inconsistent
PQ
-decision algorithm, we say attribute set
Q
is partly dependent on
attribute set
.
Similar with the definition of knowledge dependency, we can also use the
concept of positive region to define the dependency among attribute sets.
If (
P
-rules in the
algorithm is called as the positive region of the algorithm, denoted as
P
,
Q
) is a
PQ
-algorithm in
S
, the set of all consistent
PQ
POS
(
P
,
Q
).
POS
(
P
,
Q
) is the consistent part of an inconsistent algorithm. Apparently, if and
only if
POS
(
P
,
Q
)≠(
P
,
Q
) or
card
(
POS
(
P
,
Q
))≠
card
((
P
,
Q
)), the algorithm is
inconsistent.
For a
PQ
-algorithm, the degree of inconsistency is expressed by dependency
degree
k
, and it is defined as:
k
=
card
(
POS
(
P
,
Q
))/
card
(
P
,
Q
)
