Databases Reference
In-Depth Information
Let us remark that according to the Theorem 5 the implicational quantifier
⇒
∗
is classically definable if and only if its table
Tb
⇒
∗
of maximal
b
has only
finite number of steps. Let us suppose that we analyze a data matrix with
N
rows. We can define a new quantifier
⇒
∗,N
such that
⎧
⎨
⎩
Tb
⇒
∗
(
a
)
if
a
≤
N
Tb
⇒
∗,N
(
a
)=
Tb
⇒
∗
(
N
+1) if
a>N
⇒
∗,N
It is clear that the table
Tb
⇒
∗,N
of maximal
b
of 4ft-quantifier
has
⇒
∗,N
is classically definable.
finite number of steps and thus the 4ft-quantifier
It is also clear that it is
⇒
∗,N
⇒
∗
(
a,b
)
(
a,b
)=
for each
a
0.
In this way we can replace general implicational 4ft-quantifier by a clas-
sically definable 4ft-quantifier that is equivalent to the given quantifier what
concerns behavior on the given data matrix. This approach can be used also
for the additional 4ft-quantifiers. The construction of the corresponding for-
mula of the classical predicate calculus is described in [10].
≤
N
and for each
b
≥
5 Conclusions
We have presented a simple criterion of classical definability of important
types of association rules. This criterion is based on the table of critical fre-
quencies that is itself important tool for verification of association rules. The
presented criterion depends on the class of association rules (i.e. the class of
4ft-quantifiers) we deal with.
Let us remark that there are further interesting and practically useful
relations of tables of critical frequencies, classes of association rules, logical
properties of association rules and properties of association rules in the data
with missing information. They are partly published in [2, 5-7] and in more
details investigated in [4, 8]. An overview of related results is in [11].
Acknowledgement
The work described here has been supported by the grant 201/05/0325 of
the Czech Science Foundation and by the project IGA 25/05 of University of
Economics, Prague.
