Databases Reference
In-Depth Information
and the expression
(
⇒
p,B
x
)(
P
1
(
x
)
∧
P
4
(
x
)
,P
2
(
x
)
∧
P
3
(
x
))
is an example of a closed formula. The values of formulas of OPC are defined
in Tarski style, see [2].
The formula (
⇒
p,B
x
)(
P
1
(
x
)
∧
P
4
(
x
)
,P
2
(
x
)
∧
P
3
(
x
)) corresponds to the
association rule
P
1
∧
P
4
⇒
p,B
P
2
∧
P
3
defined on
- data matrix with Boolean columns (i.e. monadic predicates)
P
1
,
P
2
,
P
3
,
P
4
,
...
. It means that the association rules with Boolean attributes
can be understood as formulas of observational predicates calculi. The asso-
ciation rule with categorial attributes defined in Sect. 2 can be also seen as
an association rule defined on
{
0,1
}
- data matrix with Boolean columns
corresponding to monadic predicates, see below.
The attribute
A
with categories
{a
1
,...a
k
}
can be represented by
k
Boolean attributes
A
(
a
1
)
,...A
(
a
k
). Remember that the Boolean attribute
A
(
a
1
) is true for the object
o
if and only if the value of
A
for the object
o
is
a
1
. Thus the basic Boolean attribute
A
(
a
1
,a
2
) corresponds to the disjunction
A
(
a
1
)
{
0,1
}
∨
A
(
a
2
) etc. It means that the rule
A
(
a
1
,a
2
)
∧
B
(
b
3
)
⇒
p,B
C
(
c
4
,c
5
)
can be seen as
(
A
(
a
1
)
∨
A
(
a
2
))
∧
B
(
b
3
)
⇒
p,B
C
(
c
4
)
∨
C
(
c
5
)
and also as
(
⇒
p,B
x
)((
A
(
a
1
)(
x
)
∨ A
(
a
2
)(
x
))
∧ B
(
b
3
)(
x
)
,C
(
c
4
)(
x
)
∨ C
(
c
5
)(
x
))
It means that the association rules we deal with are formulas we can get
from formulas of classical monadic predicate calculus by adding 4ft-quantifiers.
Thus a natural question arises what association rules can be expressed by
“classical” quantifiers
. It is shown that the answer is related to the classes
of association rules, see Sect. 7.
∀
,
∃
6 Deduction Rules
Deduction rules concerning association rules are not only theoretically inter-
esting but also practically important. The deduction rules of the form
ϕ
≈
ψ
ϕ
≈
ψ
