Databases Reference
In-Depth Information
5 Logical Calculi of Association Rules
Logical calculi formulae of which correspond to association rules are defined
in [12] according to the following principles:
•
The association rule is the expression
ϕ ≈ ψ
where the Boolean attributes
ϕ
and
ψ
are built from the basic Boolean attributes by usual Boolean
connectives.
•
The basic Boolean attribute is the expression of the form
A
(
α
). Here
α
is
the subset of the set of all possible values of the attribute
A
, see Sect. 2.
•
The expression
A
(
a
1
)
∧
B
(
b
3
,b
4
)
≈
C
(
c
1
,c
11
,c
21
)
∨
D
(
d
9
,d
10
,d
11
)
is an example of the association rule. Here
A
and
B
are the names of the
attributes and
a
1
,
b
3
,
b
4
are the names of its possible values, analogously
for
C
and
D
, see also Sect. 2. Each attribute has a finite number of possible
values called categories.
•
The set of all association rules related to the given data matrix is given by:
-
The set of attributes and by the sets of possible values for each attribute
-
The set of all 4ft quantifiers
-
Usual Boolean connectives
•
The association rules are interpreted and evaluated in corresponding data
matrices. The corresponding data matrix has one column for each attribute
and only possible values of the attribute can occur in the corresponding
column.
•
The evaluation function
Val
with values 1 (truth) and 0 (false) is defined.
The value
Val
(
ϕ
≈
ψ,
M
)
of the association rule
ϕ
≈
ψ
in the data matrix
M
is done by the four-fold
table 4
ft
(
ϕ
,
ψ
,
M
)of
ϕ
and
ψ
in
M
and by the condition associated to
the 4ft-quantifier
≈
.
The above described calculi of association rules are special case of observa-
tional calculi defined and studied in [2] (main principles are summarized also
in [12]). Important observational calculi are defined in [2] by modifications of
predicate calculi. The modification consists in
1. Allowing only finite models that correspond to analysed data in the form
of
- data matrices.
2. Adding generalized quantifiers that make possible to express general re-
lations of two or more derived predicates.
{
0,1
}
The resulting calculi are called
observational predicate calculi
(OPC for short).
The 4ft-quantifier
is a special case of the generalized quantifier. An example
of open formula of observational predicate calculus is the formula
≈
(
⇒
p,B
x
)(
P
1
(
x
)
∧
P
4
(
y
)
,P
2
(
x
)
∧
P
3
(
y
))
