Databases Reference
In-Depth Information
Tabl e 2 .
Classes of association rules defined by truth preservation conditions
Class
Truth preservation condition
Examples
a
≥ a ∧ b
≤ b
Implicational
TPC
⇒
⇒
p,Base
⇒
!
p,α,Base
a
≥ a ∧ b
≤ b ∧ c
≤ c
⇔
0
.
9
,
0
.
1
⇔
p,Base
Double implicational
TPC
⇔
a
≥ a ∧ b
+
c
≤ b
+
c
Σ
-double implicational
TPC
Σ,⇔
⇔
p,Base
⇔
!
p,α,Base
a
≥ a ∧ b
≤ b ∧ c
≤ c ∧ d
≥ d
Equivalency
TPC
≡
∼
α,Base
≡
p,Base
a
+
d
≥ a
+
d ∧ b
+
c
≤ b
+
c
Σ
-equivalency
TPC
Σ,≡
≡
p,Base
≡
!
p,α,Base
M
that is better from the point of view of implication. This expectation is
ensured for implicational quantifiers by the above given definition.
It is easy to prove that the 4ft-quantifier
⇒
p,Base
of founded implication
is implicational. It is proved in [2] that the 4ft-quantifier
!
p,α,Base
⇒
of lower
critical implication is also implicational.
There are several additional important classes of association rules defined
by truth preservation conditions, see [2, 4, 9, 11]. Overview of these classes
and some examples are given in Table 2. The quantifiers
!
⇒
p,Base
,
⇒
p,α,Base
,
⇔
p,Base
,
≡
p,Base
,and
∼
α,Base
used in Table 2 as examples are defined in
⇔
0
.
9
,
0
.
1
is explained below.
Sect. 2, the quantifier
3.2 Double Implicational Quantifiers
The class of double implicational quantifiers is defined in [11] by the
truth
preservation condition TPC
⇔
for double implicational quantifiers
:
TPC
⇔
=
a
≥
b
c
a
∧
≤
b
∧
≤
c
.
We can see a reason for the definition of double implicational quantifier in
an analogy to propositional calculus. If
u
and
v
are propositions and both
u → v
and
v → u
are true, then
u
is equivalent to
v
(the symbol “
→
”ishere
a propositional connective of implication). Thus we can try to express the
relation of equivalence of attributes
ϕ
and
ψ
using a “double implicational”
4ft-quantifier
⇔
∗
such that
ϕ
⇔
∗
ψ
if and only if both
ϕ
⇒
∗
ψ
and
ψ
⇒
∗
ϕ
,
⇒
∗
is a suitable implicational quantifier.
If we apply the truth preservation condition for implicational quantifier
TPC
⇒
to
ϕ
where
⇒
∗
ψ
, we obtain
a
≥
b
≤
⇒
∗
ϕ
,we
a
∧
b
.Ifweapplyitto
ψ
obtain
a
≥
c
≤
c
,(
c
is here instead of
b
, see Table 1). This leads to
the truth preservation condition for double implicational quantifiers
TPC
⇔
see Table 2.
The class of
Σ
-double implicational quantifiers is defined to contain useful
quantifiers
a
∧
!
⇔
p,Base
and
⇔
p,α,Base
. They deal with the summa
b
+
c
in the
