Databases Reference
In-Depth Information
It is easy to prove that the function
Tb
⇒
∗
is a non-negative and non-decreasing
function that satisfies
⇒
∗
(
a,b
) = 1 if and only if
b<Tb
⇒
∗
(
a
)
for all integers
a
≥
0and
b
≥
0. It means e.g. that
a
+
b
i
p
i
(1
a
+
b
p
)
a
+
b−i
−
≤
α
∧
a
≥
Base
if and only if
b<Tb
⇒
!
p,α,Base
(
a
)
i
=
a
where
Tb
⇒
!
p,α,Base
is a table of maximal b for implicational quantifier
!
p,α,Base
of lower critical implication.
We define analogously tables of critical frequencies for
Σ
-double impli-
cational quantifiers and for
Σ
-equivalency quantifiers. The table of critical
frequencies for
Σ
-double implicational quantifier
⇒
⇔
∗
is defined as a
table of
⇔
∗
. It is the function
Tb
⇔
∗
assigning a value
Tb
⇔
∗
(
a
)
+
maximal b+c for
∈N
to each
a
0 such that
Tb
⇔
∗
(
a
)=min
{b
+
c|⇔
∗
(
a,b,c
)=0
}
.
It is easy to prove that the function
Tb
⇔
∗
is a non-negative and non-decreasing
function that satisfies
⇔
∗
(
a,b,c
) = 1 if and only if
b
+
c<Tb
⇔
∗
(
a
)
for all integers
a
≥
0.
The table of critical frequencies for
Σ
-equivalency quantifier
≥
0,
b
≥
0, and
c
≥
≡
∗
is defined
≡
∗
. It is the function
Tb
≡
∗
as a
table of maximal b+c for
that assigns a value
+
to each
E
Tb
≡
∗
(
E
)
∈N
≥
0suchthat
|≡
∗
(
a,b,c,d
)=0
Tb
≡
∗
(
E
)=min
{
b
+
c
∧
a
+
d
=
E
.
It is easy to prove that the function
Tb
≡
∗
is a non-negative and non-decreasing
function that satisfies
≡
∗
(
a,b,c,d
) = 1 if and only if
b
+
c<Tb
≡
∗
(
a
+
d
)
for all integers
a
0.
The tables of critical frequencies are practically useful, they can be used
to avoid complex computation in the above outlined way. However these ta-
bles also describe the behavior of 4ft quantifiers in a very close way. It means
that we can use the tables of critical frequencies to study the properties of 4ft
quantifiers. Simple property of the table of maximal b for implicational quan-
tifier can be used to decide if the corresponding association rule is definable
in the classical predicate calculus or not, see Sect. 7.
We have shown tables of critical frequencies for implicational,
Σ
-double
implicational and
Σ
-equivalency quantifiers. It is shown in [7] that it is pos-
sible to define also a reasonable table of critical frequencies for symmetrical
quantifiers with property F. This table can be used to avoid complex compu-
tation related to Fisher's quantifier. We will not explain it here in details.
≥
0,
b
≥
0,
c
≥
0, and
d
≥
