Database Reference
In-Depth Information
(A)
∈
TA
and
A
=
−
TA
=
Let us prove Claim 2. From the fact that
=
{
−−−−−−−→
T(
=
−−−→
TA
(because
n
is fi-
nite, so that we have a finite union, which can be represented by a finite-length term
of
Σ
R
algebra). From the fact that
A
has infinite relations, we have that
SK
{
(A)
}
)
(A)
π
i
(
(A))
|
1
≤
i
≤
n
=
ar(
(A))
}∈
⊆
val(A)
and hence
−
SK
⊆
A
.Let
S
={
a
1
,...,a
m
}=
dom
, thus a finite subset
of values in the finite set
dom
, then
−
SK
can be obtained by a term
A
WHERE
((name
val(A)
\
SK
⊆
a
m
))
of
Σ
R
algebra and, consequently,
−
SK
=
a
1
)
∧···∧
(name
=
∈
TA
,
and hence from
R
A
∈
TA
, we must have that
(c)
T(
{
−
SK
}
)
⊆
T(
{
R
A
}
)
and hence
−
R
A
=
−
SK
.
Consequently, for an infinite
A
(thus with infinite relations), we set
(A)
=
(A
\
−
SK)
TIMES
R
A
.
Notice that from Proposition
69
above we obtain that, for any
A
∈
Ob
DB
(that
is,
A
Υ
), the set of views which can be obtained from
A
is equal to the set of
view that can be obtained from a single relation
R
A
=
⊆
(A)
, and
R
A
∈
Υ
(i.e.,
n
=
ar(R
A
)
is finite).
We can have a number of different functions
that satisfy the constraint 1, but
and every database
A
for any two such functions
,
∈
Ob
DB
,
T(
(A))
=
TA
=
T(
(A))
, that is,
(A)
are observationally equivalent (hence isomor-
phic objects in the
DB
category). Consequently, it is not important which one of
these functions will be chosen for a transformation of a database-instance
A
(A)
and
∈
Ob
DB
∈
into a single, equivalent to it finitary relation
(A)
Υ
.
Example 45
Let us consider the following continuation of Example
44
, when
A
=
T(can(
I
,D))
is an infinite database, with two infinite binary relations
{⊥
,
R
1
,R
2
}=
can(
I
,D)
⊆
T(can(
I
,D))
represented in the following table:
R
1
=
can(
I
,
D
)
R
2
=
can(
I
,
D
)
R
11
r
s
R
21
a,b
b,ω
0
ω
1
,ω
3
ω
1
,ω
2
b,ω
1
ω
1
,ω
2
ω
3
,ω
5
ω
3
,ω
4
ω
1
,ω
3
ω
3
,ω
4
ω
5
,ω
7
ω
5
,ω
6
ω
3
,ω
5
ω
5
,ω
6
ω
7
,ω
9
ω
7
,ω
8
...
...
...
...
From point 2 of Proposition
69
above,
A
∈
TA
, and for each
a
∈
val(A)
,
R
={
a
}
can be obtained by the term of the
Σ
R
algebra
A
WHERE
(name
=
a)
, so that
R
∈
TA
and, consequently,
R
={
ω
0
,ω
0
,ω
0
,ω
0
}
is obtained by the
Σ
R
-term
))), thus
R
∈
{
ω
0
}
TIMES (
{
ω
0
}
TIMES (
{
ω
0
}
TIMES
{
ω
0
}
TA
and hence,
R
11
R
1
WHERE
(nome
1
=
b)
∈
a)
∧
(name
1
=
TA
;
R
21
R
2
WHERE
(nome
1
=
b)
∈
TA.
