Database Reference
In-Depth Information
R
1
=
r
can(
I
,
D
)
R
2
=
s
can(
I
,
D
)
a,b
b,ω
0
b,ω
1
ω
1
,ω
2
ω
1
,ω
3
ω
3
,ω
4
ω
3
,ω
5
ω
5
,ω
6
...
...
Let us consider a unary infinite relation
=
0
,
1
,...
=
,...
R
ω
4
i
|
i
=
ω
0
,
ω
4
,
ω
8
such that
R
≤
v
R
2
≤
v
A
}=
−
SK
=
R
={
a
,
b
} ∪ {
ω
i
|
i
=
0
,
1
,...
∪
, that is,
R
∈↓
v
A
. However, there is no
finite
term
t
of
Σ
R
(SPRJU)
algebra such that
{
a
,
b
}
TA
. In fact, we can obtain
R
only as an
infinite
union of terms
π
2
(
R
2
WHERE name
2
=
t
A
=
R
, so that
R/
∈
0
,
4
,
8
,...
, and not
by
π
2
(
R
2
WHERE C
) because there is no unary select operation _
WHERE C
in
Σ
R
with the condition
C
equal to an infinite string '
(name
2
=
ω
0
)
∨
(name
2
=
ω
4
)
ω
i
), for
i
=
', and by using negation (complements) we cannot obtain
a finite string for a condition
C
as well. Consequently, with finite-length
Σ
R
-algebra
terms we cannot obtain
R
and hence
∨
(name
2
=
ω
8
)
∨···
↓
v
A
TA
.
Instead, for the complete lattice in the next Lemma
22
, we have the behavioral
equivalence
TA
=↓
(R
1
TIMES R
2
)
. Notice that in this case, differently from the
case above,
R
{
}
{
R
2
}
. In fact, the view '
R WHERE name
1
=
R
2
because
T
R
T
ω
0
'in
T
(analogously to
the case above, it can't be obtained by finite-length
Σ
R
-term from
R
2
).
{
R
}
equal to
{
ω
4
i
|
i
=
1
,
2
,...
}
is not a view in
T
{
R
2
}
Consequently, we need an alternative method of an
equivalent
transformation of
a database-instance
A
into a single finitary relation
R
A
(with a finite
n
=
ar(R
A
)
),
with possibly an infinite set of tuples, in order to guarantee that
TA
=↓
R
A
,as
follows:
Proposition 69
There exists a function
:
Ob
DB
→
Υ
such that for each A
∈
0
)
⊥
⊥
Ob
DB
,
(T A)
(A) with
(
and
:
1.
(A)
∈
T(
{
(A)
}
)
=
T(A)
,
with finite k
=
ar(
(A)) and R
Υ
(
Υ
)
.
2.
If A has infinite relations then
−
SK
⊆
A with
{
−
SK,A
}⊆
T(A)
.
Proof
It is easy to see that for every finite set (a database)
A
∈
Ob
DB
,
that is, when
n
=|
A
|
is finite, so that the relation
R
=×
C
(A)
(the Cartesian product of relations
in
A
) is finitary (i.e.,
ar(R)
=
R
i
∈
A
ar(R
i
)
is finite), we can define
(A)
R
(an
A
).
alternative is to define
(A)
