Database Reference
In-Depth Information
Definition 29
Weak power-view operator
T
w
:
Ob
DB
−→
Ob
DB
is defined as fol-
lows: for any simple database
A
in
DB
category,
R
∀
1
≤
k
≤|
R
|
∀
d
i
∈
π
k
(R)
Va l(d
i
)
T
w
(A)
|
R
∈
T(A)
and
|
|
where
is the number of attributes of the view (relation)
R
, and
π
k
is a
k
th projec-
tion operator on relations. For
A
R
=
1
≤
j
≤
m
A
j
,m
=
1
≤
j
≤
m
T
w
(A
j
)
.
≥
1,
T
w
(A)
w
' for databases and a weak observational
We define a partial order relation '
≈
w
'(by
A
≈
w
B
iff
A
w
B
and
B
w
A
) as in Definition
19
equivalence relation '
by substituting operator
T
by
T
w
.
The following properties are valid for the weak partial order
w
w.r.t. the partial
order
(we denote '
A
≺
B
'iff
A
B
and not
A
≈
B
):
=
1
≤
j
≤
m
A
j
,m
=
1
≤
i
≤
k
B
i
,k
Proposition 12
Let A
≥
1
and B
≥
1
be any
two databases
(
objects in
DB
category
),
then
:
1.
T
w
(A)
≺
A
,
if A is a database without Skolem constants
;
T
w
(A)
A
,
otherwise
;
w
B
(
thus
,
A
≈
≈
w
B
);
2.
A
B implies A
B implies A
=
=
=
⊆
=
3.
T
w
(T
w
(A))
T(T
w
(A))
T
w
(T A)
T
w
(A)
TA
,
thus
,
each object D
=
≈
w
A
.
T
w
(A) is a closed object
(
i
.
e
.,
D
TD
)
such that D
Proof
Claim 1. From
T
w
(A)
⊆
TA
(
T
w
(A)
=
TA
only if
A
is without Skolem
constants).
Claim 2. If
A
≺
B
then
TA
j
⊂
TB
σ(j)
with
T
w
(T A
j
)
⊆
T
w
(T B
σ(j)
)
, for all
1
≤
w
B
.
Claim 3. It holds from the definition of the operator
T
and
T
w
:
T
w
(T
w
(A))
j
≤
m
, i.e.,
A
=
T(T
w
(A))
because
T
w
(A)
is the set of views of
A
without Skolem constants and
due to Claim 1.
T
w
(T A)
={
R
|
R
∈
TT(A)
and
∀
1
≤
k
≤|
R
|
∀
(d
i
∈
π
k
(R))Va l(d
i
)
}=
{
R
|
R
∈
TA
and
∀
1
≤
k
≤|
R
|
∀
(d
i
∈
π
k
(R))Va l(d
i
)
}=
T
w
(A)
, from
T
=
TT
. Let us
show that
T
w
(T
w
(A))
=
T
w
(A)
. For every view
v
∈
T
w
(T
w
(A))
, from
T
w
(T
w
(A))
=
T(T
w
(A))
⊆
TA
, it holds that
v
∈
TA
and hence, from the fact that
v
is without
Skolem constants,
v
∈
T
w
(A)
. The converse is obvious.
Notice that from point 2, the partial ordering '
' is a stronger discriminator for
databases than the weak partial order '
w
', i.e., we can have two objects
A
≺
B
that
are weakly equivalent,
A
T
w
(B)
and
B
is a database
with Skolem constants). Let us extend the notion of the type operator
T
w
into the
notion of the endofunctor of
DB
category:
≈
w
B
(for example, when
A
=
(T
w
,T
w
)
Theorem 7
There
exists
a
weak
power-view
endofunctor T
w
=
:
DB
such that the object component T
w
DB
−→
is equal to the type operator T
w
and
:
1.
For any simple morphism f
B
,
the arrow component T
w
is defined by
:
A
−→
∈
f and ψ
T
w
(f )
T
w
(f )
=
{
id
R
:
R
→
R
|
R
}
