Database Reference
In-Depth Information
α
∗
(
where
A
=
A
)
={
α(r
)
}
is the “equality” database with the built-in relation
α(r
)
=
R
=
.
Corollary 15
If there is the canonical model of a GAV data integration system
I
with FK constraints
,
described in this section and expressed by the sketch cate-
gory
Sch
(
)
,
then its functorial model α
∗
:
I
I
−→
DB
(
where the mapping-
interpretation α in Definition
11
is an extension of Tarski's interpretation I
T
)
is
provided by the corresponding arrows
:
Sch
(
)
α
∗
(
M
SG
T
)
f
M
=
:
D
−→
ret(
I
,D),
α
∗
(
M
G
T
G
)
f
in
=
:
ret(
I
,
D
)
−→
can(
I
,
D
),
α
∗
(
T
GA
)
f
Σ
G
=
:
can(
I
,
D
)
−→
A
,
where α
∗
(
α
∗
(
S
)
=
D is the instance of the source database
S
,
G
T
=
G
T
)
=
ret(
I
,D) is the retrieved global database
,
G
=
α
∗
(
G
)
=
can(
I
,D) is the univer-
sal
(
canonical
)
instance of the global schema with the integrity constraints
,
and
•
f
M
={
q
⊥
}∪{
:
×···×
→
|
f
α(r
1
)
α(r
l
)
α(r)
where (r
1
,...,r
l
) is the set of all
relational symbols in the query ρ(r),(ρ(r)
r)
∈
M
}
,
where α(r
i
)
=
r
i
D
,
1
≤
i
≤
l
,
and α(r)
=
r
ret(
I
,D)
.
For each map (ρ(r)
r)
∈
M
(
i
.
e
.,
atgd
∀
x
(ρ(r)(
x
)
⇒
r(
x
))
)
and its op-
erad q
O(r
1
,...,r
l
,r)
,
where e is an operad's expression ob-
tained from the query formula ρ(r)(
x
)
,
we obtain the function
=
(e
⇒
(_)(
x
))
∈
f
=
α(q)
:
α(r
1
)
×···×
α(r
l
)
→
α(r).
•
f
in
={
,
where each function in
r
is an
injection
.
Thus
,
we obtain the monomorphism f
in
:
in
r
:
r
ret(
I
,D)
→
r
can(
I
,D)
|
r
∈
S
T
}
ret(
I
,D)
→
can(
I
,D)
.
•
f
Σ
G
={
g
1
,...,g
n
,q
⊥
}
where each g
i
is
:
