Database Reference
In-Depth Information
The condition that
α
is a model is satisfied if both functions
α(v
1
)
:
α(r
q
1
)
→
R
=
and
α(v
2
)
R
=
).
This condition means that
α
satisfies the implications in
Φ
which are equiv-
alent to the implications
:
α(r
q
2
)
→
R
=
are injections (we have that for every
α
,
α(r
)
=
∀
x,z(s(x,z)
⇒
r(x,f
2
(x,z)))
(they are algebraically expressed by the relations in points 1 and
2 above). As we have examined in Sect.
4.2.4
, the canonical model
can(
∀
x,y(r(x,y)
⇒
s(y,f
1
(x,y)))
and
,D)
in
this case is infinite. The sketch category derived by this data integration system is
denoted by
Sch
(
I
I
)
.
where is the “truth” database
A
=
α
∗
(
A
)
={
α(r
)
}
with the built-in relation
=
α(r
)
.
Thus, in order to pass from the logic framework into the
Σ
R
-algebra framework,
we need to transform the two tgds in points 1 and 2 above into the guarded set of
equations with
Σ
R
terms. Let
R
R
=
ret(
I
,D)
be the exten-
sions of
r
and
s
in the retrieved (from the source database
D
) database
ret(
I
,D)
,
respectively, which are imported into the global database by injective mapping
M
G
T
G
:
G
T
→
G
=
r
ret(
I
,D)
and
S
=
r
. Hence, the final extensions of
r
and
s
are determined by the
solution of the following flattened guarded system of equations with variables (re-
lational symbols) in
X
={
r
X,
1
,...,r
X,
6
}
, such that
ar(r
X,
3
)
=
ar(r
X,
5
)
=
3 and all
other relational symbols have the arity equal to 2:
r
X,
1
≈
r
X,
4
UNION
r
X,
7
;
r
X,
2
≈
r
X,
6
UNION
r
X,
8
;
f
1
nr(
1
),nr(
2
)
;
r
X,
3
≈
EXTEND
r
X,
2
ADD
at(
3
), name
1
AS
name
1
=
r
X,
3
nr(
2
),nr(
3
)
;
r
X,
4
≈
f
2
nr(
1
),nr(
2
)
;
r
X,
5
≈
EXTEND
r
X,
1
ADD
at(
3
), name
2
AS
name
2
=
