Database Reference
In-Depth Information
Based on Definition
17
, for any single
query q
i
(
x
i
)
over the global schema
G
with relational symbols in
a tuple of
variables, we can define the lifted query
q
L
(
x
i
)
equivalent to the conjunctive for-
mula
q
i
(
x
i
)
{
r
i
1
,...,r
ik
}⊆
S
G
, where
x
i
=
x
1
,...,x
k
∧
Va l(x
1
)
∧···∧
Va l(x
k
)
and hence with the schema mapping operad's
operation
q
i
=
O(r
q
,r
q
)
is
the identity operation. This lifted query can be represented by an atomic mapping
v
i
·
q
G,i
where
q
G,i
∈
O(r
i
1
,...,r
ik
,r
q
)
and
v
i
=
1
r
q
∈
MakeOperads
∀
x
i
q
i
(
x
i
)
∧
h
V
(x
1
)
.
1
∧···
M
GG
={
q
i
,
1
r
∅
}=
=
∧
h
V
(x
k
)
.
1
⇒
r
q
(
x
i
)
:
G
→
G
=
where
r
q
∈
S
G
is a new introduced relational symbol for this query with
k
=
ar(r
q
)
=|
(the number of variables in the tuple
x
i
) and
h
V
is the functional
symbol of the characteristic function of the built-in unary pr
ed
icate
Va l
, so that for
every Tarski's interpretation
I
T
we obt
ai
n the fixed function
h
V
=
x
i
|
I
T
(h
V
)
:
U
→
2
such that for each
d
∈
U
=
dom
∪
SK
,
h
V
(d)
=
1iff
d
∈
dom
; 0 otherwise.
Analogously, this lifted query over
G
F
can be represented by an atomic mapping
G
F
G
F
=
q
i
,
1
r
∅
=
MakeOperads
∀
x
i
q
i
(
x
i
)
∧
h
V
(x
1
)
.
1
∧···
M
=
∧
h
V
(x
k
)
.
1
⇒
r
q
(
x
i
)
:
G
F
→
G
F
=
where
r
q
∈
S
G
F
is a new introduced relational symbol for this query.
For a given mapping interpretation
α
with
α
∗
(
G
)
=
can(
I
,D)
and hence
α
∗
(
G
)
=
T can(
,D)
,
α
r
q
=
α(r
q
)
=
q
i
(
x
i
)
∧
I
Va l(x
k
)
can(
I
,D)
Va l(x
1
)
∧···∧
q
i
(
x
i
)
Va l(x
k
)
can
F
(
I
,D)
.
=
∧
Va l(x
1
)
∧···∧
Hence, for this R-algebra
α
we obtain the morphism
α
∗
(
M
h
q
={
f,q
⊥
}=
)
:
can(
I
,D)
→
T can(
I
,D)
GG
and
h
q
F
={
f
F
,q
⊥
}=
α
∗
(
M
G
F
G
F
)
:
can
F
(
I
,D)
→
T can
F
(
I
,D)
, with the func-
tions:
f
:
r
i
1
can(
I
,D)
×···×
r
ik
can(
I
,D)
→
α(r
q
),
α
r
q
.
f
F
:
r
i
1
can
F
(
I
,D)
×···×
r
ik
can
F
(
I
,D)
→
Consequently, we obtain that the images of the functions
f
and
f
F
are equal to the
same computed certain query answer, that is,
im(f )
α(r
q
)
α(r
q
)
,so
that the following diagram in
DB
(on the right), based on the T-coalgebra homo-
morphism
f
in
:
(can
F
(
I
,D),h
q
F
)
→
(can(I,
D
),h
q
)
, commutes.
=
im(f
E
)
=
=
