Database Reference
In-Depth Information
Then, for a mapping-interpretation (an R-algebra)
α
such that it is a model (i.e.,
satisfies all the constraints in
Σ
A
) of the source schema
A
=
(S
A
,Σ
A
)
and defines
α
∗
(S
A
)
its database instance
A
=
={
α(r
i
)
|
r
i
∈
S
A
}
and, analogously, a model of
with
B
=
α
∗
(S
B
)
such that it satisfies also the schema mapping
B
M
AB
. Con-
sequently,
(A,B)
∈
M
AB
by fixing Tarski's interpretation for the functional symbol
f
1
in this SOtgd (de-
noted by
I
T
(f
1
)
). We obtain the
function α(q
1
)
Inst(
M
AB
)
and
α
satisfies the SOtgd of the mapping
=
α(v
1
)(α(q
A,
1
))
:
α(
Emp
)
→
α(
SelfMgr
)
, such that for any tuple
a
∈
α(
Emp
)
:
α(q
A,
1
)
=
=
;
a
a
if
a
I
T
(f
1
)(a)
otherwise.
Relation
α(r
q
)
is equal to the image of
α(q
A,
1
)
, so that for any
a
∈
α(r
q
)
we have
α(v
1
)
=
a
a
if
a
∈
α(
SelfMgr
)
;
otherwise.
Hence, the function
α(v
1
)
is an injection for the inclusion
α(r
q
)
⊆
α(
SelfMgr
)
.
We consider that every relation has the empty tuple as well; an empty relation in
this case is considered as a relation that has only the empty tuple.
Remark
Note that in the case when
(B,D)
∈
Inst(
M
BD
)
as well (i.e., when the
schema mapping
M
BD
is satisfied by
B
and
D
), the resulting mapping
M
AD
is
also satisfied and, as a consequence, the function
α(v
1
)
is an
inclusion
.If
M
AD
is
not
satisfied then
α(v
1
)
is not an inclusion. Consequently, the function
α(v
1
)
distin-
guishes when the mapping is satisfied or not, while the function
α(q
A,
1
)
represents
the computation of the query
Emp
(x
e
)
∧
(x
e
.
=
f
1
(x
e
))
for the instance-database
A
(so that
α(r
q
)
=
Emp
(x
e
)
∧
(x
e
.
=
f
1
(x
e
))
A
is the image of the function
α(q
A,
1
)
)
that corresponds to the left-hand side of the implication of the operad's operation
q
1
.
The example above introduced the important properties of mapping-interpreta-
tions (R-algebras) and the way of recognizing when they are
models
of the schema
mappings (that is, when they satisfy the schema mappings). Thus, we can formalize
this property of mapping-interpretations by the following corollary:
Corollary 4
be a schema mapping
.
Then
,
for a given R-
algebra α that is a mapping-interpretation
,
the function α(v
i
) of each operad's
operation q
i
=
Let
M
AB
:
A
→
B
v
i
·
q
A,i
∈
M
AB
=
MakeOperads(
M
AB
) is an injection iff the map-
α
∗
(
α
∗
(
ping
M
AB
is satisfied by the instances A
=
A
) and B
=
B
)
(
i
.
e
.,
when
(α
∗
(
),α
∗
(
A
B
))
∈
Inst(
M
AB
)
).
That is
,
M
AB
is satisfied iff for each q
i
=
v
i
·
q
A,i
∈
O(r
1
,...,r
k
,r
B
) in
M
AB
,
with q
A,i
∈
O(r
1
,...,r
k
,r
q
)
,
v
i
∈
O(r
q
,r
B
)
,
it holds
that the image of the function f
=
α(q
A,i
)
:
R
1
×···×
R
k
→
α(r
q
)
(
where for each
ar(r
i
)
1
≤
i
≤
k
,
R
i
=
U
\
α(r
i
) if the place symbol (_ )
i
∈
q
i
is preceded by negation
¬
⊆
operator
;
α(r
i
) otherwise
)
is a subset of α(r
B
)
,
i
.
e
.,
im(f )
α(r
B
)
.
Proof
Let
φ
Ai
(
x
)
⇒
r
B
(
t
)
be an implication
χ
in the normalized SOtgd of the map-
ping
M
AB
and
t
a tuple of terms with variables in
x
=
x
1
,...,x
m
. Then, based
on Definition
11
for the mapping-interpretation
α
,wehave:
