Database Reference
In-Depth Information
This corollary is a direct proof that the power-view database operator
T
,intro-
duced in Sect.
1.4.1
, represents the
observational
point of view for the instance-
databases.
In fact, a conjunctive query is an implication from the body of a query
q
i
(
x
)
∈
L
A
with relational symbols in
{
r
i
1
,...,r
ik
}
(a subset of relational symbols in a schema
A
) into the head
r
q
(
x
)
, that is, a formula
q
i
(
x
)
⇒
r
q
(
x
)
that can be represented
equivalently by an operad's operation
q
A,i
=
O(r
i
1
,...,r
ik
,r
q
)
of the mapping
M
:
A
→
r
q
(
x
))
.
Thus, for a given R-algebra
α
that is a mapping-interpretation of operads (Defi-
nitions
10
and
11
) such that the instance
A
T
A
with an SOtgd
∀
x
(q
i
(
x
)
⇒
α
∗
(
=
A
A
)
of the schema
satisfies the
SOtgd of the mapping
above,
Out(q
i
(
x
))
=
q
i
(
x
)
A
is the image of the function
α(q
A,i
)
:
α(r
i
1
)
×···×
α(r
ik
)
→
α(r
q
)
ob-
tained from the operad's operation
q
A,i
=
M
={∀
x
(q
i
(
x
)
⇒
r
q
(
x
))
}:
A
→
T
A
O(r
i
1
,...,r
ik
,r
q
)
, that is,
Out
q
i
(
x
)
=
α(q
A,i
)(
d
1
,...,
d
m
)
|
≤
j
≤
k
=
α(r
q
).
d
j
∈
α(r
ij
),
for 1
α
∗
(
Consequently, the AOT without side-effects of the instance database
A
)
is
the observational point of view for the basic database mappings and closely interre-
lated with mapping-operads.
Note that a single view-mapping at instance-database level can be defined as a T-
coalgebra
f
=
A
α
∗
(MakeOperads(
TA
that, obviously,
is not
a function but a set of functions. These arguments will be analyzed in detail
after the definition of the database category
DB
, and hence it will be demonstrated
that
f
is a morphism in such a database category.
=
M
))
={
α(q
A,i
),q
⊥
}:
A
→
2.4.3 Strict Semantics of Schema Mappings: Information Fluxes
As it was explained previously, the SOtgds represent the logical language syntax
for the mapping compositions and use the existentially quantified functional sym-
bols whose interpretation introduces the data values not contained in the active data
domain of the source schema.
In fact, these functional symbols, introduced in the new algorithm (defined in
Sect.
2.3
) for composition of mappings, are used in order to replace the data con-
tained in 'intermediate' databases (the hidden databases between the source and
target database schemas) in order to guarantee the logical equivalence of
composed
SOtgd and
the set
of SOtgds used in such a composition for the determination of the
target database. This fact in the data-exchange setting guarantees that each query
over a target database will give the same resulting view regardless of whether we
are using the set of intermediate mappings from the source into this target database,
or the single SOtgd of the resulting composition obtained from the algorithm in
Sect.
2.3
.
From this logical point of view, the composed SOtgd represents the
complete
information of all 'intermediate' databases used in this composition and includes
the strict subset of information of the source database that is mapped into the target
database.
