Database Reference
In-Depth Information
Remark
We can conclude this section with an important conclusion: The “alge-
braization” of the SOtgds in the database-mapping graph
G
, by transforming its
edges into the mapping-operads of the sketch
Sch
(G)
, specified by embedding
γ
Sch
(G)
, gives to us the possibility to relax the strong logical requirement
that the solutions for such database-mapping system have to be only the
models
of
its logical theory based on SOtgds (requirement which is satisfied by the Strong
Data Integration semantics). In fact, in the algebraic setting for a given mapping-
operad
M
AB
:
A
→
B
:
G
→
, for each
v
i
·
q
i
∈
M
AB
and a given mapping interpretation
(a functor)
α
∗
∈
DB
Sch
(G)
, a logical model is obtained if each
α(v
i
)
is an
injective function. But in such an algebraic setting, all works well also if it is not
an injection. Thus, not necessarily the solutions of the database-mapping system
must always be models, and in such an algebraic setting the Weak Data Integration
semantics is well defined, and we can use it without limitations of logical theories.
In such a weakened setting, we can see the implications used in SOtgds for the
database-mappings as a kind of default rules, and not as axioms of a logical the-
ory. Such a framework offers a great variety for practical employment, differently
from the Strong Data Integration, especially for a very Big Data Integration with
thousands of peer databases. Limitations of the Strong Data Integration are very
clear: it does not permit the independent change of the schemas of peer databases
because the introduction of new integrity constraints in their schemas can invalidate
the model of the whole database-mapping system.
Int(G)
⊆
In the following example, we will demonstrate a significant difference between
Strong and Weak Data Integration semantics:
Example 38
Let us consider the case of Example
27
in Sect.
4.2.4
,usedforthe
two binary relations in a global schema
, and consider now the case when each
of these two relations belongs to two different schemas (peers)
G
A
=
(
{
r
}
,
∅
)
and
B
=
(
{
r
}
,
∅
)
, with the schema mappings
M
AB
={
Φ
}:
A
→
B
and
M
BA
={
Ψ
}:
