Database Reference
In-Depth Information
M
AD
:
A
→
D
Analogously, the composition
, obtained by the composition of
M
AC
and
M
CD
, can be equivalently represented by the SOtgd:
(ii)
⇒
Teaching
(f
2
(x
n
,x
c
),x
c
)))
.
Clearly, by using the semantics of inter-schema mappings defined in [
5
], we obtain
that these two composed mappings
∃
f
2
(
∀
x
n
∀
x
c
(
Takes
(x
n
,x
c
)
M
AD
are different.
As we mentioned, both SOtgds (i) and (ii) are not strict mappings from
M
AD
and
A
into
D
. Since
f
1
in (i) encapsulates the data from
B
as well, it substantially remains a
mapping from
A
and
B
into
D
. Analogously,
f
2
in SOtgd in (ii) encapsulates the
data from
C
as well, so it substantially remains a mapping from
A
and
C
into
D
.The
differences from the intermediate databases
B
and
C
explain why these two SOtgds
are different (i.e., are not logically equivalent).
In spite of that, the strict information that is transferred (only) from
A
into
D
is
equal for SOtgds (i) and (ii) and corresponds for each instance
A
of the schema
to
the projection of its relation
Takes
(x
n
,x
c
)
over the attribute
x
c
(see the complete
proof in Examples
6
and
15
). Consequently, in our semantics for composed map-
pings, these two composed mappings have to be equal (differently from the data
exchange framework presented in [
5
]) and hence we need a new formal definition
for this strict inter-schema mapping composition.
A
Consequently, the main difference between the data-exchange framework and
our more general approach is the following: In a data-exchange framework, the
mapping from a source to a target database determines, for a given source database
instance, the target database instance, so that two equal mappings “produce” two
equal solutions for the target database. From this point of view, two equal inter-
schema mappings have to be logically equivalent [
5
].
In our more general framework, each database in a schema mapping system is
relatively independent and can be consistently modified locally, by requiring only
that after this local modification of this database all other instances of the correlated
database schemas in this mapping system have to be (minimally) updated in order
to obtain a new instance of this mapping system where all inter-schema mappings
are satisfied again (as explained in Chap.
7
dedicated to operational semantics for
database mappings).
Consequently, we do not require the strong determination of the target instance
by a given mapping, but only that this target instance
satisfies
the mappings as well,
and hence this general framework satisfies the OWA. Hence in given composed
mappings from a source schema to a given target schema, in order to verify the
equality of two different mappings between these two databases, we are not in-
terested in contributions of the intermediate databases to the target database, but
only in the strict information contribution from a given source database-instance
into the target database-instance. Such an information contribution of the source
database can be only filtered (thus decremented) by intermediate databases and not
incremented. This strict contribution can be represented only at the instance-level
semantics of a composed mapping for a given instance of the source schema and
hence the
equality of two mappings
in our framework can be considered only at this
instance-level
. What is common for both semantic approaches to composition of
