Database Reference
In-Depth Information
In the categorial semantics of mappings and their composition, we are only in-
terested in this strict subset of information that is mapped (only)
from the source
database.
Differently from the data-exchange settings, we are not interested in the 'mini-
mal' instance
B
of the target schema
B
such that, for a given instance (model)
A
A
of the source schema
,
(A,B)
is an instance of the SOtgd of the schema mapping
M
AB
. In our more general framework, we do not intend to determine 'exactly',
'canonically' or 'completely' the instance of the target schema
B
(for a fixed in-
stance
A
of the source schema
). Such a setting is more general and the target
database is
only partially
determined by the source database
A
: the other part of
this target database can be, for example, determined by another database
A
C
(that
is not any of the 'intermediate' databases between
), or by the software
programs which update the information contained in this target database
A
and
B
.
In other words, the Data-exchange (and Data integration) settings are only spe-
cial particular cases of this
general framework
of database mappings, where each
database can be mapped from other sources, to map a part of its own information
into other targets, and to be locally updated as well.
This last feature of the local update of a given database, in the database-mapping
systems, will be considered in full detail in Chap.
7
dedicated to Operational se-
mantics for database mappings. The process of an update of a database-mapping
system begins with one local update of a given database and, after that, this update
has to be propagated through the whole network of inter-mapped databases in order
to guarantee that every (atomic) schema mapping in this network has to be satisfied
at the end of this update-processing.
In our case, two equal compositions are not necessarily logically equivalent for-
mulae, and their equality is considered only at the instance-database level.
Let us suppose that
B
M
AB
1
:
A
→
B
1
is an atomic mapping, based on the set of
tgds that 'transfer', for a given instance
A
of
A
,theset
S
of views from
A
to
B
1
(based on the left-hand side of implications in the tgds), and let
M
B
1
B
n
:
B
1
→
B
n
be a (possibly non-atomic) mapping composed of a set of atomic mappings
M
B
i
B
i
+
1
:
B
i
→
B
i
+
1
,for1
2. Let
S
i
be the set of views of the
i
th
atomic mapping for a given instance
B
i
of the schema
≤
i
≤
n
,
n
≥
B
i
(based on the left-hand
side of implications in the tgds of this atomic mapping). This second (possibly com-
posed) mapping
M
B
1
B
n
filters the information contained in the set of all possible
views
TS
that can be obtained from the set of relations in
S
.Aview
v
∈
TS
will
∈
be propagated into the target database
B
n
if
v
TS
i
for all intermediate atomic
≤
≤
mappings, i.e., for 1
n
.
Consequently, the
strict semantics
of a composed mapping
i
M
B
1
B
n
◦
M
AB
1
:
A
→
B
n
represents the subset of all views in
TS
that are transferred to the target
database
B
n
, that is, it is the intersection
TS
∩
TS
1
∩···∩
TS
n
and hence equal to
TS
∩
TS
R
where
TS
R
=
TS
1
∩···∩
TS
n
is the strict semantics of the composed
mapping
M
B
1
B
n
.
Based on these considerations, we are now able to define an abstraction of this
transferred information from a given source to a target schema, called
information
flux
.
