Database Reference
In-Depth Information
of the instance-databases of
B
M
P
n
−
1
M
P
1
M
P
n
α
∗
===
⇒
α
n
====
⇒
α
n
−
1
···===
⇒
α
1
where P
i
is a sequential composition of
RA
arrows in the Application Plan consid-
ered as a program of categorial machine
M
RC
which executes the inserting of the
tuples t
i
⊆
M
AB
⇒
∈S
B
will be simply denoted by α
∗
===
α
1
,
B
t
i
in
for each (V
i
,t
i
)
M
BA
,
so that α
1
(
where
M
BA
=
M
P
n
◦···◦
P
1
is associated to the arrow
B
) is the fi-
nal updated database instance of
B
.
Hence
,
we extend these changes to the mapping-
interpretation
(
a functor
)
α
1
:
Sch
(G)
→
DB
such that for each
C
=
B
in V
G
,
α
1
(
α
∗
(
)
.
This new mapping-interpretation α
1
∈
C
)
C
Int(G) satisfies a mapping
M
AB
:
A
→
B
,
t
i
=
if for each (V
i
,t
i
)
∈S
B
t
i
,
but not necessarily of all mapping system G
.
is empty then there is no a transition
α
∗
==
M
AB
⇒
α
1
; other-
Proof
Note that if
S
B
,if
t
i
=
t
i
then this set of tuples in
t
i
for a
wise, for each
(V
i
,t
i
)
=
(α(r
i
),t
i
)
∈S
B
given mapping
M
AB
is just the set of tuples for which this mapping is not satisfied.
Thus, by inserting them into the relation
α(r
i
)
(i.e., view
V
i
), this mapping will be
satisfied, and hence after all insertions in
α
∗
===
M
AB
⇒
α
1
,
α
1
will satisfy
M
AB
.This
, so that
α
1
process is done for each outgoing arrow from the vertex
B
will satisfy
all outgoing arrows from the vertex
A
(the arrow
M
AB
is one of them). However,
if
t
i
⊂
(for example, the primary-key constraints for
the relations can be broken by inserting all tuples in
t
i
) then
t
i
for at least one
(V
i
,t
i
)
∈S
B
M
AB
is not satisfied,
and hence the interpretation
α
1
B
is not necessarily a model of
.
M
AB
⇒
Note that, based on this proposition, the transition
α
∗
====
α
1
(caused by
a number of insertions of tuples in tables of a database
A
) denotes the transition
DB
Sch
(G)
(which is a model for all
vertices (i.e., the database schemas) in
V
G
, but it does not necessarily satisfy all
inter-schema mappings in
E
G
) into a new mapping-interpretation
α
1
∈
from a mapping-interpretation
α
∗
∈
Int(G)
⊆
Int(G)
⊆
M
AB
⇒
DB
Sch
(G)
. Hence, this transition
α
∗
===
α
1
does not necessarily satisfy
all
inter-
schema mappings in
E
G
. In order to guarantee that the insertion of the new tuples in
the relation of
B
does not make its integrity constraints inconsistent, we can adopt a
strategy to divide such relations into two relations (with the same attributes: the first
one can be “locally” updated by the legacy software for this schema by applying to
it all necessary integrity constraints; the second “imported” relations will be used
only to collect the tuples from another databases in
G
and for this relation we do
not apply any integrity constraint, so that the insertion of the new tuples provided
by the inter-schema mapping into this database will not render it inconsistent.
In this way, each schema in a database-mapping system can have two partitions:
one original (or “local”) with all integrity constraints and updated only by the local
legacy programs that fully respect the integrity constraints, and another “imported”
partition composed by only relations used to receive the information from another
