Database Reference
In-Depth Information
This new mapping-interpretation α
1
∈
Int(G) satisfies
M
BA
:
B
→
A
if for each
,
t
i
=
(V
i
,t
i
)
∈S
B
t
i
,
but not necessarily of all mapping system G
.
OP
BA
⇒
M
α
1
; otherwise,
for each
(V
i
,t
i
)
∈S
B
,if
t
i
=
t
i
then this set of tuples in
t
i
for a given mapping
is empty then there is no transition
α
∗
===
Proof
Note that if
S
B
M
BA
is just the set of tuples for which this mapping is not satisfied. Thus, by eliminating
them from the view
V
i
this mapping will be satisfied and hence after all deleting
in
α
∗
====
OP
BA
⇒
M
α
1
,the
α
1
M
BA
. However, if
t
i
⊂
will satisfy
t
i
for at least one
(V
i
,t
i
)
∈S
B
then
M
BA
is not satisfied. This process is done for each ingoing arrow
. The interpretation
α
1
into the vertex
A
is a model of
A
and
B
, but not necessarily
of all mapping system
G
.
OP
BA
⇒
M
Note that, based on this proposition, the transition
α
∗
===
α
1
(caused by a
number of deletions of tuples in tables of a database
A
) denotes the transition from
a mapping-interpretation
α
∗
∈
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
∈
Int(G)
⊆
DB
Sch
(G)
which
Int(G)
⊆
is a model for all vertices (or database schemas) in
V
G
.
Remark
Notice that the accepted tuples
t
i
to be deleted can be a strict subset of
the “exact” set
t
i
because the local DBMS of the schema
can decide to protect
from deleting its own tuples managed by its local legacy application during such
a backward chaining. Obviously, if
t
i
⊂
B
t
i
than after the deleting of
t
i
from
B
,the
mapping
M
BA
will remain unsatisfied.
OP
BA
⇒
α
1
does not necessarily satisfy all inter-schema
mappings in
E
G
. Consequently, the process of the updates propagates backward
(w.r.t. the directed graph
G
of a database mapping system) from
M
Hence, this transition
α
∗
===
A
B
into
, and then
can continue to propagate backward from
to another database schemas in
E
G
.
This backward propagation in
G
can be equivalently seen as a forward propa-
gation in the inverted graph
G
OP
, and this is a reason that we are using the label
B
OP
BA
⇒
M
OP
BA
in the “atomic” transition
α
∗
===
α
1
, where the mapping
M
M
BA
:
B
→
A
OP
is represented by the edge
M
BA
:
A
→
B
.
OP
BA
⇒
M
α
1
as a
DB-atomic transition
of
an LTS, based on the opposite graph
G
OP
, which represents the complete process
of a backward chaining. In the case when the initial model of database mapping
system
G
contains only finite extensions of all database schemas in
G
, the complete
process will always produce a finite LTS tree (the branching is caused by a number
of ingoing inter-schema-mappings in a given schema), also when
G
is not an acyclic
graph: it hold from the fact that process of deleting tuples from finite databases
cannot be infinite.
Hence, we may see the transition
α
∗
====
