Database Reference
In-Depth Information
local modifications of the database schemas, caused by the propagation of the given
initial modification of this particular database schema which becomes the root of an
LT S .
Definition 54
Let
G
=
(V
G
,E
G
)
be a graph of a given schema-database mapping
program (from Definition
14
in Sect.
2.6
and Proposition
16
in Sect.
4.1.2
). Then,
•
For a given vertex
A
∈
V
G
we define a subgraph
G
A
⊆
G
composed of arrows
M
A
_
∈
E
G
such that
dom(
M
A
_
)
=
A
and put them in the list
L
with
n
≥
0
A
elements. Then, for a state
(A,α
∗
)
, i.e., an instance-database
A
α
∗
(
=
A
)
,we
define the set
S
(A,α
∗
)
as follows:
1. Set
S
to be the empty set, the pointer
i
=
0 to elements in
L
, and
α
1
=
α
.
A
2. Set
α
=
α
1
and
i
=
i
+
1. If
i>n
, goto 4.
M
AB
from the list
L
A
. If there exists a transition
3. Take the
i
th arrow
M
AB
⇒
α
∗
====
α
1
(from Proposition
32
) such that
α
1
=
α
∗
then insert into
S
,α
1
)
the tuple
a,(
B
, where
a
=
Δ(α, MakeOperads(
M
AB
))
.Goto2.
4.
S
(A,α
∗
)
=
S
.
V
G
we define a subgraph
G
OP
G
OP
•
For a given vertex
A
∈
A
⊆
composed of
OP
G
OP
OP
arrows
M
_
A
∈
such that
dom(
M
_
A
)
=
A
and put them in the list
L
A
0 elements. Then, for each state
(A,α
∗
)
, i.e., an instance-database
A
with
n
≥
=
α
∗
(
)
, we define the set
S
OP
(A,α
∗
)
as follows:
1. Set
S
to be the empty set, the pointer
i
A
=
0 to elements in
L
, and
α
1
=
α
.
A
=
=
+
2. Set
α
α
1
and
i
i
1. If
i>n
, goto 4.
OP
BA
3. Take the
i
th arrow
M
from the list
L
A
. If there exists a transition
OP
BA
⇒
M
α
∗
====
α
1
(from Proposition
31
) such that
α
1
=
α
∗
then insert into
S
,α
1
)
the tuple
a,(
B
, where
a
=
Δ(α, MakeOperads(
M
BA
))
.Goto2.
4.
S
OP
(A,α
∗
)
=
S
.
Note that if
S
(A,α
∗
)
is not empty (i.e.,
1) then for each element
a,
B
,α
1
∈
S
(A,α
∗
)
or
a,
B
,α
1
∈
S
OP
|
S
(A,α
∗
)
|≥
(A,α
∗
)
DB
Sch
(G
OP
DB
Sch
(G
A
)
)
) is a model of
the mapping-interpretation
α
1
∈
(or
α
1
∈
A
and
α
1
(
α
∗
(
a database
B
A
)
=
A
)
. We recall that in the definition of
S
(A,α
∗
)
,all
mapping arrows in
G
are satisfied (from Proposition
32
) w.r.t. the extensions of
all database schemas in
G
A
.
An analogous result holds for
G
OP
A
A
as well (from Proposition
31
). Based on this
definition, we can define the semantics for the database-mapping processes by the
equations, as follows:
,α
∗
,
DB-process algorithm
DBprog(G,
A)
Input.
A graph
G
=
(V
G
,E
G
)
of a database mapping program, a current model
α
∗
of this system, the vertex
A
A
is a modi-
fication of its instance-database equal to
A
after such initial update which causes
a modification of
α
∗
by setting
α
A
∈
V
G
(a database schema) where
∗
(
A
)
=
A
. This process happens during an
