Database Reference
In-Depth Information
repetitive
loops. This control is affected by the universal machine
M
U
DB
during the
execution of a program specified by a graph
G
. Note also that each path of the
process ends with a leaf
A
equal to the instance-database which is the target of the
last action
a
n
(a kernel of information flux transferred into
A
) so that
a
n
⊗
=
A
Ta
n
∩
=
Ta
n
and hence it preserves the meaning of the paths of each process
described after Definition
53
.
TA
Remark
The set of program equations
E
in
P
with the process variables in
X
,pro-
vided by the algorithm above, composes a
guarded system
of equations that has the
unique (maximal fixed point) final coalgebra solution. In this guarded system, the
term
n
(p
1
,...,p
n
)
(equivalent to
p
1
(...(p
n
−
1
p
n
)...)
) is considered as a
flat-
tened
one-depth term with variables in
X
. Note that this process stops the execution
when the administrator stops explicitly the execution by the operation '
nil
' (and re-
sets all modifications produced by this program), or (for default) when the execution
reaches the end or by the fact that the backward/forward process does not modify
the databases further, or because the infinite loops are interrupted (it can happen
only if the graph
G
is cyclic). Each equation in the system of equations
E
, obtained
from this algorithm, has only one depth term on the right sides and the constants
(leafs). Thus, such equations are 'flattened'. Notice that in the case of a node with
n
≥
2 outgoing branches (described in point 3 of this algorithm), the direct trans-
formation of transitions in
Y
into the equations would produce the non-flattened
equation
p
i
=
(a
1
.p
i
+
1
,...,a
n
.p
i
+
n
)
, so, in order to obtain the flattened system
of equations (in point 3 of this algorithm), we introduced the set of extra idle pro-
cess transitions
p
i
ass(p
i
+
k
)
p
i
+
k
for
k
=
1
,...,n
, that
do not change
the states (i.e.,
with
ass(p
i
+
k
)
=
ass(p
i
)
for all
k
=
1
,...,n
).
Let us consider a simple example:
Example 36
Let us consider the simple graph (a database-mapping program)
G
with three edges,
M
AB
3
:
A
→
B
3
, and
with a model
α
∗
of
Sch
(G)
in the case of a Strong Data Integration semantics. Let
us insert a number of tuples into an instance-database
A
=
M
AB
1
:
A
→
B
1
,
M
AB
2
:
A
→
B
2
and
α
∗
(
A
)
such that the
α
1
(
updated database-instance is equal to
A
)
that satisfy all three inter-schema
mappings above (and hence
α
1
is a new model of the database mapping system in
G
), with the new updated databases for the schemas
=
A
α
1
(
B
i
,
B
i
=
B
i
),i
=
1
,
2
,
3,
and
a
1
=
Act
(and analogously, for
a
2
and
a
3
).
Thus, our database process-program
P
, obtained from the
DBprog(G,
Δ(α, MakeOperads(
M
AB
1
))
∈
,α
∗
,
A
A)
algorithm has the set of process variables
p
0
,p
1
,...,p
7
∈
X
, with the graph
of the mapping
ass
:
X
→
S
equal to the set
p
0
,
0
,(p
1
,A),(p
2
,A),(p
3
,A),(p
4
,A),(p
5
,B
1
),(p
6
,B
2
),(p
7
,B
3
)
,
⊥
so that
ass(p
i
)
1
,
2
,
3.
The system of guarded
flattened
equations of the program
P
provided by this
algorithm is:
=
A
for
i
=
1
,...,
4, and
ass(p
i
+
4
)
=
B
i
for
i
=
