Database Reference
In-Depth Information
L
A
,
where a
∈
Act
⊆
Ob
DB
,
is substituted by an atomic
DB
morphism
(
in Definition
α
∗
(MakeOperads(
18
)
f
a
=
M
BC
))
:
→
B
C where α is a mapping-interpretation
(
in Definition
11
)
such that α
∗
(
in a G and
,
in the
case of the forward propagation
,
a
=
Δ(α, MakeOperads(
M
BC
))
.
Then we define the bijection
B
)
=
B is a model of schema
B
DB
:
D
P
S
→
Sub(
DB
) such that for each LTS tree
L
A
∈
D
P
S
,
D
(
L
A
)
=
D
L
A
.
@
@
Consequently
,
[
p
k
]
=[
p
m
]
⇐⇒
p
k
∼
p
m
iff A
=
ass(p
k
)
B
=
ass(p
m
)
and the category
DB
(
L
A
) is equal to the category
DB
(
L
B
)
.
a
B
, derived
Proof
For any LTS 'insertion' tree
L
A
∈
D
P
S
, a transition
A
from an inter-schema mapping
M
AB
:
A
→
B
, can be represented in
DB
by
B
such that
f
a
=
the arrow
f
a
:
A
→
Ta
. In fact, from the definition of
DB
,
a
B)
is mapped into an atomic morphism
f
a
=
α
∗
(MakeOperads(
(A
M
AB
))
,
where
MakeOperads(
M
AB
)
is an arrow in the small category
DB
(
L
A
)
, with
M
AB
))
, so that
f
a
=
a
=
Δ(α, MakeOperads(
Ta
.
Thus, we have the inclusion embedding
In
:
DB
(
L
A
)
→
DB
. Analogous result
holds for 'deletion' (backward propagation) trees as well.
Notice that this result can be obtained from Definition
51
, too.
The equivalence of categories
DB
(
L
A
)
and
DB
(
L
B
)
is obtained from the
@
@
fact that the trees
L
A
=[
p
k
]
and
L
B
=[
p
m
]
are bisimilar, i.e.,
trace equivalent
with
A
=
ass(p
m
)
(all corresponding arrows (transitions) in these
categories that compose the traces have equal information fluxes).
ass(p
k
)
B
=
DB
into a particular subcategory of
DB
and demonstrates that
DB
is an adequate cate-
gory both for denotational and operational semantics for database mapping systems.
Consequently, any tree (i.e., oriented graph) in the
D
P
S
can be mapped by
Example 39
For the database mapping process in Example
36
we obtain the fol-
lowing LTS tree
L
A
in
D
P
S
where
A
=
ass(p
1
)
, and its small category
DB
(
L
A
)
in
Sub(
DB
)
:
where
f
a
i
=
1
,
2
,
3.
That is, it is similar to the transition tree in Example
36
(the tree with the process
variables).
Ta
i
,i
=
