Database Reference
In-Depth Information
The commutativity
T
AA
=
M
A
∅
A
◦
M
AA
∅
comes from the fact that
AA
=
M
A
∅
A
◦
M
AA
∅
=
M
AA
∅
)
and, in this case, from the fact that
Φ
is logically equivalent to
Φ
M
and to
Φ
E
∧
Compose(
M
A
∅
A
,
Φ
M
.
Note that for both obtained mappings
Va r (
M
AA
∅
)
, and hence
they have also empty information fluxes for each mapping-interpretation
α
. That is,
Flux(α,
M
AA
∅
)
=
=
Va r (
M
A
∅
A
)
=∅
0
.
Flux(α,
M
A
∅
A
)
=⊥
will
produce the pair of two trivial mappings as well, that is, we will not produce new
mappings by their decomposition. If we apply the decomposition to the graph in
the previous example, by using the original mappings instead of derived mapping-
operads, we will obtain the following graph:
The decomposition of any trivial mapping (in Example
7
)
M
={
r
∅
⇒
r
∅
}
where
M
A
∅
A
∅
=
Id
A
∅
={
r
∅
⇒
r
∅
}:
A
∅
→
A
∅
is the identity mapping for the
empty schema
A
∅
(see Example
7
).
2.6
Database Schema Mapping Graphs
In this section, we will summarize all the results of this chapter by a formal transla-
tion of the logic theory, of a given database mapping system, into a graph's formal-
ism which represents a categorification provided in the next two chapters, that is, a
translation of the logic schema-mapping theory into a small sketch category with a
functorial semantics for its models.
We recall that any tuple-generating
mapping
from a schema
A
B
into a schema
is represented by a set of tgds
∀
x
(φ
A
(
x
)
⇒∃
z
(ψ
B
(
y
,
z
)))
where
y
⊆
x
, by taking
out the universal quantification
∀
x
from the head of this tgd logical sentence, that
is, by the
view mapping q
A
(
y
)
y
1
φ
A
(
y
,
y
1
)
with
x
equal to the tuple of all variables in
y
and
y
1
) is a conjunctive query over
the schema
⇒
q
B
(
y
)
where
q
A
(
y
)
(equivalent to
∃
A
∃
and
q
B
(
y
)
(equivalent to
z
ψ
B
(
y
,
z
)
) is a conjunctive query over the
schema
B
. These view-based mappings are interpreted as follows:
•
When a sentence
q
A
(
d
)
is true for a tuple of values
d
then
q
B
(
d
)
has to be a
true sentence as well, so that the information
d
of an instance-database
A
(of the
schema
A
) is “transferred” by this mapping into the instance-database
B
(of the
B
schema
).
