Database Reference
In-Depth Information
α
∗
(
Proof
The interpretation of a given schema
A
is an instance
A
=
A
)
of this
database, that is, an object in
DB
; for every interpretation
α
∗
(
0
.
A
∅
)
=⊥
From the monoidal property we have the equation
A
⊕
A
∅
=
A
in the al-
S
Alg
. By the above homomorphism,
α
∗
(
gebra
=
)
is equal to the isomorphism
in
DB
,
α
∗
(
⊕
)
=
rn
, so that
α
∗
(
A
⊕
A
∅
)
=
α
∗
(
A
)
rn
α
∗
(
A
∅
)
=
A
rn
⊥
0
A
.
A
A
∅
=
A
From the monoidal property we have the equation
†
in the algebra
S
Alg
. By the above homomorphism,
α
∗
(
†
)
=
‡, so that, based on Definition
27
,
α
∗
(
α
∗
(
)
‡
α
∗
(
0
, and from Corollary
13
A
‡
0
A
†
A
∅
)
=
A
A
∅
)
=
A
‡
⊥
⊥
A
is an
isomorphism in
DB
.
(S
A
,Σ
A
)
be the nonempty database schema where
S
A
is a set of rela-
tional symbols with a given list of attributes and
Σ
A
=
Let
A
=
Σ
tgd
Σ
egd
A
A
∪
is the set of its
integrity constraints which can be an empty set as well.
We can represent this schema database by a logic-graph composed by the
integrity-constraints mapping
AA
:
A
→
A
(“truth” arrow), where
A
=
(
)
is the “FOL-identity” schema (introduced in Sect.
2.2
and the sketch's
graphs by Definition
14
) such that for every
α
,
α
∗
(
{
r
}
,
∅
A
)
={
α(r
)
}={
I
T
(r
)
}=
R
=
(identity relation from Sect.
1.3
, Definition
11
, proof of Lemma
2
and Example
12
).
For the mapping-interpretations
α
∗
which are also the models of a given database
schema (or database inter-schema mapping), the information flux of
AA
is empty
0
(from Corollary
6
) as well, i.e.,
α
∗
(MakeOperads(
AA
))
=⊥
={⊥}
(
⊥={}
is an empty relation with the empty tuple
) and hence there is no information
transferring which explains their pure “logical” semantics in the database mapping
systems. We recall that when we pass from a “logic-mapping” graph
G
into the
sketch-graph
G
then we substitute “logical” edge
M
∈
G
by an algebraic edge
MakeOperads(
M
)
∈
G
.
Proposition 15
Given a nonempty database schema
A
=
(S
A
,Σ
A
) with a set of
integrity constraints Σ
A
=
Σ
tgd
A
∪
Σ
egd
A
,
we define the sketch-graph G(
A
) derived
from this schema
,
composed of the vertices
A
A
=
{
r
}
∅
and
(
,
)
(
more precisely
,
by S
A
and
{
r
}
,
respectively
),
and the following edges
:
•
Identity mapping-operad for
A
,
M
AA
={
1
r
|
r
∈
S
A
}∪{
1
r
∅
}:
A
→
A
,
and for
the “FOL-identity” schema
A
,
M
A
A
={
1
r
,
1
r
∅
}:
A
→
A
.
•
Integrity-constraint mapping-operad
:
-
If Σ
eg
A
=∅
and Σ
tg
A
=∅
,
then Φ is equal to
EgdsToSOtgd
Σ
eg
A
∧
TgdsToConSOtgd
Σ
tg
A
;
If Σ
egd
and Σ
tgd
,
then Φ is equal to EgdsToSOtgd(Σ
egd
A
-
A
=∅
A
=∅
)
;
If Σ
eg
A
=∅
and Σ
tg
A
=∅
,
then Φ is equal to TgdsToConSOtgd(Σ
tgd
A
-
)
;
Hence
,
we define the mapping-operad
T
AA
=
AA
)
:
A
→
A
AA
={
}
MakeOperads(
where
Φ
.
