Database Reference
In-Depth Information
Let Φ be the SOtgd equal to EgdsToSOtgd(Σ
egd
A
Corollary 6
) and Ψ be the SOtgd
equal to TgdsToConSOtgd(Σ
tgd
A
(S
A
,Σ
eg
A
∪
Σ
tgd
A
) for a given schema
A
=
)
.
Then
,
0
for every R-algebra α
,
the information flux Flux(α, MakeOperads(
{
Φ
∧
Ψ
}
))
=⊥
is empty
.
Proof
From Definition
7
, the integrity constraints are representable by an atomic
mapping
.
From the fact that each abstract operad-oper
a
t
io
n
q
i
in
MakeOperads(
AA
={
Φ
∧
Ψ
}:
A
→
A
{
Φ
}
)
and
in
MakeOperads(
(
0
,
1
)
, we have no free variables on
the right-hand side of these implications, so that
x
i
in Definition
13
for the infor-
mation flux is empty and
Va r (
T
AA
)
{
Ψ
}
)
is of the form
e
⇒
r
=∅
, so that, from Definition
13
of its kernel,
0
0
0
Δ(α,
T
AA
)
=⊥
and hence
Flux(α,
T
AA
)
=
T
⊥
=⊥
is empty.
Note that this corollary confirms that, for any database schema
A
=
(S
A
,Σ
A
)
Σ
tg
A
, we can define the integrity-constraints mapping, as it was
demonstrated by Example
12
, by a schema mapping
Σ
eg
A
∪
where
Σ
A
=
AA
={
Φ
∧
Ψ
}:
A
→
A
(where
Φ
is the SOtgd equal to
EgdsToSOtgd(Σ
egd
A
)
and
Ψ
is the SOtgd equal
to
TgdsToConSOtgd(Σ
tgd
A
A
=
{
r
}
∅
)
and
(
,
)
) and, consequently, by equivalent
operads-mapping
T
AA
=
{
}
∪
{
}
:
A
→
A
MakeOperads(
Φ
)
MakeOperads(
Ψ
)
with the
empty
information flux.
We have no mapping from
A
into other schema mappings, so that this integrity-
constraint mapping does not participate in any significant composition with other
mappings in a given database mapping system. Moreover, from the fact that the in-
formation flux of composed mappings is equal to the intersection of the information
fluxes of all atomic arrows which compose this mapping, such a composed mapping
which contains an atomic integrity-constraint mapping will always have an empty
flux. Consequently, the role of the integrity-constraint mappings is only “logical”,
used to express the integrity constraints for schemas, and to verify if for a given
mapping-interpretation
α
in Definition
11
they are satisfied (as it was specified by
Corollary
4
). The extension of the database mapping systems with the integrity-
constraints mappings will not modify its original semantic structure, but in this ex-
tended framework not only the inter-schema mappings but also the schema integrity
constraints will be the “first objects” of big data integration theory and will be pre-
sented in a uniform elegant manner.
Let us consider the following example, in the case when the schema mappings
are satisfied by a given mapping-interpretation
α
:
Example 15
Let us consider Example
2
for composition of the atomic mappings
M
AB
:
A
→
B
,
M
BD
:
B
→
D
,
M
AC
:
A
→
C
and
M
CD
:
C
→
D
, where
M
AB
=
∀
x
c
Takes
(x
n
,x
c
)
⇒
Takes1
(x
n
,x
c
)
,
x
n
∀
x
c
Takes
(x
n
,x
c
)
⇒
Student
x
n
,f
1
(x
n
)
,
∀
x
n
∀
M
BD
=
∀
y
Takes1
(x
n
,x
c
)
x
n
∀
x
c
∀
