Database Reference
In-Depth Information
0
otherwise,
=
T
{
π
x
i
e
[
(
_
)
j
/r
i,j
]
1
≤
j
≤
k
A
|
q
i
=
v
i
·
q
A,i
=
(e
⇒
(
_
)(
t
i
))
∈
1
≤
i
≤
m
such that
y
i
=∅;⊥
M
AB
,q
i
∈
O(r
i
1
,
1
,...,r
i
k
,k
,r
i
)
and
y
i
=∅}
,if
α(v
i
)
is an injection for all 1
≤
i
≤
m
0
such that
y
i
=∅;⊥
otherwise,
Flux(α,
M
AB
)
.
Thus, from (i) and (ii),
Flux(α,
M
E
)
=
Flux(α,
M
AB
)
.
The instance mappings
in(f )
and
ep(f )
are well defined because
C
∩
Flux(α,
M
M
)
=
⊆
TC
(i.e., all
relations in
C
are the relations in
TC
as well).
In Example
16
, we considered the decomposition of a mapping between two
database schemas. Let us now consider the case of a decomposition of the integrity-
constraint mapping for a given schema:
A
=
Example 17
Let as consider a schema
(S
A
,Σ
A
)
with the integrity con-
Σ
egd
Σ
tgd
A
straints in
Σ
A
=
A
∪
, which can be represented (see Example
12
)by
a schema mapping
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
)
and
A
=
(
{
r
}
,
∅
)
is an auxiliary schema, and, consequently, by equivalent operads-
mapping
MakeOperads
{
Φ
∧
Ψ
}
:
A
→
A
with the
empty
information flux.
That is, from Corollary
6
, for each mapping-interpretation
α
,
Flux
(α,
T
AA
)
T
AA
=
=
0
, because each operad's operation
q
i
∈
⊥
T
AA
has a form
e
⇒
(
_
)(
0
,
1
)
without
free variables on the right-hand side of this implication and hence, from Defini-
tion
13
,
Va r (
T
AA
)
.
Thus, from the fact that
Va r (
T
AA
)
=∅
=∅
and the Decomposition algorithm, we
obtain that
(Φ
E
,Φ
M
)
=
DeCompose(
AA
)
where
Φ
E
is composed of im
pl
i
ca
-
tions
q
Ai
(
x
i
)
(
0
,
1
)
.
Thus, from the fact that
r
∅
is a tautology (truth propositional letter, Definitions
1
and
8
), each implication in
Φ
E
,
q
Ai
(
x
i
)
⇒
r
and
Φ
M
is composed of implications
(r
∅
∧
ψ
Ai
(
x
i
))
⇒
r
∅
⇒
r
∅
is a tautology and hence
Φ
E
is a
tautology as well, so that we can substitute it by a trivial tautology
r
∅
⇒
r
∅
.
From the fact that
r
is a tautology, each impli
cati
on
(r
∅
∧
ψ
Ai
(
x
i
))
⇒
r
(
0
,
1
)
∅
is equivalent t
o th
e implication
ψ
Ai
(
x
i
)
⇒
r
(
0
,
1
)
, that is, to the implication
⇒
q
Ai
(
x
i
)
r
(
0
,
1
)
(from the fact that
q
Ai
(
x
i
)
is logically equivalent to
ψ
Ai
(
x
i
)
).
Consequently,
Φ
M
is equal to the normalized
Φ
, so that SOtgd
Φ
M
and the original
SOtgd
Φ
are logically equivalent.
Consequently,
Φ
E
∧
Φ
M
is logically equivalent to the original SOtgd
Φ
.
Thus, we obtain a trivial mapping (see Example
7
for the trivial mappings with
the empty database schema
A
∅
)
M
AA
∅
={
Φ
E
}={
r
∅
⇒
r
∅
}:
A
→
A
∅
and
M
A
∅
A
={
Φ
M
}:
A
∅
→
A
.
Consequently, the obtained mapping-operads are
M
AA
∅
=
MakeOperads(
{
Φ
E
}
)
=
{
1
r
∅
}:
A
→
A
∅
, and
M
A
∅
A
=
MakeOperads(
{
Φ
M
}
)
:
A
∅
→
A
. This decompo-
sition can be represented by the following commutative diagram
