Database Reference
In-Depth Information
From Corollary
35
,
T
=
Γ
for the poset
W
=
Υ
, so that
F
(W)
=
F
(
Υ
)
={↓
R
|
R
∈
Υ
}={
T
{
R
}|
R
∈
Υ
}={
TA
|
R
=
(A),A
⊆
Υ
}={
TA
|
A
∈
Ob
DB
}=
Ob
DB
sk
. Consequently, we obtain that
L
alg
=
F
0
,
Υ
,
¬
(
Υ
),
⊆
,Γ
∩
,Γ
∪
,
⇒
,
⊥
is equal to the Heyting algebra
F
(W)
(for
W
Υ
) in Corollary
34
. Notice that
the operations of negation and disjunction are not standard (set-complement and
set-union). Instead, they are modified by the power-view operator
T
and hence are
(
multi
)
modal
operators.
=
Notice that we can consider each closed object
TA
∈
Ob
DB
sk
as a representative
element of the equivalence class
[
A
]={
B
∈
Ob
DB
|
TB
=
TA
}
, with
A
TA
↓
. Because of
that, the
DB
algebraic logic
L
alg
(of the weak monoidal topos) is factored by the
equivalence relation equal to the isomorphism of objects in
DB
category.
From the fact that the bottom element 0 is equal to the empty relation
(T A)
↓
(A)
∈
F
(
Υ
)
, that is,
A,T A,
{
(T A)
}
,
↓
(A)
∈[
A
]
⊥
,in
Kripke semantics of this logic for each formula
φ
, from the fact that
M
|=
⊥
φ
,for
the set
A
=
φ
={
a
∈
Υ
|
M
|=
a
φ
}∈
F
(
Υ
)
of this logic, we obtain a standard
result
Ob
DB
sk
.
Consequently, the logic of the weak monoidal topos determines the databases
up to the behavioral (observational) equivalence (which is an isomorphism of the
simple objects in the
DB
category), that is, as a set of all possible views (all pos-
sible observations) of a given database (associated to a given logical formula
φ
).
From this point of view, this weak monoidal topos logic is an abstract-behavior
logic which does not determinate a model of a given database schema, but the ab-
stract behavior of such a model, defined by the set of all observations (views) which
can be obtained by using a
Σ
R
(SPJRU) Codd's relational algebra.
This result is coherent with the fact that for each model (instance-database)
A
⊥∈
A
TA
∈
=
α
∗
(
A
A
, the closed object
TA
is an “abstract behavioral
model” of a database
A
, represented by the set of all observations (i.e., views) which
can be obtained from this database. From the fact that both matching and merging
database-lattice operators are based on the power-view closure operator
T
,wehave
seen that the logical connectives of the weak monoidal topos are closed by
Γ
)
of a database schema
=
T
,
so that the obtained database logic is abstract in the same way.
From the fact that for each different database set of individual symbols
dom
(and
universe
U
=
∪
={
}
for
marked null values, used in Data Integration with incomplete information as pre-
sented in Sect.
4.2.4
) we obtain a different
DB
category with different total objects
Υ
, the weak monoidal topos logic corresponds to this class of all
DB
categories:
dom
SK
with infinite set of Skolem constants
SK
ω
0
,ω
1
,...
Definition 64
We define the WMTL (weak monoidal topos database logic) by the
following class of complete Heyting algebras in Proposition
71
:
HA
DB
={
L
alg
|
for every
dom
}⊂
HA
.
