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= in OP
h q E Th OP
q E Tf OP
Tf in ) OP
Consequently, its information flux is
T e (f in ;
in
( from the fact that in h q = in OP
h q E = Th OP
in h q =
q E = h q E
and from Corollary 15
h q E f in = Tf OP
= h q E =
= β 1 and hence T e (f in ;
Tf in ) OP
)
( from Example 26 )
in
=
β 1 .
Let us now show how the query-rewriting algorithms can be represented in DB
category. First define the set of all queries over canonical model of the global
schema, which is represented in DB by the set of query-morphisms S can( I ,D) =
{
,D)) such that h q
.
Then the query-expansion algorithm can be represented by the function Exp Q :
S can( I ,D)
h q |
h q ={
f,q
}∈
DB (can(
I
,D),Tcan(
I
T w (can(
I
,D))
}
DB (ret( I ,D),Tret( I ,D)) such that for each query-morphism over
global database h q ={
S can( I ,D) we obtain an equivalent query-morphism
over retrieved database Exp Q (h q )
f,q
}∈
={
f E ,q }∈
DB (ret(
I
,D),Tret(
I
,D)) , with
im(f ) and hence h q =
= h q E .
=
im(f e )
Exp Q (h q )
4.2.4 Fixpoint Operator for Finite Canonical Solution
The database instance can(
G =
(S G G ) can be an infinite database (see Example 27 bellow) and hence impos-
sible to materialize for the real applications. Thus, in this subsection, we introduce
a new approach to the canonical instance-database, closer to the data exchange ap-
proach [ 4 ]. It is not restricted to the existence of query-rewriting algorithms and
hence can be used in order to define a Coherent Closed World Assumption for data
integration systems also in the absence of query-rewriting algorithms [ 12 ].
The construction of the finite canonical instance-database for
I
,
D
) which is a model of the global schema
that does not
satisfy all the integrity constraints of the logical theory for a data integration sys-
tem
G
)
described in [ 2 ]. The difference lies in the fact that in the construction of this
revisited canonical database for a global schema (which is not a model of the
global schema), denoted by can F (
I = G
,
S
,
M
is similar to the construction of the canonical model can(
I
,
D
I
,
D
) , the fresh marked null values (from the
set SK
of Skolem constants) are used instead of terms involving
Skolem's functions in the SOtgds obtained from the foreign key constraints in Σ tgd
={
ω 0 1 ,...
}
G ,
following the idea of construction of the restricted chase of a database described
in [ 8 ]. Thus, for a given universe
SK , we permit to use these Skolem
constants for primary keys as well, differently from standard relational databases
where we have only one NULL value that cannot be used for primary key attributes
of a relation. Here it is possible just because we have the marked null values ω i ,
so that we can use them also for the primary-key attributes because ω i =
U =
dom
ω j for all
i = j .
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