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and the set of mapping tgds in
M
are satisfied (i.e., the SOtgd
TgdsToSOtgd(
M
)
is
a true sentence) and hence from Proposition
17
α
∗
∈
Mod(G
I
)
is a functorial model
. Such a functor
α
∗
:
of the data integration system
DB
, between the
schema integration level (theory) and the instance-level (which is a model of this
theory) is just a particular R-algebra for operads.
I
Sch
(
I
)
−→
Remark
A solution for a data integration/exchange system does not always ex-
ist (if there exists a failing finite chase, see [
2
,
4
] for more information). How-
ever, if it exists then it is a
canonical universal solution
and in that case there
exists also a mapping model-functor of the theorem above. But there always are
the functors (which are mapping-interpretations)
α
∗
such that
D
α
∗
(
)
which
do not satisfy the mapping
M
SG
. Moreover, in Sect.
4.2.4
, we will consider one
of them for which the global schema will give the same certain answers to the
queries as an infinite canonical model. The results for Data Exchange systems
in [
4
] are restricted to a class of tgds (the
weakly-acyclic
tgds) that ensures that
the models of the target schema are always finite (otherwise the target schema could
not be materialized). So, the problem of dealing with infinite models is not ad-
dressed.
In this topic, we do not make such a restriction so that we are able to consider
the infinite models when
dom
is finite as well: these infinite models are obtained
by using the tgds with existentially quantified right side, by introducing an infinite
sequence
SK
=
S
SK
.
Consequently, if a model is infinite it means that we are using all Skolem constants,
by introducing them in the fixed ordering
ω
0
,ω
1
,ω
2
,...
, in a given set of
cyclic
tgds.
So, this theorem can be abbreviated by:
“given a data mapping graph
={
ω
0
,ω
1
,...
}
of Skolem constants in the universe
U
=
dom
∪
, there is a model-functor
α
∗
:
I
=
G
,
S
,
M
Sch
(
I
)
−→
DB
if there exists a universal (canonical) solution for a correspond-
ing data integration/exchange problem”.
Obviously, we can have other model-functors
α
∗
:
DB
that are dif-
ferent from the canonical one. In what follows, for the GAV system with key con-
straints, we will consider in more details its canonical (universal) model.
Notice that in the complex integration/exchange systems composed of a number
of integrations (sub)systems
Sch
(
I
)
−→
we can use their represen-
tation by a database mapping system graphs in Definition
14
.
In the rest of this chapter, we will consider Data Integration applications by us-
ing GAV mappings with the (PK) and (IC) integrity constraints over the global
schema. Unfortunately, PKs and ICs interact reciprocally so that the (decision)
problem of query answering in this setting becomes undecidable [
2
]. In fact, we
have the following table for the complexity of query answering (for the union of
conjunctive queries) in GAV systems with sound semantics and with PK and IC
constraints:
I
={
G
k
,
S
k
,
M
k
|
k
∈
N
}
