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where
q
and
q
are two queries of the same arity, over the source schema
S
and
S
G
over the target schema
G
, respectively. Intuitively, an assertion
q
S
q
speci-
G
fies that the concept represented by the query
q
over the sources corresponds to
the concept in the target schema represented by the query
q
S
.
G
•
Queries
q
C
(
x
)
, where
x
(x
1
,...,x
k
)
is a nonempty list of variables, over the
global schema are
conjunctive queries
. We will use, for every original query
q
C
(
x
)
, only a
lifted query
over the global schema, denoted by
q
, such that
q
=
:=
Va l(x
k
)
.
In order to define the semantics of a data integration system, we start from the data at
the sources, and specify which are the data that satisfy the global schema. A
source
instance database D
for
q
C
(
x
)
∧
Va l(x
1
)
∧···∧
I
=
G
,
S
,
M
is constituted by one relation
r
D
for
each source relational symbol
r
in
(sources that are not relational may be suitably
presented in the relational form by wrapper's programs). We call a
global database
for
S
I
,orsimplya
database
for
I
, any (instance) database for
G
. A database
G
for
I
is said to be
legal
with respect to
D
if:
•
G
satisfies the integrity constraints of
G
;
•
G
satisfies
M
with respect to
D
.
•
We restrict our attention to sound views only, which are typically considered the
most natural ones in a data integration setting [
7
,
10
].
In order to obtain an answer to a lifted query
q
from a data integration system, a
tuple of constants is considered as an answer to this query only if it is a
certain
answer, i.e., it satisfies the query in
every legal
global database.
We may try to infer all the legal databases for
and compute the tuples that
satisfy the lifted query
q
in all such legal databases. However, the difficulty here
is that, in general, there is an infinite number of legal databases. Fortunately, we
can define another
universal
(
canonical
) database
can(
I
,D)
that has the interest-
ing property of faithfully representing all legal databases. The construction of the
canonical database is similar to the construction of the
restricted chase
of a database
described in [
8
].
I
4.2.2 GLAV Categorial Semantics
Let us consider the following Global-and-Local-As-View (GLAV) case when each
dependency in
M
is a
tuple-generating dependency
(
tgd
), introduced in Sect.
1.4.2
,
x
∃
zq
G
(
x
,
z
)
∀
=⇒ ∃
y
q
S
(
x
,
y
)
(4.2)
where the formula
q
(
x
)
is a conjunction of atomic formulas over
S
and
q
(
x
,
z
)
is
S
G
a conjunction of atomic formulas over
.
In the Global-As-View (GAV) approach, for each relation
r
in the global (medi-
ated) schema), we write a query over the source relations specifying how to obtain
r
's tuples from the sources. However, adding sources to the data integration system
is not trivial. In particular, given a new source, we need to figure out all the ways in
which it can be used to obtain tuples for each of the relations in the global schema.
G
