Database Reference
In-Depth Information
L
A
of the union of
a finite set
S
of conjunctive
finite-length
queries
q(
x
)
(with the same head with a
finite tuple of variables
x
) so that
R
Input sorts for a given instance-database
A
is a countable set
=
q(
x
)
∈
S
=
ev
A
(q(
x
))
q(
x
)
A
is the relation
(a materialized view) obtained by applying this query to
A
.
Each query (FOL formula introduced in Sect.
1.4
) has an equivalent finite-length
algebraic term
of the SPJRU algebra (or equivalent to it, SPCU algebra, Chaps. 4.5,
5.4in[
1
]) as shortly introduced in the previous section, and hence the power
view-operator
T
can be defined by the initial SPRJU algebra of ground terms (see
Sect.
5.1.1
). We define this fundamental
idempotent power-view
operator
T
, with
the domain and codomain equal to the set of all instance-databases, such that for
any instance-database
A
, the object
TA
=
T(A)
denotes a database composed of
the set of
all views
of
A
. The object
TA
, for a given instance-database
A
, corre-
sponds to the carrier of the quotient-term Lindenbaum algebra
, i.e., the set of
the
equivalence classes
of queries (such a query is equivalent to a term in
L
A
/
≈
T
P
X
of
an SRRJU relational algebra
Σ
R
, formally given in Definition
31
of Sect.
5.1
, with
the select, project, join and union operators, with relational symbols of a database
schema
A
). More precisely,
TA
is “generated” from
A
by this quotient-term al-
L
A
,
ev
A
:
L
A
TA
, which
is surjective function. From the factorization theorem, there is a unique bijection
is
A
:
L
A
/
≈
→
gebra
L
A
/
≈
and a given evaluation of queries in
TA
such that
commutes, where the surjective function
nat
≈
:
L
A
→
L
A
/
≈
is a natural repre-
sentation for the equivalence of the queries
≈
. For every object
A
,
A
⊆
TA
and
TA
T(TA)
, i.e., each (element) view of database instance
TA
is also an element
(view) of a database instance
A
. It is easy to verify that
T
corresponds to the closure
operator
Sg
on
A
(introduced in Sect.
1.2
)forthe
Σ
R
relational “select-project-
join+union” (SPJRU) algebra, but with relations instead of variables (i.e., relational
symbols). Notice that when
A
has a finite number of relations, but at least one rela-
tion with an infinite number of tuples, then
TA
has an infinite number of relations
(i.e., views of
A
) and hence can be an infinite object.
=
1.4.2 Introduction to Schema Mappings
The problem of sharing data from multiple sources has recently received significant
attention, and a succession of different architectures has been proposed, beginning
with federated databases [
61
,
73
], followed by data integration systems [
9
,
10
,
38
],
data exchange systems [
20
,
21
,
25
] and Peer-to-Peer (P2P)) data management sys-
tems [
11
,
24
,
29
,
34
,
49
,
50
].