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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 ].
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