Database Reference
In-Depth Information
2
Composition of Schema Mappings: Syntax and
Semantics
2.1
Schema Mappings: Second-Order tgds (SOtgds)
Based on previous considerations and weak points in the schema-mapping develop-
ments presented in the introduction (Sect.
1.4.2
), a different semantics for composi-
tion that is valid for every class of queries has been proposed in [
5
]. The suggested
semantics does not carry along a class of queries as a parameter, and the authors
have shown that the set of formulae defining a composition is unique up to logical
equivalence.
On the negative side, however, they demonstrated that the composition of a finite
set of tgds with a finite set of
full
tgds (a tgd is full if no existentially quantified
variables occur in the tgd) may not be definable by any set (finite or infinite) of
tgds. Moreover, the composition of a finite set of tgds with a finite set of tgds is not
definable in least-fixed point logics.
Based on these negative results, a class of existential second-order formulae with
function symbols (introduced as a result of Skolemization of the existentially quan-
tified variables) and equalities, the so-called second-order tgds (SOtgds), has been
introduced in [
5
] as follows:
Definition 6
[
5
]Let
a target schema. A second-order
tuple-generating dependency (SOtgd) is a formula of the form:
A
be a source schema and
B
f
∀
x
1
(φ
1
⇒
ψ
1
)
∧···∧
∀
x
n
(φ
n
⇒
ψ
n
)
,
∃
where
1. Each member of the tuple
f
is a functional symbol;
2. Each
φ
i
is a conjunction of:
•
Atomic formulae of the form
r
A
(y
1
,...,y
k
)
, where
r
A
∈
S
A
is a
k
-ary rela-
tional symbol of schema
A
and
y
1
,...,y
k
are variables in
x
i
, not necessarily
distinct;
•
The formulae with conjunction and negation connectives and with built-in
predicate's atoms of the form
t
.
=
t
,
, where
t
and
t
are
∈{
,<,>,...
}
the terms based on
x
i
,
f
and constants.
