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that can be semantically represented by a binary relation of the function-graph) are
interpreted over relations on the active domain.
Since SOtgds have existentially quantified function symbols, one needs suffi-
ciently many elements in the universe in order to interpret these function symbols
by values that are not contained in the source database instance as well. In [
5
], it
was shown that as long as we take the universe to be finite but sufficiently large,
the semantics of SOtgds remains unchanged from an infinite universe semantics.
The most natural choice for the universe, of an instance
(A,C)
of a composition
M
AC
:
A
→
C
(obtained from mappings
M
AB
:
A
→
B
and
M
BC
:
B
→
C
with
a “intermediate” database
), seems to be the active domain. But for the semantics
of composition of mappings in [
5
] the authors did not use this simple choice when
considering the necessity in the composition of instances
(A,B)
and
(B,C)
to take
on values in the missing middle instance
B
for quantified functions in SOtgds.
The approach for what is the
proper
semantics of a given SOtgd mapping be-
tween
B
in our setting is more general than in the OWA data-exchange setting
used in [
5
]: in our case, we consider that a more adequate semantics of a mapping
from
A
and
B
has to be a
strict mapping
semantics, which considers the part of in-
formation that is mapped (only) from
A
into
B
A
into
C
, without the information taken from
B
another intermediate databases (as
in this case), and we need to derive it formally
from the SOtgd.
In fact, in the former approach to the semantics of composition of mappings, the
authors are using a non-well-defined “extended” semantics for mapping instances
(
U
,A,C)
instead of
(A,C)
, where
is an unspecified but “sufficiently large” uni-
verse, “fixed and understandable from the context” [
5
], that includes the active do-
main
A
and
C
.
U
Remark
In what follows, we introduce the characteristic functions for predicates
(i.e., relations) during the composition of mappings and hence assume that the set of
the classic truth logical values
2
={
0
,
1
}
is a subset of the universe
U
=
dom
∪
SK
.
With the syntax choice for SOtgds in [
5
], the authors encapsulate the neces-
sary information contained in the intermediate database
into the semantics of the
quantified functional symbols. In the interpreted functions of the composed map-
ping SOtgd, the information is mixed from both database instances
A
and
B
. Con-
sequently, we will not consider two mappings between a given source and target
databases equal if they are logically equivalent (as in data-exchange setting pre-
sented in [
5
]), but only if the
strict semantics
of the SOtgds of these two mappings
are equal. Let us clarify these different approaches to the semantics of inter-schema
mappings:
B
Example 2
Let us consider the following four schemas
A
,
B
,
C
and
D
(an ex-
tension of Example 2.3 in [
5
]). Schema
consists of a single binary relational
symbol
Takes
that associates student names with courses they take, i.e.,
S
A
=
{
Takes
(x
n
,x
c
)
A
}
,
Σ
A
=∅
. Schema
B
consists of one binary relational symbol
