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for example, at copying settings, then one expects, intuitively, the target to be a copy of the
source rather than an extension of a copy, which OWA permits.
To capture this intuition, we propose a different semantics. It is based not on the OWA,
but rather on the closed-world assumption , or CWA. Under the CWA semantics, solutions
are closed for adding any new facts which are not implied by the source instance and
the mapping. Since data exchange is about transferring data from source to target, then,
intuitively, the semantics for the data exchange problem should be based on the exchanged
data only, and not on data that can be later added to the instances. In other words, the
semantics should be based on those solutions that contain no more than what is needed to
satisfy the specification.
Defining such CWA solutions is not as straightforward as defining solutions under the
OWA. The definition in fact is quite different in spirit. Note that the definition of OWA-
solutions is declarative: they are instances T such that S and T together satisfy the con-
straints of the mapping. In contrast, the definition of CWA solutions is procedural :itex-
plains how to construct T from S using the mapping and nothing else.
This procedural definition relies on a notion of a justification for a tuple with nulls that
may appear in the target. We will present this definition later, but for the time being we shall
useaprecise characterization of solutions under the CWA as our working definition. This
characterization is easy to state. For the sake of simplicity, in this section we only deal
with mappings without target dependencies. We will later extend the study of the CWA
semantics to mappings with target dependencies in Section 8.3 .
Recall that an instance T is a homomorphic image of T if there is a homomorphism
h : T
T such that h ( T ) - the instance obtained from T by replacing each null
by h (
)
- is exactly T .
Definition 8.8 (Working definition of CWA solutions) Let
Σ st ) beamap-
ping, S a source instance and T a solution for S .Then T is a CWA solution for S if
M
=(R s , R t ,
T is a universal solution for S ,and
T is a homomorphic image of the canonical universal solution for S .
by S OL CWA
M
We denote the set of CWA solutions for S under
M
( S ).
The definition makes sense since canonical universal solutions are unique when map-
pings have no target dependencies (see Proposition 6.10 ).
If
is a copying mapping and S is a source instance, then S OL CWA
M
( S ) contains a
copy of S and nothing else (since the canonical universal solution is a copy of S and thus
the only homomorphism defined on it is the identity). Of course, this observation will be
crucial when we later show that a semantics based on the CWA solutions solves query
answering anomalies of the OWA.
Note also that the core of the universal solutions is a CWA solution (since, by definition,
it is a homomorphic image of the canonical universal solution, and by Proposition 6.15 it
is also a universal solution). Moreover, every CWA solution contains the core (since it is
universal).
M
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