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unqualified number restrictions are 'harmless' provided that we do not use both
axioms (
≥
kR
B
), for
k
≥
2, and
R
R
in the same TBox (consult [4] for a
more precise condition).
The results discussed so far guarantee that if our ontology is formulated in a
DL with the help of such and such constructs then it can safely be used for OBDA
with databases. A different approach to understanding the phenomenon of FO-
rewritability has recently been suggested [47]: it attempts to classify all TBoxes
T
in some master DL, say
ALCI
, according to the complexity of answering CQs
over
.
Finally, we note that another family of ontology languages suitable for OBDA
with databases has been designed by the datalog community. We refer the reader
to the recent papers [10,9] for a survey. Connections of query answering via DL
ontologies with disjunctive datalog and constraint satisfaction problems have
been established in [7].
In the next section, we shall see how one can construct FO-rewritings of CQs
and
T
OWL 2 QL
TBoxes.
3 Tree-Witness Rewriting
A standard architecture of an OBDA system over relational data sources can be
represented as follows:
rewriting
unfolding
CQ
q
+
FO
q
+
SQL
mapping
TBox
T
+
ABox virtualisation
The user is given an
OWL 2 QL
TBox
T
and can formulate CQs
q
(
x
)inthe
signature of
ABox
A
data
D
q
(
T
. The system rewrites
q
(
x
)and
T
into an FO-query
x
)such
q
(
that (
T
,
A
)
|
=
q
(
a
)iff
A|
=
a
), for any ABox
A
and any tuple
a
of individuals
q
is called a
PE-rewriting
if it is a PE-query and an
NDL-
rewriting
if it is an NDL-query.
The rewriting
in
A
. The rewriting
q
(
and has to be further
transformed into the vocabulary of the data source
D
before being evaluated.
For instance,
x
) is formulated in the signature of
T
q
(
x
) can be unfolded into an SQL query by means of a mapping
M
to the vocabulary of
D
. We consider unfolding in
Section 6, but before that we assume the data to be given as an ABox (say, as
a universal table in a database or as a triple store) with a trivial mapping.
A number of different rewriting techniques have been proposed and imple-
mented for
relating the signature of
T
(PerfectRef [54], Presto/Prexto [62,61], Rapid [15]) and
its extensions ([35], Nyaya [22], Requiem/Blackout [52,53], Clipper [17]). In this
section, we discuss the tree-witness rewriting [32].
OWL 2 QL
