=Paper=
{{Paper
|id=Vol-2663/invited-2
|storemode=property
|title=Rewritability Results for OMQs with Closed Predicates
|pdfUrl=https://ceur-ws.org/Vol-2663/invited-2.pdf
|volume=Vol-2663
|authors=Magdalena Ortiz
|dblpUrl=https://dblp.org/rec/conf/dlog/Ortiz20
}}
==Rewritability Results for OMQs with Closed Predicates==
https://ceur-ws.org/Vol-2663/invited-2.pdf
Rewritability Results for OMQs
with Closed Predicates ?
Magdalena Ortiz
Institute of Logic and Computation, TU Wien, Austria
ortiz@kr.tuwien.ac.at
Abstract. The semantics of Description Logic (DL) ontologies adopts
the open-world assumption (OWA) from classical first-order logic (FOL),
which makes them well suited to reason about incomplete knowledge.
However, there are many applications where incomplete and complete
data co-exist, and the standard OWA, which assumes that all data is
incomplete, allows to draw less inferences than desired. Adding closed-
predicates to ontology-mediated queries (OMQs) is an elegant and pow-
erful way to reach better conclusions leveraging partial completeness,
but unfortunately, it is well-known that OMQs with closed predicates
are coNP-hard in practically every non-trivial setting, and hence they
are not FOL-rewritable. We recall here a small selection of recent re-
sults on rewritability of OMQs with closed predicates: two polynomial
rewritings into extensions of Datalog for restricted classes of conjunctive
queries mediated by ontologies in very expressive DLs like ALCHOI and
ALCHOIF, and a recent approach that identifies an FOL-rewritable
family of OMQs that includes full conjunctive queries and DL-Lite on-
tologies, while restricting in a novel yet non-trivial way the use of closed
predicates.
1 Motivation and Background
Like classical first-order logic (FOL), description logics (DLs) adopt the open-
world assumption (OWA) [5]. A model of a DL ontology O is a structure that
satisfies the axioms in O, and which can freely interpret as true or false the as-
sertions whose truth or falsity is not implied by O. A DL ontology typically has
many models, many of which can contain facts that are not relevant to the mod-
eled problem but are not ruled out by the ontology. With this classical semantics,
DLs are well suited to reason about incomplete knowledge, and they have been
successfully applied in many areas where problems of interest can be modeled
and solved by means of different reasoning services, which are usually defined in
terms of logical entailment in a way that ‘irrelevant models’ are immaterial.
An important reasoning service that is found at the core of many modern
applications of DLs (such as the prominent ontology-based data access (OBDA)
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 Magdalena Ortiz
[29] paradigm) is that of ontology-mediated query (OMQ) answering [8], that
allows is to retrieve all certain answers to a given query over a possibly incomplete
dataset—or ABox, in DL jargon—taking into account not only the explicit data,
but also all additional facts that can be inferred from the data using the ontology.
In an OMQ we pair a given query q with a domain ontology O. To evaluate it
over a possibly incomplete dataset A, we adopt the certain answer semantics,
where a tuple is an answer to (q, O) if it is an answer to q in every model of A
and O.
Example 1. Consider the following example, inspired by Example 1 in [25]. Sup-
pose that we have a possibly incomplete dataset A of car models and their motors
(e.g., in an online car marketplace).
Car(c1). hasMotor(c1, 2KD). DieselMotor(2KD). ToyotaMotor(2KD).
Car(c2). hasMotor(c3, 2SDI). DieselMotor(2SDI). ToyotaMotor(1GZ).
ToyotaCar(c2). PetrolMotor(1GZ).
Car(c3).
To find all cars with an internal combustion motor, one can leverage an ontology
O that includes, among others, the following axioms stating that petrol and
diesel motors are internal combustion (IC) motors, and Toyota cars have Toyota
motors.
PetrolMotor v ICMotor ToyotaCar v ∃hasMotor.ToyotaMotor
DieselMotor v ICMotor
Let q(x) = ∃y. Car(x) ∧ hasMotor(x, y) ∧ ICMotor(y). Then, by posing the OMQ
(q, O) over A, one obtains as certain answers c1 and c3, the two cars that are
known to have an IC motor. Note that we know that c2 has a Toyota motor,
but we do not know whether it is an IC motor.
In the OMQ community, rewritability results play a fundamental role (see,
e.g., [2, 4, 9, 11, 20] and their references). We say that a family of OMQs Q is
rewritable into a target query language L, if for every Q = (q, O) in Q, there
is a query rew (Q) in L such that, for every A, the certain answers of Q over
A coincide with the (usual) answers of rew (Q) over A. The rewritability into
a target L means—at least in principle—that OMQ answering in the class Q
can be realized using off-the-shelf query answering engines for L, and indeed,
rewritings have played a crucial role the successful implementation of OMQs.
FOL-rewritability is particularly appealing, since it means that OMQ answering
can be realized using existing RDBMSs. For example, for the OMQ above we can
use the following FOL-rewriting that retrieves the same answers when evaluated
over A, without the need to reason about O.
rew (q, O)(x) = ∃y Car(x) ∧ hasMotor(x, y) ∧
ICMotor(y) ∨ DieselMotor(y) ∨ PetrolMotor(y)
The importance of rewritings is not restricted to reusing technologies for OMQ
answering. In fact, many existing rewriting algorithms produce rewritings that
Rewritability Results for OMQs with Closed Predicates 3
are too large or complex to evaluate with standard query engines. However, even
if this is the case, rewritings can play an important role also in foundational
research. Establishing the (non-)existence of rewritings, as well as bounds on
their sizes, are central tools to study the relative expressiveness of different OMQ
languages, and to compare them to traditional query languages.
1.1 OMQs with Closed Predicates
There are many applications of DLs in general, and of OMQs in particular, where
the OWA results in a semantics that is weaker than desired, as intended conse-
quences are not entailed because they are falsified in some unintended model [2,
26]. For example, if all Toyota motors are IC motors, then we want to obtain in
the example above also c2 as an answer. To overcome this challenge, variations of
standard DLs with different forms of (partial) closed-world assumption (CWA)
have been extensively studied since the early days of DLs and keep receiving sig-
nificant attention, e.g., [6, 10, 12, 14–18, 21, 27]. In the setting of OMQ answering,
we may find ourselves facing a scenario where the data is incomplete, but some
parts of it are known to be complete, and standard OMQs do not have the
expressive means to harness the partial data completeness for obtaining better
inferences.
Closed predicates are a simple and appealing way to evaluate OMQs in set-
tings where incomplete and complete information co-exist. An OMQ with closed
predicates (OMQCs) (q, O, Σ) extends an an (q, O) with a set of predicates Σ
that are to be viewed as complete in the data [15, 2, 26]. The certain answers
(q, O, Σ) over A are thus the tuples that are an answer to q in every intended
model of A and O, that is, every structure extending A to satisfy O, but without
extending the extensions of the predicates in Σ.
Example 2. Assume that the list of Toyota motors in A is imported from a
complete list provided by the manufacturer.1 We can express this information
by saying that ToyotaMotor is a closed predicate, and harness this information
to retrieve also c2. The set of certain answers to the OMQ (q, O, {ToyotaMotor})
over A is {c1, c2, c3}.
Several authors have studied OMQCs [2, 15, 24–26, 28], but unfortunately,
most works are marked by negative complexity results. Indeed, reasoning with
closed predicates has a high computational cost [15, 24, 28]. In the setting of
OMQs, the biggest obstacle is that closed predicates cause intractability in data
complexity: OMQC answering is coNP-hard, even when the ontology is in very
weak DLs like DL-Litecore or EL, and the query is in very restricted classes
of conjunctive queries (CQs) [15, 26]. This, of course, immediately implies that
there do not exist any data independent rewritings of OMQCs into query lan-
guages with tractable data complexity, such as FOL or Datalog. Only severely re-
stricted classes of CQs and DL-Lite ontologies have been shown to be rewritable
1
Our toy example has only two motors, but a complete database of all Toyota motors
can be easily found online.
4 Magdalena Ortiz
into equivalent FOL queries [25, 26]. Moreover, for DL-Lite it has been shown
that a class of OMQCs is FOL-rewritable only if it satisfies a convexity condition
which, as it turns out, implies that the presence of closed predicates is irrelevant
[24]. For a fine-grained non-uniform analysis of what makes OMQC answering
(in)tractable, please see [26].
However, despite this rather discouraging landscape, we have seen interesting
progress in rewritability results for OMQCs in the last couple of years, e.g., [2, 3,
7, 19, 23, 25]. Here we will review a selection of these works, focusing on two types
of results. First, we discuss two polynomial rewritings for OMQCs that allow for
very expressive DLs as ontology languages, and use as target languages variants
of Datalog. Then we will briefly recall a recent approach that identifies an FOL-
rewritable family of OMQs that includes full CQs and DL-Lite ontologies, but
restricts in a novel yet non-trivial way the use of closed predicates.
2 Rewriting instance OMQCs in very expressive DLs
Since we know that OMQC answering is coNP-hard in basically all interesting
cases [24], it makes sense to consider as target for rewritings more expressive
query languages that also exhibit such data complexity.
Moreover, OMQCs are a non-monotonic query language.
Example 3. Recall that the certain answers to (q, O, {ToyotaMotor}) over A are
{c1, c2, c3}. Assume that we received and updated list of Toyota motors that
includes ToyotaMotor(1PMSM), which is not listed as an IC motor. The cer-
tain answers to (q, O, {ToyotaMotor}) over A ∪ {ToyotaMotor(1PMSM)} are now
{c1, c3}, since c2 can now have 1PMSM as motor and thus no IC motor.
Therefore, any potential rewriting must consider as target a query language that
is also non-monotonic. This makes extensions of Datalog with negation a natural
target [13].
Indeed, rewritings into Datalog with negation have been developed for very
expressive DLs like ALCHOI [2] and ALCHOIF [19], considering as query
languages instance queries (IQs) and other restricted classes of CQs. Crucially,
both works produce Datalog programs of polynomial size. We also stress that
both of these translations are data independent, hence the presence of nominals
in the underlying DL does not help us simulate the closed predicates.
In the former work [2], we characterize the satisfaction of the ontology as a
two-player game (inspired by type elimination algorithms), and then we design
a Datalog query that can decide the existence of a winning strategy for the
game. The resulting Datalog program uses a restricted form of negation and
falls in a fragment that can be evaluated in deterministic single exponential time,
hence it also provides a query answering algorithm that is worst-case optimal in
combined complexity. If there are no closed predicates—that is, for an instance
query mediated by a plain ALCHOI ontology—the translation yields a positive
disjunctive Datalog program of polynomial size, and if the input is in a Horn
fragment, a modified technique produces a plain Datalog program. These ideas
Rewritability Results for OMQs with Closed Predicates 5
have been used also for tuple-generating dependencies (TGDs) in the absence
of closed predicates, showing that guarded TGDs can be polynomially encoded
into Datalog, and disjunctive guarded TGDs into disjunctive Datalog [1].
For the DL ALCHOIF, in contrast, IQ answering (without closed predicates)
is already co-nexptime-hard, and the ideas discussed above are not applicable.
Therefore in our most recent work [19] we characterize models in terms of match-
ing set of mosaics, and show that the existence of the latter can be reduced to
integer programming. Then we define a carefully crafted program in Datalog
with negation that finds solutions to dynamically generated systems of integer
inequalities. This translation yields optimal co-np and co-nexptime-hard upper
bounds in data and combined complexity, respectively. We note that the latter
bound is new, and it was not obvious whether such a bound was possible in the
presence of closed predicates, and of the tricky combination of nominals, inverse
roles and role functionality.
Both rewritings consider IQs and restricted families of CQs. Unfortunately,
allowing full CQs precludes the existence of a polynomial sized rewritings under
standard complexity assumptions, even if we restrict the ontology language. In-
deed, in the presence of closed predicates, OMQ answering with CQs is 2ExpTime-
hard already for EL and DL-LiteR [28], while entailment in Datalog with nega-
tion is in coNExpTimeNP .
3 An FOL-rewritable family of ontology-mediated CQs
Finally, we discuss a recent positive FOL-rewritability result that applies to
full CQs. We have already made clear that OMQCs are not FOL-rewritable for
any standard DL. However, we identified an interesting setting where OMQCs
that pair arbitrary CQs and DL-LiteR ontologies are FOL-rewritable, but the
use of closed predicates is restricted: these may not occur in the ontology, but
instead, we allow complex ABox assertions where closed predicates may occur
[3]. Although somewhat non-standard at first sight, such a setting is simple and
intuitive, and captures the typical examples that are used to argue in favor of
closed predicates in OMQs.
The key idea underlying this rewriting is is that, if the closed predicates do
not occur in the TBox, then one may need to do some form of case-by-case
reasoning over models, but this can be encoded into a (non-monotonic) FOL
query that uses universal quantification to iterate over all possible extensions of
the closed predicates in the models of each input dataset. The resulting query
can be large and take a rather complex form, but does not seem to fully rule out
the possibility of implementation.
Example 4. Consider a modified version of the example above, which is in fact
a minor modification of Example 1 in [25]. Since ToyotaMotor is closed, it is
disallowed in the ontology, and hence we consider O0 :
PetrolMotor v ICMotor DieselMotor v ICMotor
6 Magdalena Ortiz
We can, however, explicitely say that c2 has a Toyota motor by assuming
the ABox assertion ∃hasMotor.ToyotaMotor(c2). The techniques in [3] essentially
rewrites the OMQ (q, O) from Example 1 into (an equivalent form of) the FOL
query:
∃y Car(x) ∧ hasMotor(x, y) ∧
ICMotor(y) ∨ DieselMotor(y) ∨ PetrolMotor(y) ∨
(ToyotaMotor(y) ∧ ∀y 0 (ToyotaMotor(y 0 )) →
(ICMotor(y 0 ) ∨ DieselMotor(y 0 ) ∨ PetrolMotor(y 0 )))
Essentially, the disjunction in rew (q, O)(x) now has an additional disjunct that
allows to substitute y by any Toyota motor in every dataset in which all Toyota
motors are IC motors. Note that this query correctly retrieves {c1, c2, c3} when
evaluated over A, but only {c1, c3} when evaluated over
A ∪ {ToyotaMotor(1PMSM)}.
The similar version of this example given in [25] is not FOL-rewritable: the axiom
ToyotaCar v ∃hasMotor.ToyotaMotor in the ontology makes it unsafe.
We note that the algorithm in [3] considers a slightly more general framework.
Rather than adding complex ABox assertions, one can make a set of hypothesis
(which are instances of certain assumption patterns that are complex concepts
given as part of the input). The output for such a query are conditional an-
swers that pair instantiated hypothesis with tuples that are answers given the
hypothesis.
4 Outlook
Reconciling the open- and closed-world assumptions is one of the most fascinat-
ing challenges that the OMQ and DL communities face. Much has been achieved,
but we need to pay more attention to understanding the way complete and in-
complete information coexist in real-world applications. There is plenty of room
for meaningful characterizations of sets of intended models that are more useful
that the full set of open-world models that is prevalent in DL reasoning, without
fully giving up the open-world inference and the possibility of reasoning about
incomplete aspects of the world [18]. Many great ideas have been proposed over
the decades in areas like hybrid languages that combine rules and ontologies
(e.g., [7, 27, 22]), and in circumscription [10] and other forms of non-monotonic
reasoning in DLs [14, 6]. However, some of these ideas have not been pursued
sufficiently or left aside for too long.
Acknowledgments The works reported here were supported by the Austrian
Science Fund (FWF) projects P30360, P30873 and W1255. They were carried out
with different co-authors, to whom I am grateful: special thank you to Shqiponja
Ahmetaj, Medina Andreşel, Sanja Lukumbuzya, and Mantas Šimkus.
Rewritability Results for OMQs with Closed Predicates 7
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