=Paper=
{{Paper
|id=Vol-1350/paper-55
|storemode=property
|title=A
higher-order semantics for OWL 2 QL ontologies (Extended
abstract)
|pdfUrl=https://ceur-ws.org/Vol-1350/paper-55.pdf
|volume=Vol-1350
|dblpUrl=https://dblp.org/rec/conf/dlog/LenzeriniLP15
}}
==A
higher-order semantics for OWL 2 QL ontologies (Extended
abstract)==
https://ceur-ws.org/Vol-1350/paper-55.pdf
A higher-order semantics for OWL 2 QL ontologies
Maurizio Lenzerini, Lorenzo Lepore, Antonella Poggi
Dipartimento di Ingegneria Informatica, Automatica e
Gestionale “Antonio Ruberti”- Sapienza Università di
Roma
Via Ariosto 25, I-00183 Roma, Italy
lastname @dis.uniroma1.it
Recent OBDA projects have pointed out that one of the drawbacks of OWL 2 is
the lack of metamodeling and metaquerying capabilities, i.e., features for specify-
ing and reasoning about metaconcepts and metaproperties [1]. Roughly speaking,
a metaconcept is a concept whose instances can be themselves concepts, and a
metaproperty is a relationship between metaconcepts. Although OWL 2 provides
syntactic support for metamodeling through punning (by which the same name
can be used to denote ontology elements of different categories, such as a class
and an individual), we argue that the official semantics of OWL 2, the so-called
Direct Semantics (DS), treats punning in a way that is not adequate for rep-
resenting metaconcepts and metaproperties. The reason is simply that proper
metamodeling requires that the same element plays the role of individual and
class (or, class and relation), while DS sanctions that an individual and a class
with the same name are different elements. This is confirmed by the fact that the
Direct Semantics Entailment Regime (DSER), which is the logic-based seman-
tics of the SPARQL 1.1 query language when applied to OWL 2 QL, forces queries to
obey the so-called typing constraint, which rules out the possibility of using the
same variable in incompatible positions (for example, in individual and in class
position).
The issue of metamodeling in OWL has been investigated in several papers.
It is known that the semantics of metamodeling of OWL 2 Full leads to unde-
cidability of basic inference problems [7]. A possible solution to this problem is
to enable metamodeling in OWL 2 DL by axiomatizing the higher order knowl-
edge into first order assertions [4], but the process involves the use of complex
expressions that are not supported by OWL 2 DL tractable profiles, and therefore
seems inapplicable in OBDA. Another possible solution is the stratification of
class constructors and axioms to describe metalevels of classes and properties [8],
but such stratification poses challenges for the modeler, and rules out interest-
ing ontology patterns. Relevant to our work are recent papers aiming at devising
efficient techniques to answer SPARQL 1.1 queries posed to OWL 2 QL ontologies [2,
6, 5]. However, such papers concentrate on DSER, and therefore do not aim at
the full power of metamodeling and metaquerying.
The goal of our work is to present a new higher-order semantics for OWL 2 QL,
called HOS and inspired by [3], allowing us to effectively exploit the capabilities
for metamodeling offered by punning, and to show that, based on such semantics,
it is possible to define a new SPARQL entailment regime, called HOSER, for the
class of (meta)queries obtained by relaxing the typing constraint of DSER.
Illustrating scenario. We refer to an OWL 2 QL ontology (see an excerpt in Ta-
ble 1), whose central entity is :financial instrument.Metamodeling comes into
play in order to capture the notion of types of financial instruments, modeled
by using the metaclass :type of f i, whose instances are other classes which
are intended to be subclasses of :financial instrument (for example, :BTP and
:commercial paper, see axioms (3),(4),(6)), and by defining its properties. One
notable example of such properties is :established by (see axioms (9),(10),(11)),
which associates to each type of financial instrument the law that formally estab-
lished it (for example, the law :DR135bis established :BTP as a type of financial
instrument, see axiom (5)).Observe that syntactically we are using punning.
However, as we said before, the direct semantics of OWL 2 considers the individ-
ual named :BTP different from the class with the same name, and therefore is
inadequate for metamodeling. This is evident if one considers the query
select $x $y where {:IT0005069395 rdf:type $x . $x :established by $y}
asking for the law that established the type of financial instrument having a spe-
cific financial instrument (for example :IT0005069395) as instance. This query
is not legal under DSER, because the variable $x appears both in individual
and in class position. Note that if typing constraint is lifted, thus making the
query legal, then it would become empty under DSER is trivially empty since
DS cannot interpret any ontology element as both an individual and a class.
ClassAssertion(:BTP :IT0005069395) (1)
ClassAssertion(:commercial paper :ZAG000117292) (2)
ClassAssertion(:type of Italian f i :BTP) (3)
ClassAssertion(:type of foreign f i :commercial paper) (4)
ObjectPropertyAssertion(:established by :BTP :DR135bis) (5)
SubClassOf(:BTP :financial instrument) (6)
SubClassOf(:commercial paper :financial instrument) (7)
SubClassOf(:financial instrument DataSomeValuesFrom(:duration)) (8)
SubClassOf(:type of f i ObjectSomeValuesFrom(:established by :law)) (9)
SubClassOf(:type of Italian f i ObjectSomeValuesFrom(:established by :Italian law)) (10)
SubClassOf(ObjectSomeValuesFrom(ObjectInverseOf(:established by)) :type of f i) (11)
SubClassOf(:type of Italian f i :type of f i) (12)
SubClassOf(:type of foreign f i :type of f i) (13)
Table 1: The ontology
Higher-Order Semantics for OWL 2 QL. A vocabulary V for an ontology is
constituted by a tuple V = (VN , VC , VOP , VDP , VDT , L), where VN is a set of
IRIs that includes the reserved vocabulary R of OWL 2 QL, VC , VOP , VDP , VDT are
subsets of VN , and L is the set of OWL 2 QL literals. The symbols in VN are called
entity names, since they are used to name all the entities of the ontology, while
VC (resp., VOP , VDP , VDT ) is the subset of VN used to name those entities play-
ing the role of class (resp., object properties, data properties, datatypes). Any
symbol in VN \ R may denote an entity that simultaneously plays the role of (i)
an individual, (ii) either a class or a datatype, and (iii) either an object property
or a data property. We denote by VI the set VN \(VC ∪VOP ∪VDP ∪VDT ), i.e., the
set of entity names that can only play the role of individuals in the ontology. The
symbols in VN , also called atomic expressions, constitute the building blocks for
denoting the entities of an ontology. Entities, however, can be denoted by using
general expressions. The set ExpV of expressions over V is the set VI ∪ ExpOP V ∪
ExpC V ∪ V DP ∪ VDT , where Exp OP
V = V OP ∪ { ObjectInverseOf (e 1 ) | e 1 ∈ V OP },
and ExpC OP
V = VC ∪ {ObjectSomeValuesFrom(e1 e2 ) | e1 ∈ ExpV , e2 ∈ VC } ∪
{DataSomeValuesFrom(e1 e2 ) | e1 ∈ VDP , e2 ∈ VDT }. Expressions over V are then
used in the axioms forming any ontology defined over the vocabulary V . HOS is
based on the notion of interpretation structure, which plays the role of “interpre-
tation domain” in classical logic. Given a vocabulary V = (VN , VC , VOP , VDP ,
VDT ), an interpretation structure for V is a tuple Σ = (∆, ·I , ·E , ·R , ·A , ·T )
Axiom a I |= a if
I I
SubClassOf(e1 e2 ) ·E is defined for both e o and e o
1 2
I I
and (e o )E ⊆ (e o )E
1 2
I I
ObjectPropertyDomain(e1 e2 ) ·R is defined for e o , ·E is defined for e o
1 2
I I
and ∀hd, d0 i ∈ (e o )R : d ∈ (e o )E
1 2
E Io Io I
ClassAssertion(e1 e2 ) · is defined for e , (e ) = T
1 2
I I
and e o ∈ (e o )E
2 1
I I I
ObjectPropertyAssertion(e1 e2 e3 ) ·R is defined for e o , (e o )I = (e o )I = T
1 2 3
I I I
and he o , e o i ∈ (e o )R
2 3 1
A Io Io I
DataPropertyAssertion(e1 e2 v) · is defined for e , (e ) = T
1 2
Fig. 1: Interpretation I I
and he o , v Io i ∈ (e o )A
2 1
structure Table 2: Satisfaction of OWL 2 QL axioms
where: (i) ∆ is the disjoint union of two sets: ∆o , the object domain, and ∆v ,
the value domain; and (ii) ·E : ∆o → P(∆o ), ·R : ∆o → P(∆o × ∆o ), ·A : ∆o →
P(∆o × ∆v ), and ·T : ∆o → P(∆v ) are partial functions, and ·I : ∆o → {T, F} is
a total function such that for each d ∈ ∆o , if ·E , ·R , ·A , ·T are all undefined for d,
then dI = T. Thus, the interpretation structure is a mathematical representation
of a world made up by domain elements (the members of the set ∆) which can
be either objects or values. Objects are polymorphic, in the sense that each of
them can simultaneously be an individual(this is the case where function ·I is T),
a set (this is the case where the partial function ·E is defined), a binary relation
(·R is defined), an attribute (·A is defined), and a value type (·T is defined).
Also, an object that is a set (resp., relation, attribute, value type) is associated
to its extension through ·E (resp., ·R , ·A , ·T ). Figure 1 shows a representation
of an interpretation structure, where α is an individual, a set and a relation,
whereas β is a set, but not an individual or a relation. An interpretation I for
V is a pair, hΣ, Io i, where Σ is an interpretation structure for V , and Io is
the interpretation function for I, i.e., a function that maps every expression in
ExpV into an object in ∆o , and every literal in L into a value in ∆v , according to
R
a set of suitable conditions. (e.g., ((ObjectInverseOf(e1 ))Io )R = ((eI1 o ) )−1 ). To
define the semantics of axioms, we refer to the notion of satisfaction of an axiom
with respect to an interpretation I. Some of the rules defining such notion are
specified in Table 2 (where e, e1 , e2 , e3 and v are expressions).
SPARQL higher-order semantics entailment regime Defining a SPARQL en-
tailment regime requires to specify (α) which is the semantics used to interpret
the queried ontology, (β) which queries are legal, and (γ) a definition for the
notion of answer to queries (called solution in SPARQL jargon). As for (α), we
adopt HOS. As for (β), we extend the class of queries which are legal for DSER,
by relaxing the typing constraint. As for (γ), we propose to interpret as pure
existentials, variables that occur in the body of the query but not in the target,
thus following the classical notion of existential variables in logic. Observe that
the query mentioned in the scenario is legal for HOSER and that by evaluat-
ing it over the scenario ontology, it would return the answer h:BTP, :DR135bisi.
One of the results of our work is an algorithm, based on “blind metagrounding”,
showing that query answering over OWL 2 QL ontologies under HOSER has the
same data complexity as under DSER. Nevertheless, such algorithm can be in-
efficient in practice. Hence, we devised a new algorithm that is polynomial with
respect to extensional assertions and more efficient for a huge class of OWL 2 QL
ontologies (so called ISA-closed) that are very common in practice.
References
1. D. Allemang and J. Hendler. Semantic Web for the Working Ontologist: Effective
Modeling in RDFS and OWL. Elsevier, 2011.
2. M. Arenas, G. Gottlob, and A. Pieris. Expressive languages for querying the seman-
tic web. In Proceedings of the 33rd ACM SIGMOD-SIGACT-SIGART Symposium
on Principles of Database Systems, PODS’14, Snowbird, UT, USA, June 22-27,
2014, pages 14–26, 2014.
3. G. De Giacomo, M. Lenzerini, and R. Rosati. Higher-order description logics for
domain metamodeling. In Proc. of AAAI 2011, 2011.
4. B. Glimm, S. Rudolph, and J. Völker. Integrated metamodeling and diagnosis in
OWL 2. In Proc. of ISWC 2010, volume 6496 of LNCS, pages 257–272. Springer,
2010.
5. I. Kollia and B. Glimm. Optimizing SPARQL query answering over OWL ontologies.
J. Artif. Intell. Res. (JAIR), 48:253–303, 2013.
6. R. Kontchakov, M. Rezk, M. Rodriguez-Muro, G. Xiao, and M. Zakharyaschev.
Answering SPARQL queries over databases under OWL 2 QL entailment regime.
In The Semantic Web - ISWC 2014 - 13th International Semantic Web Conference,
Riva del Garda, Italy, October 19-23, 2014. Proceedings, Part I, pages 552–567,
2014.
7. B. Motik. On the properties of metamodeling in OWL. J. of Logic and Computation,
17(4):617–637, 2007.
8. J. Z. Pan, I. Horrocks, and G. Schreiber. OWL FA: A metamodeling extension of
OWL DL. In Proceedings of the OWLED*05 Workshop on OWL: Experiences and
Directions, Galway, Ireland, November 11-12, 2005, 2005.