=Paper=
{{Paper
|id=Vol-1807/06_ISW-LOD2016_55_60
|storemode=property
|title=OWL Extended with Meta-modelling
|pdfUrl=https://ceur-ws.org/Vol-1807/06_ISW-LOD2016_55_60.pdf
|volume=Vol-1807
|authors=Regina Motz,Edelweis Rohrer,Paula Severi,Ignacio Vidal
|dblpUrl=https://dblp.org/rec/conf/iberamia/Motz16
}}
==OWL Extended with Meta-modelling==
https://ceur-ws.org/Vol-1807/06_ISW-LOD2016_55_60.pdf
OWL extended with Meta-modelling
Regina Motz1 , Edelweis Rohrer1 , Paula Severi2? and Ignacio Vidal1
1
Instituto de Computación, Facultad de Ingenierı́a, Universidad de la República,
Uruguay
2
Department of Computer Science, University of Leicester
Abstract. In this paper we explain how we extended the Web Ontol-
ogy Language (OWL) with meta-modelling. In order to express meta-
modelling, we extended the OWL syntax with a new constructor that
allows us to equate individuals with classes. In order to perform the main
task of checking consistency of ontologies that have meta-modelling, we
extended the free and open source reasoner for the Semantic Web called
Pellet.
Keywords: OWL, Meta-modelling, Reasoner, Consistency checking
1 Motivation
In different application scenarios the
need of linking ontologies of differ-
ent domains arises. But the way how
these ontologies are related is not al-
ways the same. Sometimes it is re-
quired to map classes or individuals
of two ontologies, or link individuals
of two ontologies through a new prop-
erty. In these cases, elements of the
same granularity are mapped, i. e.,
classes to classes or individuals to in-
dividuals. But there are some scenar-
ios where mapping elements of differ- Fig. 1. Two ontologies on Hydrography
ent granularity is needed, for instance
when the same real object is repre-
sented as an individual in one ontology and as a class in other ontology. This
kind of relation between ontologies is called meta-modelling and is the main
motivation of our work. Our extension of OWL comes up from a real-world ap-
plication on geographic objects that requires to reuse existing ontologies and
relate them through meta-modelling [1]. Figure 1 describes a simplified scenario
of this application in order to illustrate the meta-modelling relationship. It shows
two ontologies separated by a horizontal line. The two ontologies conceptualize
?
Daphne Jackson fellowship sponsored by EPSRC and the University of Leicester.
55
_______
Copyright © 2016 for this paper by its authors. Copying permitted for private and academic purposes.
2 OWL with metamodelling
the same entities at different levels of granularity. In the ontology above the
horizontal line, rivers and lakes are represented as individuals while in the one
below the line they are classes. If we want to integrate these ontologies into a
single ontology (or into an ontology network) it is necessary to interpret the
individual river and the class River as the same real object. Similarly for lake
and Lake. Our solution consists in equating the individual river to the class
River and the individual lake to the class Lake. These equalities are called
meta-modelling axioms and in this case, we say that the ontologies are related
through meta-modelling. In Figure 1, meta-modelling axioms are represented by
dashed edges. After adding the meta-modelling axioms for rivers and lakes, the
class HydrographicObject is now also a meta-class because it is a class that con-
tains an individual which is also a class. The kind of meta-modelling we consider
in this paper can be expressed in OWL Full, which allows to equate any two
resources, for instance a class to an individual or a property to an individual.
However, it cannot be expressed in OWL DL. The fact that it is expressed in
OWL Full is not very useful since the meta-modelling provided by OWL Full is
so expressive that leads to undecidability [2]. OWL 2 DL has a very restricted
form of meta-modelling called punning where the same identifier can be used as
an individual and as a class [3]. These identifiers are treated as different objects
by the reasoner and it is not possible to detect certain inconsistencies. We next
illustrate two examples where OWL would not detect inconsistencies because
the identifiers, though they look syntactically equal, are actually different.
Example 1. If we introduce an axiom expressing that HydrographicObject is a
subclass of River, then OWL reasoner will not detect that the interpretation of
River is not a well founded set (it is a set that belongs to itself). That is, the
interpretation of River (which is equal to river by meta-modelling) belongs to
that of HydrographicObject and, by the introduced axiom, the interpretation of
HydrographicObject is a subset of that of River.
Example 2. We add two axioms, the first one says that river and lake as indi-
viduals are equal and the second one says that the classes River and Lake are
disjoint. Then OWL reasoner does not detect that there is a contradiction.
2 Extending OWL with Meta-modelling Axioms
In order to express meta-modelling, we extended OWL with meta-modelling
axioms. A meta-modelling axiom a =m A is an equation between an individual a
and an atomic class A. The semantics of the above axiom is that the individual a
and the class A have the same interpretation. For example, the meta-modelling
axiom that equates the individual river with the class River is expressed in
OWL/XML as follows.
56
OWL extended with Meta-modelling 3
The individual river is interpreted as the set {queguay, santaLucia} since this
is exactly the interpretation of the class River. Extending the syntax is the
easy part. The most difficult part consists in extending the Tableau algorithm
in order to check consistency of ontologies that have meta-modelling [4, 5]. For
an introduction on the basics of the Tableau algorithm, we suggest the reader to
see [3]. The Tableau algorithm constructs a graph to represent a possible model
of the knowledge base. The nodes of the initial graph are the individuals of
the ontology and the edges are the properties between the individuals. Equality
between individuals is recorded using a ≈ b and each node x has associated a
set of class expressions L(x). In [4, 5] we extended the Tableau algorithm for
checking consistency of ontologies with meta-modelling by adding:
1. new expansion rules: to deal with equalities and inequalities of individuals
with meta-modelling which are given in Table 1.
2. a new condition to deal with circularities with respect to the membership ∈
relation. This condition detects the presence of non well founded sets.
Equality Rule:
If a =m A, b =m B, a ≈ b and A ≡ B does not belong to the Tbox then,
add A ≡ B to the Tbox.
Inequality Rule:
If a =m A, b =m B, a 6≈ b and there is no z, A u ¬B t B u ¬A ∈ L(z) then,
create a new node z with L(z) = {A u ¬B t B u ¬A}.
Close Rule:
If a =m A, b =m B and neither a ≈ b nor a 6≈ b then,
add either a ≈ b or a 6≈ b.
Table 1. Expansion rules for meta-modelling
We explain the intuition behind the new expansion rules. If a =m A and
b =m B then the individuals a and b represent classes. Any equality at the level
of individuals should be transferred as an equality between classes and similarly
with the difference. A particular case of the application of Equality Rule is when
a =m A and a =m B. In this case, the algorithm also adds A ≡ B. In the In-
equality Rule, the inequality a 6≈ b should be transferred to the level of classes
as A 6≡ B. However, we cannot add A 6≡ B because the negation of ≡ is not di-
rectly available in the language. So, what we do is to replace it by an equivalent
statement, i.e. add an element z that witnesses this difference.
The rules for equality and inequality are not sufficient to detect all inconsisten-
cies coming from meta-modelling. The idea is that we also need to transfer the
equality A ≡ B between classes as an equality a ≈ b between individuals and
we also need to transfer the semantic consequences, e.g. O |= A ≡ B. Unfortu-
nately, a recursive call of the form O |= A ≡ B is not possible. Otherwise we
57
4 OWL with metamodelling
will be captured in a vicious circle since the problem of finding out the semantic
consequences is reduced to the one of satisfiability. The solution to this problem
is to explicitly try either a ≈ b or a 6≈ b. This is exactly what the close-rule does.
The close-rule adds either a ≈ b or a 6≈ b. It is similar to the choose-rule which
adds either C or ¬C [3].
Our Tableau algorithm for meta-modelling has a Tbox rule:
Tbox Rule:
If C is a TBox statement and C 6∈ L(x), then add C to L(x).
where C is a TBox axiom C v D codified as ¬C t D.
It is not so straightforward to extend the code of Pellet [6, 7] to include the new
expansions rules for meta-modelling. This is because we have to adapt our new
rules to cope with the optimizations of Pellet [8–10]. The main complication
is given by the equality rule which has the unusual characteristic of changing
the Tbox, since we add new TBox axioms. This lead us to re-run actions and
optimizations on the TBox, which were originally executed at the beginning,
each time the equality rule is applied, because of the addition of new axioms.
First of all, Pellet divides the Tbox T in two disjoint sets: an unfoldable part Tu
that contains unique, acyclical definition axioms and a general Tbox Tg = T \Tu .
Then, it tries to absorb subclass axioms from the Tg into the Tu .
After that, in the rule application step, Pellet uses the expansion rules of lazy
unfolding [8] shown below.
Unfolding Rule I:
If A v C ∈ Tu , A ∈ L(x) and C 6∈ L(x) then add C to L(x).
Unfolding Rule II:
If ¬A v C ∈ Tu , ¬A ∈ L(x) and C 6∈ L(x) then add C to L(x).
d
The general Tbox Tg is expressed as > = Tg , which is the conjunction of all Tg
axioms. Then, Pellet also applies lazy unfolding to the nodes x that have > in
L(x) by adding uTg to L(x). But this works only if we ensure that > is in L(x)
for all nodes x of the graph. The initialisation step in Pellet guarantees that >
is added to L(a) for all the individuals a of the ontology and every expansion
rule that creates a new node z (such as ∃-rule) adds > to L(z).
We also apply optimization techniques to our meta-modelling rules. The op-
timized rules for meta-modelling are shown in Table 2. The Optimized Inequality
Rule adds > to L(z) to the new node z that it creates. The Optimized Equality
Rule adds A ≡ B to the Tg and applies the algorithm of absorption in order
to absorb the new axiom A ≡ B. This axiom could be absorbed completely or
partially. Some part of this axiom may pass to the Tu and another part may re-
main in the Tg . Finally, the Optimized Equality Rule forces the application of the
unfolding rule for >, which ensures that the parts not absorbed of the axioms
A ≡ B be applied to all nodes.
In order to ensure termination, the expansion rules should not be applied more
58
OWL extended with Meta-modelling 5
than once under the same conditions. For the Optimized Equality Rule, we keep
record of the fact that we applied this rule and equated A with B by means of
A ≈ B. It would be wrong to have the condition “A ≡ B does not belong to
Tg ” because parts of that axiom could have been absorbed. For the Optimized
Inequality Rule, we keep record of the fact that we applied this rule and set A
different from B by means of A 6≈ B. This is more efficient than having the al-
ternative condition “there is no root node z such that A u ¬B t B u ¬A ∈ L(z)”
of Table 1.
Optimized Equality Rule:
If a =m A, b =m B, a ≈ b and A ≈ B does not hold then
1. add A ≡ B to the Tg .
2. apply the algorithm of absorption to (Tu , Tg ).
3. apply the unfolding rule for >.
Optimized Inequality Rule:
If a =m A, b =m B, a 6≈ b and A 6≈ B does not hold then
create a new root node z with L(z) = {A u ¬B t B u ¬A, >}
Table 2. Optimized equality and inequality rules
The implementation of Pellet extended with meta-modelling can be found in
http://www.cs.le.ac.uk/people/ps56/pelletM.xml.
3 Conclusions and Related Work
OWL 2 DL has a very restricted form of meta-modelling called punning [3]. In
this approach, the same identifier can be used simultaneously as an individual
and as a concept, but they are semantically treated as different real objects. So,
it does not detect certain inconsistencies as the ones illustrated in Examples 1
and 2. Moreover, this approach is not natural for reusing ontologies. For these
scenarios, it is more useful to assume the identifiers be syntactically different
and allow the user to equate them by using axioms of the form a =m A.
In the literature there are other approaches proposed to deal with meta-modelling
in Description Logic [2, 11–17]. The approaches which define fixed layers or levels
of meta-modelling [11, 13, 15, 16] impose a very strong limitation to the ontology
engineer. Our approach allows the user to have any number of levels or layers
(meta-concepts, meta meta-concepts and so on). The user does not have to write
or know the layer of the concept because the reasoner will infer it for him. In
this way, axioms can also naturally mix elements of different layers and the user
has the flexibility of changing the status of an individual at any point without
having to make any substantial change to the ontology. In a real scenario of
evolving ontologies, that need to be integrated, not all individuals of a given
concept need to have meta-modelling and hence, they do not have to belong to
59
6 OWL with metamodelling
the same level in the hierarchy.
The key feature in our semantics is to interpret a and A as the same object when
a and A are connected through meta-modelling, i.e., if a =m A then aI = AI .
This allows us to detect inconsistencies in the ontologies which is not possible
under the Hilog semantics [2, 14–17]. Our semantics also requires that the do-
main of the interpretation be a well-founded set. A domain such as ∆I = {X}
where X = {X} is a set that belongs to itself which cannot represent any real
object from our usual applications in Semantic Web.
References
1. Servicio Geográfico Militar del Uruguay. Catálogo de objetos y sı́mbolos ge-
ográficos. At http://www.sgm.gub.uy/index.php/documentos/doc download/92-
catalogo-de-objetos-geograficos-y-simbolos-v1. Last date accessed September 2015.
2. B. Motik, On the properties of metamodeling in OWL, in: International Semantic
Web Conference, 2005, pp. 548–562.
3. P. Hitzler, M. Krötzsch, S. Rudolph, Foundations of Semantic Web Technologies,
Chapman & Hall/CRC, 2009.
4. R. Motz, E. Rohrer, P. Severi. Reasoning for ALCQ extended with a flexible meta-
modelling hierarchy, in: Semantic Technology-4th Joint International Conference,
JIST 2014. Revised Selected Papers, pp. 47–62.
5. R. Motz, E. Rohrer, P. Severi. The Description Logic SHIQ with
a flexible meta-modelling hierarchy, in Journal of Web Semantics,
doi:10.1016/j.websem.2015.05.002, open access 2015.
6. Clark, Parsia, Pellet, http://clarkparsia.com/pellet/. Last date accessed September
2015.
7. E. Sirin, B. Parsia, B. Cuenca Grau, A. Kalyanpur, Y. Katz, Pellet: A practical
OWL reasoner, Journal of Web Semantics 5(2):51-53 (2007).
8. I. Horrocks, Implementation and Optimisation Techniques., in: The Description
Logic Handbook: Theory, Implementation, and Applications, Cambridge Univer-
sity Press, 2003, pp. 306–346.
9. Ian Horrocks, Stephan Tobies: Optimisation of Terminological Reasoning. Descrip-
tion Logics 2000.
10. Alexander K. Hudek, Grant E. Weddell: Binary Absorption in Tableaux-Based
Reasoning for Description Logics. Description Logics 2006.
11. J. Z. Pan, I. Horrocks, G. Schreiber, OWL FA: A metamodeling extension of OWL
DL, in: OWLED, 2005.
12. B. Glimm, S. Rudolph, J. Völker, Integrated metamodeling and diagnosis in OWL
2, in: International Semantic Web Conference, 2010, pp. 257–272.
13. N. Jekjantuk, G. Gröner, J.Z. Pan, Modelling and reasoning in metamodelling
enabled ontologies, Int. J. of Software and Informatics 4 (3) (2010) 277–290.
14. G. De Giacomo, M. Lenzerini, R. Rosati, Higher-order description logics for domain
metamodeling, in: AAAI, 2011.
15. M. Homola, J. Kluka, V. Svátek, M. Vacura, Towards Typed Higher-Order De-
scription Logics, in: Description Logics, 2013.
16. M. Homola, J. Kluka, V. Svátek, M. Vacura, Typed Higher-Order Variant of
SROIQ - Why Not?, in: Description Logics, 2014.
17. M. Lenzerini, L. Lepore, A. Poggi, Making Metaquerying Practical for Hi(DLLiteR)
Knowledge Bases, in: On the Move to Meaningful Internet Systems: OTM 2014
Conferences, 2014, 8841 580–596.
60