In this paper we explain how we extended the Web Ontology Language (OWL) with meta-modelling. In order to express metamodelling, 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.
In di erent application scenarios the
need of linking ontologies of di
erent domains arises. But the way how
these ontologies are related is not
always the same. Sometimes it is
required to map classes or individuals
of two ontologies, or link individuals
of two ontologies through a new
property. In these cases, elements of the
same granularity are mapped, i. e.,
classes to classes or individuals to
individuals. But there are some
scenarios where mapping elements of di er- Fig. 1. Two ontologies on Hydrography
ent granularity is needed, for instance
when the same real object is
represented 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
application on geographic objects that requires to reuse existing ontologies and
relate them through meta-modelling [
Example 2. We add two axioms, the rst one says that river and lake as individuals 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.
<MetaModelling>
<NamedIndividual IRI="river"/>
<Class IRI = "River" />
</Metamodelling>
The individual river is interpreted as the set fqueguay; santaLuciag since this
is exactly the interpretation of the class River. Extending the syntax is the
easy part. The most di cult part consists in extending the Tableau algorithm
in order to check consistency of ontologies that have meta-modelling [
B does not belong to the Tbox then, Inequality Rule: If a =m A, b =m B, a 6 b and there is no z, A u :B t B u :A 2 L(z) then, create a new node z with L(z) = fA u :B t B u :Ag.
Close Rule: If a =m A, b =m B and neither a add either a b or a 6 b.
b nor a 6 b then,
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 di erence. 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 Inequality 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 directly 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 di erence.
The rules for equality and inequality are not su cient to detect all
inconsistencies 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 j= A B.
Unfortunately, a recursive call of the form O j= A B is not possible. Otherwise we
will be captured in a vicious circle since the problem of nding out the semantic
consequences is reduced to the one of satis ability. 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 [
Our Tableau algorithm for meta-modelling has a Tbox rule:
If C is a TBox statement and C 62 L(x), then add C to L(x). where C is a TBox axiom C v D codi ed as :C t D.
It is not so straightforward to extend the code of Pellet [
If A v C 2 Tu, A 2 L(x) and C 62 L(x) then add C to L(x).
If :A v C 2 Tu, :A 2 L(x) and C 62 L(x) then add C to L(x).
The general Tbox Tg is expressed as > = d 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 9-rule) adds > to L(z).
We also apply optimization techniques to our meta-modelling rules. The optimized 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 remain 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 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 di erent from B by means of A 6 B. This is more e cient than having the alternative condition \there is no root node z such that A u :B t B u :A 2 L(z)" of Table 1.
Optimized Equality Rule: If a =m A, b =m B, a b and A 1. add A B to the Tg. 2. apply the algorithm of absorption to (Tu; Tg). 3. apply the unfolding rule for >.
B does not hold then 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) = fA u :B t B u :A; >g
The implementation of Pellet extended with meta-modelling can be found in http://www.cs.le.ac.uk/people/ps56/pelletM.xml. 3
OWL 2 DL has a very restricted form of meta-modelling called punning [
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 domain of the interpretation be a well-founded set. A domain such as I = fXg where X = fXg is a set that belongs to itself which cannot represent any real object from our usual applications in Semantic Web.