=Paper=
{{Paper
|id=Vol-1248/paper6
|storemode=property
|title=Metamodelling in a Ontology Network
|pdfUrl=https://ceur-ws.org/Vol-1248/WoMO14-Paper8.pdf
|volume=Vol-1248
|dblpUrl=https://dblp.org/rec/conf/fois/RohrerSMD14
}}
==Metamodelling in a Ontology Network==
https://ceur-ws.org/Vol-1248/WoMO14-Paper8.pdf
Metamodelling in a Ontology Network
Edelweis ROHRER a , Paula SEVERI b , Regina MOTZ a and Alicia DÍAZ c
a
Instituto de Computación, Facultad de Ingenierı́a,
Universidad de la República, Uruguay
b
Department of Computer Science, University of Leicester, England
c
LIFIA, Facultad de Informática,
Universidad Nacional de La Plata, Argentina
Abstract. When designing ontology networks, ontologies can be com-
bined through mapping or metamodelling relationships, among others.
The coexistence of metamodelling with other kind of relationships in a
ontology network leads to identify a set of problems, which are the focus
of the present work.
Keywords. Ontology network, Metamodelling, Well-founded model
Introduction
Haase et al. [1] define an ontology network as “a collection of ontologies related
together via a variety of different relationships such as mapping, modularization,
version, and dependency relationships”, denominating the component ontologies
as “networked ontologies”.
Regarding ontology engineering, in different case studies the need of linking
ontologies of different domains arises. In general, the way how these ontologies
are related is not always the same. For instance, sometimes it is required to align
concepts of two ontologies, while in other cases reusing a whole ontology and
extending it can be the most suitable alternative. Sometimes it is necessary to
link individuals of two ontologies through a new role and in other cases, one of
the ontologies is the metamodel for the other.
Like the modular software development in software engineering, the fenomenon
of reusing and linking existing ontologies have also been studied. There exist sev-
eral proposals of logics which define a formal semantics of integration of modu-
lar ontologies. Serafini et al. [2] compare these formalisms, which define a new
syntax and semantics to control the interaction between the modules. These set
of formalisms are refered to as “Modular Ontology Languages” by Cuenca et al.
[3]. The most prominent ones are ε-connections [4,5], Package-based Description
Logics [6], Distributed Description Logics [7] and Integrated Distributed Descrip-
tion Logics [8]. Besides, there is a different approach that establishes some non-
standard reasoning services over the union of a set of ontologies, testing different
aspects of the semantics of the union ontology and their components. This ap-
proach, called “Modular Reuse of Ontologies” provides a mechanism to ensure a
“safe” combination of the involved domain ontologies [9].
In the present work we identify a set of possible relationships to relate on-
tologies, preliminarily presented in [10], which from the point of view of ontol-
ogy engineering result naturally distinct at the moment of building an ontology
network. This set of relationships allows the ontology engineer to conceptualize
the ontology network at a higher level of granularity, visualizing the interaction
among the involved ontologies.
Among the set of selected ontology relationships, we pay special attention to
the metamodelling relationship. The remainder relationships have been broadly
studied, but up to our knowledge, the scenario in which metamodelling coexists
with other kind of relationships in a ontology network has not been studied from
the point of view of what inconsistencies or contradictions can arise, which can
not be detected by a standard Description Logics (DL) reasoner.
There exist some works that define a different semantics for a knowledge base
when there is metamodelling, either specifying different ”layers” or ”stratums”
[11], or treating a symbol of the signature which is both an instance and a concept,
as two independent elements when reasoning [12]. But, up to our knowledge none
of them adress the problem of what contradictions can arise when there are also
different axioms involving these symbols.
In our work, we redefine the semantics of the metamodelling relationship to
interpret an instance and its corresponding concept or role in the metamodelling
relationship as the same element in the domain of discourse. Moreover, regarding
the coexistence of metamodelling and other relationships, we propose an addi-
tional condition which must be satisfied in the ontology network, when there are
metamodelling relationships.
The remainder of this paper is organized as follows. Section 1 defines an on-
tology network semantics to contemplate the metamodelling relationship. Section
2 addresses the issue of having metamodelling along with other relationships in a
ontology network. Section 3 define a condition that must be satisfied in order to
avoid a poor design choice when metamodelling. Section 4 gives an overview of
the existing literature about metamodelling in the ontology design and the theo-
retical background regarding metamodelling semantics. Finally, Section 5 analizes
some conclusions and proposes some future work.
1. Ontology Network
In this section we define four conceptually different relationships between ontolo-
gies. For three of them, mapping, link and extension, we relied on existing ap-
proaches of the literature [6,4,3]. For the remaining relationship, the metamod-
elling relationship, we redefine the semantics of the interpretation of an ontology
network. We base our definitions on the foundations of Description Logics syntax
and semantics, which can be reviewed by the reader in [13,14].
Definition 1 (Relationship between Ontologies) R is a relationship between the
ontologies O1 and O2 if R is a relation (or set of relations) between the signatures
of O1 and O2 .
Definition 2 (Extension) [3] R is an extension relationship between O1 and O2 if
the signature and axioms of O1 are included in the ones of O2 .
Definition 3 (Mapping) [6] R is a mapping relationship between O1 and O2 if
there exists a set RA of axioms that have one of the following forms:
CvD C≡D C uD v⊥ D(a) {a} ≡ {b}
where C, a ∈ O1 and D, b ∈ O2 .
Definition 4 (Link) [4] R is a link relationship between O1 and O2 if:
1. there exists a set RL of new roles called linking roles. Given RL, we define
the languages C1 and C2 of concepts generated from RL by simmultaneous
induction. The rules for C1 are as follows.
(a) Any basic concept of O1 is in C1 ,
(b) C1 is closed under t, ¬,
(c) If C ∈ C1 and R ∈ O1 then ∃R.C, and ≥ nR.C belong to C1 ,
(d) If C ∈ C2 and L ∈ RL then ∃L.C, and ≥ nL.C belong to C1 .
The rules for C2 are as follows.
(a) Any basic concept of O2 is in C2 ,
(b) C2 is closed under t, ¬,
(c) If C ∈ C2 and R ∈ O2 then ∃R.C, and ≥ nR.C belong to C2 ,
(d) If C ∈ C1 and L ∈ RL then ∃L−1 .C, and ≥ nL−1 .C belong to C2 .
2. there exists a set RA of axioms that have one of the following forms:
C1 v D1 C2 v D 2 C1 (a1 ) C2 (a2 ) L(a1 , a2 ) L−1 (a2 , a1 )
where all C1 , D1 , a1 belong to C1 , C2 , D2 , a2 belong to C2 and L is a linking
role belonging to RL.
Definition 5 (Metamodeling) We say that R is a metamodelling relationship be-
tween O1 and O2 if there exists a partial function m from the set of individuals
of O1 to the set of atomic concepts and roles (primitives and defined) of O2 . The
set RA for metamodelling relationships is the empty set.
Figure 1 illustrates link and metamodelling relationships.
Figure 1. Link and Metamodelling Relationships
Definition 6 (Ontology Network) An ontology network is a pair (O, R) such that
O = {O1 , . . . , On } is a set of ontologies and R = {R1 , . . . , Rm } is a set of ontology
relationships between them. If we name mi the partial function associated to a
metamodelling relationship Ri and there exist mi , mj such that a ∈ dom(mi ) and
a ∈ dom(mj ), (i) if mi (a) belongs to the set of atomic concepts (AC) then mj (a)
also belongs to it and (ii) if mi (a) belongs to the set of atomic roles (AR) then
mj (a) also belongs to it. The ontology associated to an ontology network is denoted
Sn Sm
by Ont(O, R) and defined as i=1 Oi ∪ i=1 RAi where RAi are the set of DL
axioms associated to the relationship Ri for all 1 ≤ i ≤ m.
The semantics of a DL is based on interpretations I = (∆I ,.I ), where the
domain ∆I is a non-empty set, and .I is the interpretation function [13]. That
is, the semantics of a single ontology is defined in terms of a single interpretation
domain ∆I . When more than one ontology interact through different relation-
ships, as in a ontology network, regarding their semantics, there are two main ap-
proaches already mentioned in the introduction:“Modular Ontology Languages”
(MOL) [2] and “Modular Reuse of Ontologies” [9]. In the former each ontology
or “module” is represented through a local language and a local semantics. The
meaning of local symbols is interpreted within each ontology, whereas the mean-
ing of symbols which are external to an ontology is given by a special semantics,
specific for each formalism in this approach. That is, a different interpretation
domain is considered for each ontology in the network. Unlike this, the “Modular
Reuse of Ontologies” approach specifies a single domain of interpretation for the
union of all ontology axioms, that is, a single domain for the ontology network.
The discussion about which approach to take is not the focus of the present work,
but we chose the second one because the “Modular Reuse of Ontologies” approach
is based on the standard DL formalism without defining a special syntax and
the specification of a single interpretation domain appears as a simpler approach,
which seems the right when a developer combines different ontologies to describe
a particular application.
Having adopted the approach of a single interpretation domain, we will de-
fine the semantics of a ontology network taking into account the relationships:
mapping, link, extension and metamodelling. For the last one, in Definition 6 we
established that an individual a in an ontology O1 has a corresponding concept or
role m(a) in another ontology O2 . This means that, in the ontology network, the
individual a must be interpreted as a set of domain elements, the interpretation
of the concept m(a), or as a binary relation, the interpretation of the role m(a),
respectively. Then, to support this new semantics, we redefine the interpretation
domain. Before giving a formal definition of it, we introduce the idea through a
very simple example.
Figure 2. Metamodelling Example
Figure 2 shows an ontology network about politics over natural resources.
The ontology “Natural Resources Politics” conceptualize the politics (preserva-
tion, purification) applied to different natural resources (mountains, rivers). The
ontology “Natural Resources” conceptualize at a lower abstraction level the ex-
isting natural resources, and moreover companies and organizations that imple-
ment the particular politics. So at an upper abstraction level, mountains and
rivers are individuals, while they are concepts at a lower level. Moreover, preser-
vation and purification are individuals in the ontology “Natural Resources Poli-
tics”, corresponding to roles (preservedBy, purifiedBy) in the ontology “Natural
Resources”. Then, in this example our interpretation domain is composed by:
(i) atomic elements: Aconcagua, Everest, Danubio, Amazonas, ON U , P RES,
P U RE, W AT ER), (ii) sets of elements: M ountain = {Aconcagua, Everest}
and River = {Danubio, Amazonas} corresponding to the individuals mountain
and river in the upper level and (iii) binary relations of sets of elements:
purif iedBy = {hDanubio, P U REi}, preservedBy = {hAmazonas, ON U i} cor-
responding to the individuals purif ication and preservation in the upper level.
First we assume at ground level that we have a set ∆I0 for our interpretation
domain ∆I that contains some atomic objects. Then, for the first level of meta-
modelling we introduce ∆I1 as the set that besides containing all elements in ∆I0
it also contains all subsets and relations on ∆I0 .
Definition 7 (Ontology Network Domain of Interpretation) Given a non empty
set ∆I0 of atomic objects, we define the ontology network domain ∆I of interpre-
tation as follows:
∆I = I I
S
n>0 ∆n , where ∆n is inductively defined as:
∆In = ∆In−1 ∪ P(∆In−1 ) ∪ P(∆In−1 × ∆In−1 )
From Definition 7, we can see that the domain of a ontology network is a well-
founded set. 1
Figure 3 shows a concrete example of the defined domain of interpretation.
Figure 3. Domain of Interpretation - Example
Definition 8 (Ontology Network Interpretation) An ontology network Interpre-
tation I is a pair I = (∆I ,.I ), where ∆I is a ontology network domain of inter-
pretation, and .I is the interpretation function that assigns:
• to every concept A a subset AI ⊆ ∆I
• to every role R a subset RI ⊆ ∆I × ∆I
• to every individual a an element aI ∈ ∆I
In the usual way, the interpretation function .I is extended to complex concepts
and roles via DL-constructors, see [14].
Considering standard DL [13,14], I is a model if it satisfies all standard DL
axioms in the ontology network, which are basically the TBox axioms of concept
subsumption, C v D, and ABox axioms of concept and role assertions, C(a) and
R(a, b). But now we also have metamodelling relationships, in which an instance
corresponds to a concept or a role. Then, the instance interpretation will coincide
with the concept or role interpretation. This lead to the following definition of
model of a ontology network.
Definition 9 (Model of a Ontology Network) An interpretation I of a ontology
network is a model if the following holds:
1. I is a model of Ont(O, R) where Ont(O, R) is the ontology associated to
the ontology network without the metamodelling.
2. Moreover, for all 1 ≤ i ≤ m, if Ri is a metamodelling relationship given
by a partial function m, for all a ∈ dom(m):
a. aI = C I when m(a) = C, C ∈ AC
b. aI = RI when m(a) = R, C ∈ AR
1 A relation S is well-founded if every non-empty subset S 0 has a minimal element. In set
theory, a set X is called a well-founded set if the set membership relation is well-founded on the
transitive closure of X.
Definition 10 (Consistency of an Ontology Network) We say that an ontology
network (O, R) is consistent if there exists a model of (O, R).
The first part of Definition 9 refers to a model which satisfy the union of the
axioms of all ontologies in the network plus the axioms expressing the relation-
ships: mapping, link and extension, which are expressed in standard DL. In the
second part of the definition, we add another condition that the model must
satisfy considering the metamodelling relationships. This condition restricts the
interpretation of an individual that has a corresponding concept or role in a
metamodelling relationship to be equal to the concept or role interpretation. In
the example of Figure 2 we have a metamodelling relationship with the following
correspondences:
m(mountain) = M ountain
m(river) = River
m(preservation) = preservedBy
m(purif ication) = purif iedBy
The correspoding interpretation of the individuals which belong to the domain of
the partial function m is:
mountainI = M ountainI = {Aconcagua, Everest}
riverI = RiverI = {Amazonas, Danubio}
preservationI = preservedBy I = {hAmazonas, ON U i}
purif icationI = purif iedBy I = {hDanubio, P U REi}
If we also had the axioms mountain = river and M ountain u River v ⊥, and
there was not a metamodelling relationship, the ontology Ont(O, R) associated to
the ontology network would be consistent. On the other hand, if we have the meta-
modelling relationship given by m(mountain) = M ountain, m(river) = River,
the ontology Ont(O, R) is no longer consistent.
We extend the notion of logical consequence to ontology networks in the obvious
way [14].
Definition 11 (Logical Consequence) We say that J is a logical consequence of
(O, R) (denoted as (O, R) |= J ) if all models of (O, R) are also models of J where
J may be any typical DL judgement (depending on the DL language of choice)
such as: C v D, a ∈ C, R v S or ha, bi ∈ R.
2. Metamodelling coexisting with other Relationships
Given the definition of model of a ontology network, which take into account the
existence of metamodelling relationships, we pose two questions: (i) is it possible
to infer new knowledge in the ontology network, from the metamodelling rela-
tionships? (ii) are the standard mechanisms of reasoning enough when there exist
metamodelling relationships?
The answer to the first question is obviously positive and shown by the previous
example where River ≡ M ountain is a semantic consequence of the ontology
network only when we take into account the metamodelling relationships.
The standard services of reasoning provided for DL make it possible to check the
consistency of a knowledge base and to infer new axioms that are not explicitly
declared. For the first part of the Definition 9 a model can be obtained using the
standard mechanisms of reasoning, which comprise all axioms expressed in DL,
so including the relationships mapping, link and extension. However, the Tableau
algorithm does not consider the metamodelling relationships, that is, does not
”know” that the interpretation of an individual in one ontology coincides with
the interpretation of a concept or role in another ontology, because of a meta-
modelling relationship.
Then, in order to try to answer the second question, we will extend the ontol-
ogy Ont(O, R), which is the ontology associated to the ontology network (O, R)
without metamodelling, with new axioms. At first, the new extended ontology
Ont∗ (O, R) is set equal to Ont(O, R), and then we extend it as follows 2 .
1. If Ont∗ (O, R) |= a = b and we have the metamodelling relationships
mi (a) = X and mj (b) = Y , with a and b individuals, X and Y both atomic
concepts or roles, and does not exists a TBox axiom X ≡ Y , we add this
axiom to Ont∗ (O, R).
2. If Ont∗ (O, R) |= X ≡ Y and we have the metamodelling relationships
mi (a) = X and mj (b) = Y , with X and Y both atomic concepts or roles,
and does not exist an ABox axiom a = b, we add this axiom.
In the above rules, we execute a DL-reasoner to obtain the entailments of the
form: X ≡ Y and a = b, for X and Y both atomic concepts or roles, a and b
both individuals. These rules have to be applied several times for each entailment
of the form X ≡ Y and a = b until no more rules 1 and 2 can be applied. This
process always terminate since we are considering an ontology network with a
finite set of atomic concepts, atomic roles and individuals. In spite of the fact
that the set of entailments may increase for we are adding new axioms each time,
from some point on it should stabilize.
We consider the example of Figure 2 with the axioms mountain = river and
M ountainuRiver v ⊥. Using the rule 1, the axiom M ountain ≡ River is added.
Now we can apply any DL-reasoner to the extended ontology and this will return
that the new knowledge base is inconsistent.
We also show an example where the set of inferences of the form a = b or
X = Y increases and new rules need to be applied that were not visible be-
fore having those new inferences. For this, we consider the example of Figure 2
with the axiom M ountain = River and a functional property hasP olitics such
2 Similar inference rules as the ones in Section 2 appear in Jekjantuk et al. [11], who analized
metamodelling in a single ontology and defined the interpretation domain fragmented in layers.
that hasP olitics(mountain, preservation) and hasP olitics(river, purif ication).
We apply first rule 2 and add that mountain = river to the ontology. Now,
since hasP olitics is functional, we have a new inference that we did not have
before, which is preservation = purif ication. Then, we apply rule 1 and add
that purif iedby = preservedby.
The following lemma is very easy to prove:
Lemma 1 If (O, R) is consistent then so is Ont∗ (O, R).
However, the converse does not hold as the following counterexample shows.
We add a mapping relationship to the ontology network of Figure 2:
RA2 = {N aturalResource v M ountain}
It is easy to see that Ont∗ (O, R) is consistent. However, (O, R) is inconsistent
because now for any model I of (O, R) we have that:
M ountainI = mountainI ∈ N aturalResourceI ⊆ M ountainI
That is, the set M ountainI is a non well-founded set, since belongs to itself.
This contradicts one of our basic prerequisite for being a model of an ontology
network: the domain of the interpretation should be well-founded (see Definition
7).
The following lemma is also easy to prove:
Lemma 2 If Ont∗ (O, R) |= J then (O, R) |= J .
However, the converse does not hold. To see this, we can apply the counterexample
given after the previous lemma.
3. Stratified Ontology Network
In order to avoid interpretations that have sets with cyclic definitions, we have
introduced the notion of domain of an ontology network (Definition 7) which is
well-founded. However, we think that in order to ensure that our ontology network
has a sensible design we need to require a stronger condition on our sets. For this,
we define the notion of stratified set.
Definition 12 (Meta Membership) Let X, Y be sets. We define that X is a meta-
member of Y (denoted as X ∈M Y ) by induction as follows:
1. X ∈M Y if X ∈ Y ;
2. X ∈M Y if hX, Zi ∈ Y or hZ, Xi ∈ Y ;
3. X ∈M Y if there exists Z such that X ∈M Z and Z ∈M Y ,
For example, river ∈M Y = {{{river}}}.
Note that the above definition can be applied to relations and pairs. If ha, bi ∈ R
and hR, bi ∈ S then ha, bi ∈M S where R and S are binary relations on sets.
Definition 13 (Stratified set) We say that a set X is stratified if for all x, y ∈ X,
we have that x 6∈M y.
Note that AI = {P eter, {Simon}} is a stratified set. This situation arises in a
ontology that has a concept with two elements and only one element has a meta-
modelling.
In the example of Figure 3, we have that:
{a, b, c, d, e, f, g, h, i, j, k} and {a, b, e, f, {{h, g, i}, {j, k}}} are stratified sets, but
{a, e, f, i, k, {e, f, g}, {j, k}} and {a, b, e, f, h, k, {{h, g, i}, {j, k}}} are not strati-
fied sets.
Definition 14 (Stratified Interpretation) An interpretation I of (O, R) is strati-
fied if for atomic concepts A, atomic roles R and individuals a, we have that AI
is a stratified set, RI ⊆ X × X for some stratified set X, and if aI is not an
atomic object then it is a stratified set or if aI is a relation then aI ⊆ X × X for
some stratified set X.
Definition 15 (Stratified Ontology Network) An ontology network (O, R) is strat-
ified if there exists a model of (O, R) which is a stratified interpretation.
Now we add a different mapping relationship to the example of Figure 2:
RA3 = {M ountain v N aturalResource}
None of the models of this ontology network is stratified. To see this, suppose we
have a model I of this ontology network then:
AconcaguaI ∈ M ountainI ⊆ N aturalResourceI
M ountainI ∈ N aturalResourceI
In the above, there is nothing that contradicts the condition of well-foundness.
However, N aturalResourceI is not a stratified set since it contains two elements
Aconcagua and M ountainI where the first element belongs to the second one.
We add the following link relationship in the ontology network of Figure 2:
RA4 = {handledBy(purif ication, P U RE)} where handledBy is a new role.
None of the models of this ontology network is stratified. This is because in any
model I of this ontology network we have that if handledBy I ⊆ X × X then X
is not a stratified set since purif icationI and P U RE I should belong to X, but
P U RE I ∈M purif icationI .
4. Related Work
Up to our knowledge, the works that address metamodelling in depth, consider
the issue for a single ontology. De Giacomo et al. [12] specifies a new formalism,
“Higher/Order Description Logics”, that allows to treat the same symbol of the
signature as an instance, a concept and a role. With respect to the semantics,
in principle they associate a domain element to each symbol of the signature,
and then, if it is treated as a concept or a role, a set of domain elements or a
binary relation is also associated to the symbol through a pair of functions IC
and IR . Regarding reasoning, given a DL L they define a “high-order” version of
it, Hi(L), mapping the same symbol of the signature to three different symbols,
which represent an instance, a concept or a role. This makes it possible to use
the standard mechanisms of reasoning for the DL L, which treat the three new
symbols as independent elements. Unlike this, in our approach we define meta-
modelling between two different ontologies, keeping the individual in one ontology
and the concept or role in the other one as different symbols of the signature.
In [11], metamodelling is addressed defining different “layers” or “stratums”
within a knowledge base, in such a way that instances in each layer belong to
the lower layer. They propose an algorithm to infer new axioms that arise from
metamodelling, but do not allow axioms different from metamodelling involving
elements of different layers. So, the problem of coexistence of metamodelling and
other relations among different layers is not addressed.
Glimm et al. [15] define two layers within the knowledge base, the “instance
layer” and the “metalayer”. Then, they add special roles to map concepts and
roles in the model (“instance layer”) to instances in the metamodel (“metalayer”),
as well as additional axioms to constrain the roles being introduced. That is, they
only consider two layers and, although they study the coexistence of metamod-
elling and other axioms, they introduce additional elements in order to represent
metamodelling through DL and do not use another formalism.
5. Conclusion and Future Work
In the present work we study the metamodelling relationship in the context of
a ontology network, when there are other relationships such as mapping, link or
extension along with metamodelling relationships. We specify a semantics for the
ontology network, redefining the domain of interpretation in such a way the inter-
pretation of an individual can coincide with that of a concept or a role. Moreover,
in our metamodelling semantics definition, we associate the same interpretation to
both symbols, since as we explained through examples, it is important to “know”
that they are the same domain element. If not, when they are in certain axioms
combined with other elements, an ontology network that is consistent without
metamodelling can becomes inconsistent.
We know that metamodelling combined with expressive DL can become un-
decidable [16]. We plan to study the problem that checks whether an ontology
network is stratified or not. We think that studying metamodelling in the context
of ontology networks is an interesting and very challenging issue.
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