Metamodelling in a Ontology Network

Abstract

When designing ontology networks, ontologies can be combined 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 ] de ne an ontology network as \a collection of ontologies related together via a variety of di erent relationships such as mapping, modularization, version, and dependency relationships", denominating the component ontologies as \networked ontologies".

Regarding ontology engineering, in di erent case studies the need of linking ontologies of di erent 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 several proposals of logics which de ne a formal semantics of integration of modular ontologies. Sera ni et al. [ 2 ] compare these formalisms, which de ne 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 "-connections [ 4,5 ], Package-based Description

Logics [6], Distributed Description Logics [7] and Integrated Distributed Descrip

tion Logics [ 8 ]. Besides, there is a di erent approach that establishes some nonstandard reasoning services over the union of a set of ontologies, testing di erent aspects of the semantics of the union ontology and their components. This approach, 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 ontologies, preliminarily presented in [ 10 ], which from the point of view of ontology 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 de ne a di erent semantics for a knowledge base when there is metamodelling, either specifying di erent "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 di erent axioms involving these symbols.

In our work, we rede ne 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 additional condition which must be satis ed in the ontology network, when there are metamodelling relationships.

The remainder of this paper is organized as follows. Section 1 de nes an ontology 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 de ne a condition that must be satis ed 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 theoretical background regarding metamodelling semantics. Finally, Section 5 analizes some conclusions and proposes some future work.

1. Ontology Network

In this section we de ne four conceptually di erent relationships between ontologies. For three of them, mapping, link and extension, we relied on existing approaches of the literature [ 6,4,3 ]. For the remaining relationship, the metamodelling relationship, we rede ne the semantics of the interpretation of an ontology network. We base our de nitions on the foundations of Description Logics syntax and semantics, which can be reviewed by the reader in [ 13,14 ].

De nition 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.

De nition 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.

De nition 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: C v D

C

D

C u D v ?

D(a) fag fbg where C; a 2 O1 and D; b 2 O2.

De nition 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 de ne 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 2 C1 and R 2 O1 then 9R:C, and (d) If C 2 C2 and L 2 RL then 9L:C, and 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 2 C2 and R 2 O2 then 9R:C, and (d) If C 2 C1 and L 2 RL then 9L 1:C, and nR:C belong to C1, nL:C belong to C1. nR:C belong to C2, 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 D2

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.

De nition 5 (Metamodeling) We say that R is a metamodelling relationship between 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 de ned) of O2. The set RA for metamodelling relationships is the empty set.

De nition 6 (Ontology Network) An ontology network is a pair (O; R) such that

O = fO1; : : : ; Ong is a set of ontologies and R = fR1; : : : ; Rmg 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 2 dom(mi) and a 2 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 by Ont(O; R) and de ned as Sin=1 Oi [ Sim=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 de ned in terms of a single interpretation domain I . When more than one ontology interact through di erent relationships, as in a ontology network, regarding their semantics, there are two main approaches 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 meaning of symbols which are external to an ontology is given by a special semantics, speci c for each formalism in this approach. That is, a di erent interpretation domain is considered for each ontology in the network. Unlike this, the \Modular Reuse of Ontologies" approach speci es 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 de ning a special syntax and the speci cation of a single interpretation domain appears as a simpler approach, which seems the right when a developer combines di erent ontologies to describe a particular application.

Having adopted the approach of a single interpretation domain, we will dene the semantics of a ontology network taking into account the relationships: mapping, link, extension and metamodelling. For the last one, in De nition 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 rede ne the interpretation domain. Before giving a formal de nition of it, we introduce the idea through a very simple example. First we assume at ground level that we have a set 0I for our interpretation domain I that contains some atomic objects. Then, for the rst level of metamodelling we introduce 1I as the set that besides containing all elements in 0I it also contains all subsets and relations on 0I .

De nition 7 (Ontology Network Domain of Interpretation) Given a non empty

set 0I of atomic objects, we de ne the ontology network domain I of interpretation as follows:

I = Sn>0

In, where

In is inductively de ned as: From De nition 7, we can see that the domain of a ontology network is a wellfounded set. 1

De nition 8 (Ontology Network Interpretation) An ontology network Interpre

tation I is a pair I = ( I ;:I ), where I is a ontology network domain of interpretation, and :I is the interpretation function that assigns: to every concept A a subset AI I to every role R a subset RI I to every individual a an element aI 2

I

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 satis es 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 de nition of model of a ontology network.

De nition 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 2 dom(m): a. aI = CI when m(a) = C, C 2 AC b. aI = RI when m(a) = R, C 2 AR 1A relation S is well-founded if every non-empty subset S0 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.

De nition 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 rst part of De nition 9 refers to a model which satisfy the union of the axioms of all ontologies in the network plus the axioms expressing the relationships: mapping, link and extension, which are expressed in standard DL. In the second part of the de nition, 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 = fAconcagua; Everestg riverI = RiverI = fAmazonas; Danubiog preservationI = preservedByI = fhAmazonas; ON U ig purif icationI = purif iedByI = fhDanubio; P U REig 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 metamodelling 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 ].

De nition 11 (Logical Consequence) We say that J is a logical consequence of

(O; R) (denoted as (O; R) j= 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 2 C, R v S or ha; bi 2 R.

2. Metamodelling coexisting with other Relationships

Given the de nition 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 relationships? (ii) are the standard mechanisms of reasoning enough when there exist metamodelling relationships? The answer to the rst 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 rst part of the De nition 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 metamodelling relationship.

Then, in order to try to answer the second question, we will extend the ontology Ont(O; R), which is the ontology associated to the ontology network (O; R) without metamodelling, with new axioms. At rst, 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) j= 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) j= 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 nite 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 ountain u River 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 before 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 2Similar inference rules as the ones in Section 2 appear in Jekjantuk et al. [ 11 ], who analized metamodelling in a single ontology and de ned the interpretation domain fragmented in layers. that hasP olitics(mountain; preservation) and hasP olitics(river; purif ication). We apply rst 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 = fN aturalResource v M ountaing 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 2 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 De nition 7).

The following lemma is also easy to prove: Lemma 2 If Ont (O; R) j= J then (O; R) j= J .

However, the converse does not hold. To see this, we can apply the counterexample given after the previous lemma.

3. Strati ed Ontology Network

In order to avoid interpretations that have sets with cyclic de nitions, we have introduced the notion of domain of an ontology network (De nition 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 de ne the notion of strati ed set.

De nition 12 (Meta Membership) Let X; Y be sets. We de ne that X is a meta

member of Y (denoted as X 2M Y ) by induction as follows: 1. X 2M Y if X 2 Y ; 2. X 2M Y if hX; Zi 2 Y or hZ; Xi 2 Y ; 3. X 2M Y if there exists Z such that X 2M Z and Z 2M Y , For example, river 2M Y = fffriverggg.

Note that the above de nition can be applied to relations and pairs. If ha; bi 2 R and hR; bi 2 S then ha; bi 2M S where R and S are binary relations on sets. De nition 13 (Strati ed set) We say that a set X is strati ed if for all x, y 2 X, we have that x 62M y.

Note that AI = fP eter; fSimongg is a strati ed set. This situation arises in a ontology that has a concept with two elements and only one element has a metamodelling.

In the example of Figure 3, we have that: fa; b; c; d; e; f; g; h; i; j; kg and fa; b; e; f; ffh; g; ig; fj; kggg are strati ed sets, but fa; e; f; i; k; fe; f; gg; fj; kgg and fa; b; e; f; h; k; ffh; g; ig; fj; kggg are not stratied sets.

De nition 14 (Strati ed Interpretation) An interpretation I of (O; R) is strati

ed if for atomic concepts A, atomic roles R and individuals a, we have that AI is a strati ed set, RI X X for some strati ed set X, and if aI is not an atomic object then it is a strati ed set or if aI is a relation then aI X X for some strati ed set X.

De nition 15 (Strati ed Ontology Network) An ontology network (O; R) is strat

i ed if there exists a model of (O; R) which is a strati ed interpretation. Now we add a di erent mapping relationship to the example of Figure 2: RA3 = fM ountain v N aturalResourceg None of the models of this ontology network is strati ed. To see this, suppose we have a model I of this ontology network then:

AconcaguaI 2 M ountainI N aturalResourceI

M ountainI 2 N aturalResourceI In the above, there is nothing that contradicts the condition of well-foundness. However, N aturalResourceI is not a strati ed set since it contains two elements Aconcagua and M ountainI where the rst element belongs to the second one. We add the following link relationship in the ontology network of Figure 2: RA4 = fhandledBy(purif ication; P U RE)g where handledBy is a new role. None of the models of this ontology network is strati ed. This is because in any model I of this ontology network we have that if handledByI X X then X is not a strati ed set since purif icationI and P U REI should belong to X, but P U REI 2M 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 ] speci es 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 de ne a \high-order" version of it, Hi(L), mapping the same symbol of the signature to three di erent 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 de ne metamodelling between two di erent ontologies, keeping the individual in one ontology and the concept or role in the other one as di erent symbols of the signature.

In [ 11 ], metamodelling is addressed de ning di erent \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 di erent from metamodelling involving elements of di erent layers. So, the problem of coexistence of metamodelling and other relations among di erent layers is not addressed.

Glimm et al. [ 15 ] de ne 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 metamodelling 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, rede ning the domain of interpretation in such a way the interpretation of an individual can coincide with that of a concept or a role. Moreover, in our metamodelling semantics de nition, 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 undecidable [ 16 ]. We plan to study the problem that checks whether an ontology network is strati ed or not. We think that studying metamodelling in the context of ontology networks is an interesting and very challenging issue.

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