Introduction

The widespread use of ontologies -formal specifications of shared conceptualisations [10] -together with the increasing availability of representations of overlapping domains of interest, raises the need to resolve heterogeneities [12] [14] by completely or partially mapping between different ontologies. With respect to [2] [17], we define ontology mapping as the process of defining formal relations between knowledge entities which represent the same or a similar semantic meaning in distinct ontologies [6] [19]. In that, ontology mapping strongly relies on identifying similarities [1] between entities across different ontologies. However, with respect to this goal, several issues have to be taken into account. The symbolic approach -i.e. describing symbols by using other symbols, without a grounding in the real world -of established representation standards such as OWL1 or RDF-S 2 which are based on first-order logic (FOL) and syllogistic reasoning [8] leads to ambiguity issues and does not entail meaningfulness, since meaning requires both the definition of a terminology in terms of a logical structure (using symbols) and grounding of symbols to a conceptual level [3] [16]. Therefore, concept representations across distinct ontologies -even those representing the same real-world entities -hardly equal another, since similarity is not an implicit notion carried within ontological representations. But manual or semi-automatic identification of similarity relationships -based on linguistic or structural similarities across ontologies [13] [7] [9] -is costly. Consequently, representational facilities, enabling to implicitly describe similarities across ontologies are required in order to support ontology interoperability.

Conceptual Spaces (CS) [8] follow a theory of describing entities at the conceptual level in terms of their natural characteristics similar to natural human cognition in order to avoid the symbol grounding issue [3] [16]. In that, CS consider the representation of concepts as vector spaces which are defined through a set of quality dimensions. Describing instances as vectors enables the automatic calculation of their semantic similarity by means of spatial distance metrics, in contrast to the costly representation of similarities through symbolic representations. However, several issues still have to be considered when applying CS. For instance, CS do not explicitly prescribe any applicable representation method. Moreover, CS provide no means to represent arbitrary relations between concepts or instances, such as part-of relations. In order to overcome the issues introduced above, we propose a two-fold knowledge representation approach which extends FOL ontologies with a conceptual grounding by refining individual symbolic concept representations as particular CS. Consequently, similarity becomes an implicit notion of the representation itself, instead of relying on manual or semi-automatic similarity detection approaches.

Conceptual Groundings for Ontological Concepts

With respect to the aforementioned issues, we argue that basing knowledge models on just one theory alone might not be sufficient. Therefore, we propose a two-fold representational approach -combining FOL ontologies with corresponding representations based on CS -to enable similarity-based reasoning across ontologies. In that, we consider the representation of a set of n concepts C of an ontology O through a set of n Conceptual Spaces CS. Hence, instances of concepts are represented as members in the respective CS. While still benefiting from implicit similarity information within a CS, our hybrid approach allows overcoming CSrelated issues by maintaining the advantages of FOL-based knowledge representations. In order to be able to represent ontological concepts within CS, we formalised the CS model into an ontology, represented through OCML [15]. Hence, a CS can simply be instantiated in order to represent a particular concept.

Referring to [8] [18], we formalise a CS as a vector space defined through quality dimensions d i of CS. Each dimension is associated with a certain metric scale, e.g. ratio, interval or ordinal scale. To reflect the impact of a specific quality dimension on the entire CS, we consider a prominence value p for each dimension [8]. Therefore, a CS is defined by

( ) { } P p CS d d p d p d p CS i i n n n ∈ ∈ = , ,..., , 2 2 1 1

, where P is the set of real numbers. However, the usage context, purpose and domain of a particular CS strongly influence the ranking of its quality dimensions what supports our position of describing distinct CS explicitly for individual concepts. Please note that we do not distinguish between dimensions and domains [8] but enable dimensions to be detailed further in terms of subspaces. Hence, a dimension within one space may be defined through another CS by using further dimensions [18]. In this way, a CS may be composed of several subspaces, and consequently, the description granularity can be refined gradually. Dimensions may be correlated. Information about correlation is expressed through axioms related to a specific quality dimension instance.

A particular member M -representing a particular instance -in the CS is described through valued dimension vectors v i like

( ) { } M v v v v M i n n ∈ = ,..., ,2 1

. With respect to [18], we define the semantic similarity between two members of a space as a function of the Euclidean distance between the points representing each of the members. However, we would like to point out that different distance metrics, such as the Taxicab or Manhattan distance [11], could be considered, dependent on the nature and purpose of the CS. Given a CS definition CS and two members V and U, defined by vectors v 0 , v 1 , …,v n and u 1 , u 2 ,…,u n within CS, the distance between V and U can be calculated as

∑ = − − − = n i v i u i i s v v s u u p v u dist 1 2 )) ( ) (( ) , (

where u is the mean of a dataset U and u s is the standard deviation from U. The formula above already considers the so-called Z-transformation or standardization [4] which facilitates the standardization of distinct measurement scales in order to enable the calculation of distances in a multi-dimensional and multi-metric space.

Representing Ontological Concepts through Conceptual Spaces

The derivation of an appropriate space CS i to represent a particular concept C i of a given ontology O is understood a non-trivial task which primarily implies the creation of a CS instance which most appropriately represents the real-world entity represented by the symbolic concept representation. We foresee a transformation procedure consisting of the following steps: S1. Representing concept properties pc ij of C i as dimensions d ij of CS i . S2. Assignment of metrics to each quality dimension d ij . S3. Assignment of prominence values p ij to each quality dimension d ij . S4. Representing instances I ik of C i as members in CS i . A specific CS is instantiated by applying a transformation function which is aimed at instantiating all elements of a CS (S1 -S3). S1 aims at representing each concept property pc ij of C i as a particular dimension instance d ij together with a corresponding prominence p ij of a resulting space CS i :

( ) { } ( ) { } P p CS d d p d p d p PC pc pc pc pc trans ij i ij in in i i i i ij in i i ∈ ∈ ⇒ ∈ , ,..., , ,..., , : 2 2 1 1 2 1

Please note that we particularly distinguish between data type properties and relations. While the latter represent relations between concepts, these are not represented as dimensions since such dimensions would refer to a range of concepts (instances) instead of quantified metrics, as required by S2. In the case of relations, we propose to maintain the relationships represented within the original ontology O without representing these within the resulting CS i . In that, the complexity of CS i is reduced to enable the maintainability of the spatial distance as appropriate similarity measure. S2 aims at the assignment of metric scales (interval scale, ratio scale, nominal scale), while S3 is aimed at assigning a prominence value p ij to each dimension d ij . Prominence values should be chosen from a predefined value range, such as 0...1. With respect to S4, one has to represent all instances I ki of a concept C i as member instances in the created space CS i . This is achieved by transforming all instantiated properties pi ikl of I ik as valued vectors in CS i .

( ) { } ( ) { } ik ikl ikn ik ik l ikl ikn ik ik M v v v v PI pi pi pi pi trans ∈ ⇒ ∈ ,..., , ,..., , : 2 1 2 1

Hence, given a particular CS, representing instances as members becomes just a matter of assigning specific measurements to the dimensions of the CS. In order to represent all concepts C i of a given ontology O, the transformation function consisting of the steps S1-S4 has to be repeated iteratively for all C i which are element of O. The accomplishment of the proposed procedure results in a set of CS instances which each refine a particular concept together with a set of member instances which each refine a particular instance.

Conclusion

In order to facilitate ontology mapping, we proposed a hybrid representation approach based on a combination of FOL ontologies and multiple concept representations in individual CS. Representing concepts following the CS theory enables representation of instances as vectors in a respective CS and consequently, the automatic computation of similarities by means of spatial distances. A CS-based representation is supported through a dedicated CS formalisation, i.e. a CS ontology, and a formal method on how to derive CS representations for individual concepts. Within proof-ofconcept prototype applications, e.g. [5], an OCML [15] representation of the proposed hybrid representational model was utilized to validate the applicability of the approach. Following our two-fold representational approach supports implicit representation of similarities across heterogeneous ontologies, and consequently, provides a means to facilitate ontology mapping. Moreover, our approach overcomes certain individual issues posed by each of the two approaches. Whereas traditional ontology mapping methodologies rely on mechanisms to semi-automatically detect similarities at the concept and the instance level, our approach just requires a common agreement at the concept level since similarity information at the instance level is implicitly defined.

However, the authors are aware that our approach requires a considerable amount of additional effort to establish CS-based representations. Future work has to investigate this effort in order to further evaluate the potential contribution of the approach proposed here. Moreover, further issues related to CS-based knowledge representations still remain. For instance, whereas defining instances, i.e. vectors, within a given CS appears to be a straightforward process, the definition of the CS itself is not trivial at all and dependent on subjective perspectives. With regard to this, CS do not fully solve the symbol grounding issue but to shift it from the process of describing instances to the definition of a CS. Nevertheless, distance calculation relies on the fact that resources are described in equivalent (or mapped) geometrical spaces. However, we would like to point out that the increasing usage of upper level ontologies and the progressive reuse of ontologies, particularly in loosely coupled organisational environments, leads to an increased sharing of ontologies at the concept level. As a result, our proposed hybrid representational model becomes increasingly applicable by further enabling similarity-computation at the instancelevel towards the vision of interoperable ontologies.