This paper deals with the problem of building a common knowledge body for a set of domain ontologies in order to enable their sharing and integration in a collaborative framework. We propose a novel hierarchical algorithm for concept fuzzy set representation mediated by a reference ontology. In contrast to the original concept representations based on instances, this enables the application of methods of fuzzy logical reasoning in order to characterize and measure the degree of the relationships holding between concepts from di erent ontologies. We presenta an application of the approach in the multimedia domain.
In collaborative contexts, multiple independently created ontologies often need to be brought together in order to enable their interoperability. These ontologies have an impaired collaborative functionality, due to heterogeneities coming from the decentralized nature of their acquisition, di erences in scopes and application purposes and mismatches in syntax and terminology.
We present an approach to building a combined knowledge body for a set of domain ontologies, which captures and exposes various relations holding between the concepts of the domain ontologies, such as their relative generality or speci city, their shared commonality or their complementarity. This can be very useful in a number of real-life scenarioss, especially in collaborative platforms. Let us imagine a project which includes several partners, each of which has its own vocabulary of semantically structured terms that describes its activity. The proposed framework would allow every party to keep its ontology and work with it, but query the combined knowledge body whenever collaboration is necessary. Examples of such queries can be: \which concept of a partner P1 is closest to my concept A", or \give me those concepts of all of my partners which are equally distant to my concept B", or \ nd me a concept from partner P2 which is a strong subsumer of my concept C", or "what are the commonality and speci city between my concept A and my partner's concept D\.
We situate our approach in a fuzzy framework, where every domain concept is represented as a fuzzy set of the concepts of a particular reference ontology. This can be seen as a projection of all domain source concepts onto a common semantical space, where distances and relations between any two concepts can be expressed under xed criteria. In contrast to the original instance-representation, we can apply methods of fuzzy logical reasoning in order to characterize the relationship between concepts from di erent ontologies. In addition, the fuzzy representations allow for quantifying the degree to which a certain relation holds.
The paper is structured as follows. Related work is presented in the next section. Background in the eld of fuzzy sets, as well as main de nitions and problems from the ontology matching domain are overviewed in Section 3. We present the concept fuzzi cation algorithm in Section 4, before we discuss how the combined knowledge body can be constructed in Section 5. Experimental results and conclusions are presented in Sections 6 and 7, respectively. 2
Fuzzy set theory generalizes classical set theory [
Work on fuzzy ontology matching can be classi ed in two families: (1)
approaches extending crisp ontology matching to deal with fuzzy ontologies and (2)
approaches addressing imprecision of the matching of (crisp or fuzzy) concepts.
Based on the work on approximate concept mapping by Stuckenschmidt [
Crisp instance-based ontology matching, relying on the idea that concept
similarity is accounted for by the similarity of their instances, has been overviewed
broadly in [
In this section, we introduce basics from fuzzy set theory and discuss aspects of the ontology matching problem. 3.1
A fuzzy set A is de ned on a given domain of objects X by the function
A : X 7 ! [0; 1];
(1)
which expresses the degree of membership of every element of X to A by
assigning to each x 2 X a value from the interval [0; 1] [
We recall several fuzzy set operations by giving de nitions in terms of Godel
semantics [
B(x); if A(x) otherwise.
B(x); (2) 3.2
An ontology consists of a set of semantically related concepts and provides in an
explicit and formal manner knowledge about a given domain of interest [
De nition 1 (Crisp Ontology). Let C be a set of concepts, is a C C, R a set of relations on C, I a set of instances, and g : C ! 2I a function that assigns a subset of instances from I to each concept in C. We require that is a and g are compatible, i.e., that is a(A0; A) $ g(A0) g(A) holds for all A0; A 2 C. In particular, this entails that is a has to be a partial order. With these de nitions, the quintuple
O = (C; is a; R; I; g) forms a crisp ontology.
Above, a concept is modeled intensionally by its relations to other concepts, and extensionally by a set of instances assigned to it via the function g: By assumption, every instance can be represented as a real-valued vector, de ned by a xed number of variables of some kind (the same for all the instances in I).
Ontology heterogeneity occurs when two or more ontologies are created
independently from one another over similar domains. Heterogeneity may be
observed on linguistic or terminological, on conceptual or on extensional level [
Instance-based, or extensional ontology matching gathers a set of approaches around the central idea that ontology concepts can be represented as sets of related instances and the similarity measured on these sets re ects the semantic similarity between the concepts that these instances populate. 3.3
Consider the ontologies O1 = (C1; is a1; R1; I1; g1) and Oref =
(X; is aref; Rref; Iref; gref): We rely on the straightforward idea that
determining the similarity sim(A; x) of two concepts A 2 C1 and x 2 X consists
in comparing their instance sets g1(A) and gref(x). For doing so, we need a
similarity measure for instances iA and ix; where iA 2 g1(A) and ix 2 gref(x).
We have used the scalar product and the cosine s(iA; ix) = hiAk;kixixik : Based on
kiA
this similarity measure for elements, the similarity measure for the sets can be
de ned by computing the similarity of the mean vectors corresponding to class
prototypes [
jgref(x)j k=1 1 jg1(A)j
X ijA; 1 jgref(x)j
X ikx : (3)
Note that other approaches of concept similarity can be employed as well, like
the variable selection approach in [
Let = fO1; :::; Ong be a set of (crisp) ontologies that will be referred to as source ontologies de ned as in Def. 1. The set of concepts C = Sin=1 Ci will be referred to as the set of source concepts. The ontologies from the set are assumed to share similar functionalities and application focuses and to be heterogeneous in the sense of some of the heterogeneity types described in Section 3.2. A certain complementarity of these resources can be assumed: they could be de ned with the same application scope, but on di erent levels, treating di erent and complementary aspects of the same application problem.
Let Oref = (X; is aref; Rref; Iref; gref) be an ontology, called a reference ontology whose concepts will be called reference concepts. In contrast to the source ontologies, the ontology Oref is assumed to be a less application dependent, generic knowledge source. As a consequence of Def. 1, the ontologies in and Oref are populated.
The fuzzi cation procedure that we propose relies on the idea of scoring every source concept by computing its similarities with the reference concepts, using the similarity measure (3). A source concept A will be represented by a function of the kind
A(x) = scoreA(x); 8x 2 X; (4) where scoreA(x) is the similarity between the concept A and a given reference concept x: Since score takes values between 0 and 1; (4) de nes a fuzzy set. We will refer to such a fuzzy set as the fuzzi ed concept A denoted by A.
In order to fuzzify the concepts of a source ontology O1, we propose the following hierarchical algorithm. First, we assign score-vectors, i.e. fuzzy membership functions to all leaf-node concepts of O1: Every non-leaf node, if it does not contain instances (documents) of its own, is scored as the maximum of the scores of its children for every x 2 X. If a non-leaf node has directly assigned instances (not inherited from its children), the node is rst scored on the basis of these instances with respect to the reference ontology, and then as the maximum of its children and itself. To illustrate, let a concept A have children A0 and A00 and let the non-empty function g (A) represent the instances assigned directly to the concept A. We compute the following similarity scores for this concept w.r.t. the set X : scoreA(x) = maxfscoreA0 (x); scoreA00 (x); scoreg (A)(x)g; 8x 2 X: (5) Above, scoreg (A)(x) conventionally denotes the similarity obtained for the concept A and a reference concept x by only taking into account the documents assigned directly to A: The algorithm is given in Alg. 1.
It is worth noting that assigning the max of all children to the parent for every x leads to a convergence to uniformity of the membership functions for nodes higher up in the hierarchy. Naturally, the functions of the higher level concepts are expected to be less \speci c" than those of the lower level concepts. A concept in a hierarchical structure can be seen as the union of its descendants, and a union corresponds to taking the max (an approach underlying the single link strategy used in clustering).
The hierarchical scoring procedure has the advantage that every x-score will be larger for a parent node than those for any of its children, and it holds that A0 (x) ! A(x) = 1 for all x and all children A0 of A. From computational viewpoint, the procedure which only scores the populated nodes is less expensive, compared to scoring all nodes one by one. 5
The construction of a combined knowledge body for a set of source ontologies aims at making explicit the relations that hold among their concepts, across these ontologies. To these ends, we apply the fuzzy set representations acquired in the previous section. In what follows, we consider two source ontologies O1 and O2 but note that all de nitions can be extended for multiple ontologies. Let C = fA1; :::; AjC1j; B1; :::; BjC2jg be the union of the concept sets of O1 (the A-concepts) and O2 (the B-concepts). We introduce several relations and operations that can be computed over C and will be used for constructing a combined reduced knowledge body that contains the concepts of interest. 5.1
The implication A0 ! A holds for any A0 and A such that is a(A0; A): We provide a de nition for a fuzzy subsumption of two fuzzi ed concepts A0 and A based on the fuzzy implication (2).
Function score(concept A, ontology Oref , sim. measure sim) begin for i = 1; :::; jXj do
sim[i] = sim(A; xi) // xi 2 X return sim end Procedure hierachicalScoring(ontology O, ontology Oref , sim. measure sim) begin 1. Let C be the list of concepts in O. 2. Let L be a list of nodes, initially empty 3. Do until C is empty: (a) Let L0 be the list of nodes in C that have only children in L (b) L = append(L; L0) (c) C = C L0 4. Iterate over L ( rst to last), with A being the current element: if children(A) = ; then score(A) = score(A; Oref ; sim) else if g (A) 6= ; then
score(A) = maxfmaxB2children(A)score(B); score(A; Oref ; sim)g else
score(A) = maxB2children(A)score(B) return score(A); 8A 2 C end Algorithm 1: An algorithm for hierarchical scoring of the source concepts.
and denoted in the following manner: is a(A0; A) = inf x2X
A0!A(x) (6) Equation (6) de nes the fuzzy subsumption as a degree between 0 and 1 to which one concept is the subsumer of another. It can be shown that is a, similarly to its crisp version, is re exive and transitive (i.e. a quasi-order). In addition, the hierarchical procedure for concept fuzzi cation introduced in the previous section assures that is a(A0; A) = 1 holds for every child-parent concept pair, i.e. the crisp subsumption relation is preserved by the fuzzi cation process.
Taking the example of a collaborative platform from the introduction, computing the fuzzy is a between two concepts allows for answering a user query regarding generality and speci city of their partners concepts with respect to a given target concept.
We provide a de nition of a fuzzy ontology which follows directly from the fuzzi cation of the source concepts and their is a relations introduced above.
C C ! [0; 1] a fuzzy is a-relationship, R a set of fuzzy relations on C, i.e., R contains relations r : Cn ! [0; 1], where n is the arity of the relation (for the sake of presentation, we only consider binary relations), X a set of objects, and : C ! F (X ; [0; 1]) a function that assigns a membership function to every fuzzy concept in C: We require that is a and are compatible, i.e., that is a(A0; A) = infx A0!A(x) holds for all A0; A 2 C. In particular, it can be shown that this entails that is a is a fuzzy quasi-order. With these de nitions, the quintuple
O = (C; is a; R; X ; ) forms a fuzzy ontology.
Above, the set X is de ned as a set of abstract objects. In our setting, these are the concepts of the reference ontology, i.e. X = X. The set C is any subset of C . In case C = C1, where C1 is the set of fuzzi ed concepts of the ontology O1, O de nes a fuzzy version of the crisp source ontology O1. In case C = C , O de nes a common knowledge body for the two source ontologies. Note that the membership values of the reference concepts entail fuzzy membership values for the documents populating the reference concepts. However, we will work directly with the concepts scores in what follows.
Based on the subsumption relation de ned above, we will de ne equivalence of two concepts in the following manner.
De nition 4 (Fuzzy -Equivalence). Fuzzy -equivalence between a concept A and a concept B, denoted by A v B holds if and only if is a(A; B) > and is a(B; A) > ; where is a parameter between 0 and 1:
The equivalence relation allows to de ne classes of equivalence on the set C : In the collaborative framework described in the introduction, this can be used for querying concepts equivalent (up to a degree de ned by the user) to a given user concept from the set of their partners concepts. 5.2
We propose several measures of closeness of two fuzzy concepts A and B. We begin by introducing a straightforward measure given by (7) (8) (9) We consider a similarity measure based on the Euclidean distance: base( A; B) = 1 mx2aXx j A(x)
B(x)j: eucl( A; B) = 1 k A
Bk2 ; where kxk2 = Px2X jxj2 1=2 is the `2-norm. Several measures of fuzzy set compatibility can be applied, as well. Zadeh's partial matching index between two fuzzy sets is given by sup-min( A; B) = sup T ( A(x); B(x)):
x2X Finally, the Jaccard coe cient is de ned by jacc( A; B) = PPxx TS(( AA((xx));; BB((xx)))) : It is required that at least one of the functions A or B takes a non-zero value for some x: T and S are as de ned in Section 3.
The similarity measures listed above provide di erent information as compared to the relations introduced in the previous subsection. Subsumption and equivalence characterize the structural relation between concepts, whereas similarity measures closeness between set elements. The two types of information are to be used in a complementary manner within the collaboration framework. 5.3
The union of two fuzzy concepts can be decomposed into three components, each quantifying, respectively, the commonality of both concepts, the speci city of the rst compared to the second and the speci city of the second compared to the rst expressed in the following manner
S(A; B) = (AB) + (A
B) + (B
A): Each of these components is de ned as follows and, respectively, accounts for: (10) (11) (12) (13) (14)
AB = T (A; B) == what is common to both concepts; A B
B = T (A; :B) == what is characteristic for A;
A = T (B; :A) == what is characteristic for B.
Several merge options can be provided to the user with respect to the values of these three components. In case AB is signi cantly larger than each of A B and B A, the two concepts can be merged into their union. In case one of A B or B A is larger than the other two components, the concepts can be merged to either A or B. 6
We situate our experiments in the multimedia domain, opposing two
complementary heterogeneous ontologies containing annotated pictures. We chose, on
one hand, LSCOM [
A text document has been generated for every image of the two ontologies, by taking the names of all concepts that an image contains in its annotation, as well as the (textual) de nitions of these concepts (the LSCOM de nitions for TRECVID images or the WordNet glosses for LabelMe images). An example of a visual instance of a multimedia concept and the constructed textual description is given in Figure 2. Several problems related to this representation are worth noting. The LSCOM keyword descriptions sometimes depend on negation and exclusion which are di cult to handle in a simple bag-of-words approach. Taking the WordNet glosses of the terms in LabelMe introduces problems related to polysemy and synonymy. Additionally, a scene often consists of several objects, which are frequently not related to the object that determines the class of the image. In such cases, the other objects in the image act as noise.
In order to fuzzify our source concepts, we have applied the hierarchical scoring algorithm from Section 4 independently for each of the source ontologies. As a reference ontology, we have used an extended version of the Wikipedia's socalled main topic classi cations (adding approx. 3 additional concepts to every rst level class), containing more than 30 categories. For each topic category, we included a set of corresponding documents from the Inex 2007 corpus.
The new combined knowledge body has been constructed by rst taking the union of all fuzzi ed source concepts. For every pair of concepts, we have computed their Godel subsumptional relations, as well as the degree of their similarities (applying the measures from Section 5.2 and the standard cosine measure). Apart from the classical Godel subsumption de ned in (6), we consider a version of it which takes the average over all x instead of the smallest value, given as is amean(A0; A) = avgx2X A0!A(x): The results for several intra-ontology concepts and several cross-ontology concepts are given in Table 1. Fig. 3 shows a fragment of the common fuzzy ontology built for LSCOM and LabelMe. The labels of the edges of the graph correspond to the values of the fuzzy subsumptions between concepts.
We will underline several shortcomings that need to be adressed in future work. Due to data heterogeneity, it appears that the fuzzy is a-structure is reected better within one single ontology, as compared to cross-ontology relations which are more interesting. Additionally, some part of relations are expressed as subsumptional (e.g. torso is a person) which is a natural e ect in view of the instance-representations. Indeed, the textual representation of images needs to be improved by accounting for the limitations discussed earlier in this section.
Note that computing the common fuzzy ontology is inexpensive, once we have in hand the fuzzy representations of the source concepts made available by the hierarchical scoring algiorithm. 7
Whenever collaboration between knowledge resources is required, it is important to provide procedures which make explicit to users the relations that hold between di erent terms of these resources. In an attempt to solve this problem, we have proposed a fuzzy theoretical approach to build a common ontology for a set of source ontologies which contains these relations, as well as the degrees to which they hold, and can be queried upon need by di erent parties within a collaborative framework.
In future work, we will investigate the impact of the choice of a reference ontology onto the concept fuzzi cation and the quality of the constructed fuzzy common ontology. Additionally, the approach will be extended with elements of OWL 2, including relations and axioms between instances which is not covered by the ontology de nition used in this work.