=Paper=
{{Paper
|id=None
|storemode=property
|title=Building A Fuzzy Knowledge Body for Integrating Domain Ontologies
|pdfUrl=https://ceur-ws.org/Vol-778/paper1.pdf
|volume=Vol-778
|dblpUrl=https://dblp.org/rec/conf/semweb/TodorovGH11
}}
==Building A Fuzzy Knowledge Body for Integrating Domain Ontologies==
https://ceur-ws.org/Vol-778/paper1.pdf
Building A Fuzzy Knowledge Body for
Integrating Domain Ontologies
Konstantin Todorov1 , Peter Geibel2 , and Céline Hudelot1
1
Laboratory MAS, École Centrale Paris
2
TU Berlin, Sekr. FR 5-8, Fakultät IV
Abstract. 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 log-
ical reasoning in order to characterize and measure the degree of the
relationships holding between concepts from different ontologies. We pre-
senta an application of the approach in the multimedia domain.
1 Introduction
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, differences 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 be-
tween the concepts of the domain ontologies, such as their relative generality
or specificity, their shared commonality or their complementarity. This can be
very useful in a number of real-life scenarioss, especially in collaborative plat-
forms. Let us imagine a project which includes several partners, each of which
has its own vocabulary of semantically structured terms that describes its activ-
ity. 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 “find me a concept from partner
P2 which is a strong subsumer of my concept C”, or ”what are the commonality
and specificity 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 fixed 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 different 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 field of fuzzy sets, as well as main definitions and
problems from the ontology matching domain are overviewed in Section 3. We
present the concept fuzzification 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 Related Work
Fuzzy set theory generalizes classical set theory [19] allowing to deal with impre-
cise and vague data. A way of handling imprecise information in ontologies is to
incorporate fuzzy reasoning into them. Several papers by Sanchez, Calegari and
colleagues [4], [5], [13] form an important body of work on fuzzy ontologies where
each ontology concept is defined as a fuzzy set on the domain of instances and
relations on the domain of instances and concepts are defined as fuzzy relations.
Work on fuzzy ontology matching can be classified in two families: (1) ap-
proaches 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 [16] and
Akahani et al. [1], Xu et al. [18] suggested a framework for the mapping of fuzzy
concepts between fuzzy ontologies. With a similar idea, Bahri et al. [2] propose
a framework to define similarity relations among fuzzy ontology components. As
an example of the second family of approaches, we refer to [8] where a fuzzy ap-
proach to handling matchinging uncertainty is proposed. A matching approach
based on fuzzy conceptual graphs and rules is proposed in [3]. To define new
intra-ontology concept similarity measures, Cross et al. [6] model a concept as
a fuzzy set of its ancestor concepts and itself, using as a membership degree
function the Information Content (IC) of concept with respect to its ontology.
Crisp instance-based ontology matching, relying on the idea that concept sim-
ilarity is accounted for by the similarity of their instances, has been overviewed
broadly in [7]. We refer particularly to the Caiman approach which relies on
estimating concepts similarity by measuring class-means distances [10].
3 Background and Preliminaries
In this section, we introduce basics from fuzzy set theory and discuss aspects of
the ontology matching problem.
3.1 Fuzzy Sets
A fuzzy set A is defined on a given domain of objects X by the function
µA : X 7−→ [0, 1], (1)
which expresses the degree of membership of every element of X to A by assign-
ing to each x ∈ X a value from the interval [0, 1] [19]. The fuzzy power set of X,
denoted by F(X, [0, 1]), is the set of all membership functions µ : X 7−→ [0, 1].
We recall several fuzzy set operations by giving definitions in terms of Gödel
semantics [15]. The intersection of two fuzzy sets A and B is given by a t-norm
function T (µA (x), µB (x)) = min(µA (x), µB (x)). The union of A and B is given
by S(µA (x), µB (x)) = max(µA (x), µB (x)) where S is a t-conorm. The comple-
ment of a fuzzy set A, denoted by ¬A, is defined by the membership function
µ¬A (x) = 1 − µA (x). We consider the Gödel definition of a fuzzy implication
(
1, if µA (x) ≤ µB (x),
µA→B (x) = (2)
µB (x), otherwise.
3.2 Ontologies, Heterogeneity and Ontology Matching
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 [7]. We
are particularly interested in ontologies, whose concepts come equipped with a
set of associated instances, defined as it follows.
Definition 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 ∈ C. In
particular, this entails that is a has to be a partial order. With these definitions,
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, defined
by a fixed number of variables of some kind (the same for all the instances in I).
Ontology heterogeneity occurs when two or more ontologies are created in-
dependently from one another over similar domains. Heterogeneity may be ob-
served on linguistic or terminological, on conceptual or on extensional level [7].
Ontology matching is understood as the process of establishing relations between
the elements of two or more heterogeneous ontologies. Different matching tech-
niques have been introduced over the past years in order to resolve different
types of heterogeneity [9].
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 reflects the semantic
similarity between the concepts that these instances populate.
3.3 Crisp Concept Similarities
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 deter-
mining the similarity sim(A, x) of two concepts A ∈ C1 and x ∈ 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 ∈ g1 (A) and ix ∈ gref (x).
A x
We have used the scalar product and the cosine s(iA , ix ) = kihiA kki
,i i
x k . Based on
this similarity measure for elements, the similarity measure for the sets can be
defined by computing the similarity of the mean vectors corresponding to class
prototypes [10]:
|g1 (A)| |gref (x)|
1 X 1 X
simproto (A, x) = s iA , ixk . (3)
|g1 (A)| j=1 j |gref (x)|
k=1
Note that other approaches of concept similarity can be employed as well, like
the variable selection approach in [17]. In the context of our study, we have used
the method that both works best and is less complex. A hierarchical application
of the similarity measure for the concepts of two ontologies is presented in [17].
4 A Hierarchical Algorithm for Concept Fuzzification
Let Ω = {O1 , ..., On } be a set of (crisp) ontologies that will be referred Sn to
as source ontologies defined as in Def. 1. The set of concepts CΩ = i=1 Ci
will be referred to as the set of source concepts. The ontologies from the set
Ω 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 defined with the same application scope, but on different levels, treating
different and complementary aspects of the same application problem.
Let Oref = (X, is aref , Rref , Iref , gref ) be an ontology, called a reference ontol-
ogy 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 Ω and
Oref are populated.
The fuzzification 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
µA (x) = scoreA (x), ∀x ∈ 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) defines a fuzzy set. We
will refer to such a fuzzy set as the fuzzified 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 mem-
bership 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 ∈ X. If a non-leaf node has directly assigned
instances (not inherited from its children), the node is first 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) = max{scoreA0 (x), scoreA00 (x), scoreg∗ (A) (x)}, ∀x ∈ X. (5)
Above, scoreg∗ (A) (x) conventionally denotes the similarity obtained for the con-
cept 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 “specific” 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 µA0 (x) → µA (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 Building a Combined Knowledge Body
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 definitions can be extended for multiple ontologies.
Let CΩ = {A1 , ..., A|C1 | , B1 , ..., B|C2 | } 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 CΩ and will be used for constructing a
combined reduced knowledge body that contains the concepts of interest.
5.1 Fuzzy Concept Relations
The implication A0 → A holds for any A0 and A such that is a(A0 , A). We provide
a definition for a fuzzy subsumption of two fuzzified concepts A0 and A based
on the fuzzy implication (2).
Function score(concept A, ontology Oref , sim. measure sim)
begin
for i = 1, ..., |X| do
sim[i] = sim(A, xi ) // xi ∈ 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 (first 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) = max{maxB∈children(A) score(B), score(A, Oref , sim)}
else
score(A) = maxB∈children(A) score(B)
return score(A), ∀A ∈ C
end
Algorithm 1: An algorithm for hierarchical scoring of the source concepts.
Definition 2 (Fuzzy Subsumption). The subsumption A0 is a A is defined
and denoted in the following manner:
is a(A0 , A) = inf µA0 →A (x) (6)
x∈X
Equation (6) defines 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 reflexive and transitive (i.e. a quasi-order). In addition, the
hierarchical procedure for concept fuzzification 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 fuzzification process.
Taking the example of a collaborative platform from the introduction, com-
puting the fuzzy is a between two concepts allows for answering a user query
regarding generality and specificity of their partners concepts with respect to a
given target concept.
We provide a definition of a fuzzy ontology which follows directly from the
fuzzification of the source concepts and their is a relations introduced above.
Definition 3 (Fuzzy Ontology). Let C be a set of (fuzzy) concepts, is a :
C × C → [0, 1] a fuzzy is a-relationship, R a set of fuzzy relations on C, i.e., R
contains relations r : C n → [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
φ : C → F(X , [0, 1]) a function that assigns a membership function to every fuzzy
concept in C. We require that is a and φ are compatible, i.e., that is a(A0 , A) =
inf x µA0 →A (x) holds for all A0 , A ∈ C. In particular, it can be shown that this
entails that is a is a fuzzy quasi-order. With these definitions, the quintuple
O = (C, is a, R, X , φ)
forms a fuzzy ontology.
Above, the set X is defined 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 CΩ . In case C = C1 , where C1 is the set of fuzzified concepts of the ontology
O1 , O defines a fuzzy version of the crisp source ontology O1 . In case C = CΩ , O
defines 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 defined above, we will define equivalence
of two concepts in the following manner.
Definition 4 (Fuzzy θ-Equivalence). Fuzzy θ-equivalence between a concept
A and a concept B, denoted by A vθ B holds if and only if is a(A, B) > θ and
is a(B, A) > θ, where θ is a parameter between 0 and 1.
The equivalence relation allows to define classes of equivalence on the set CΩ .
In the collaborative framework described in the introduction, this can be used
for querying concepts equivalent (up to a degree defined by the user) to a given
user concept from the set of their partners concepts.
5.2 Similarity Measures for Fuzzy Concepts
We propose several measures of closeness of two fuzzy concepts A and B. We
begin by introducing a straightforward measure given by
ρbase (µA , µB ) = 1 − max |µA (x) − µB (x)|. (7)
x∈X
We consider a similarity measure based on the Euclidean distance:
ρeucl (µA , µB ) = 1 − kµA − µB k2 , (8)
2 1/2
is the `2 -norm. Several measures of fuzzy set
P
where kxk2 = x∈X |x|
compatibility can be applied, as well. Zadeh’s partial matching index between
two fuzzy sets is given by
ρsup-min (µA , µB ) = sup T (µA (x), µB (x)). (9)
x∈X
Finally, the Jaccard coefficient is defined by
P
T (µA (x), µB (x))
ρjacc (µA , µB ) = Px . (10)
x S(µA (x), µB (x))
It is required that at least one of the functions µA or µB takes a non-zero value
for some x. T and S are as defined in Section 3.
The similarity measures listed above provide different information as com-
pared to the relations introduced in the previous subsection. Subsumption and
equivalence characterize the structural relation between concepts, whereas sim-
ilarity measures closeness between set elements. The two types of information
are to be used in a complementary manner within the collaboration framework.
5.3 Quantifying Commonality and Relative Specificity
The union of two fuzzy concepts can be decomposed into three components,
each quantifying, respectively, the commonality of both concepts, the specificity
of the first compared to the second and the specificity of the second compared
to the first expressed in the following manner
S(A, B) = (AB) + (A − B) + (B − A). (11)
Each of these components is defined as follows and, respectively, accounts for:
AB = T (A, B) // what is common to both concepts; (12)
A − B = T (A, ¬B) // what is characteristic for A; (13)
B − A = T (B, ¬A) // what is characteristic for B. (14)
Several merge options can be provided to the user with respect to the values
of these three components. In case AB is significantly 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 Experiments
We situate our experiments in the multimedia domain, opposing two comple-
mentary heterogeneous ontologies containing annotated pictures. We chose, on
one hand, LSCOM [14] initially built in the framework of TRECVID1 and pop-
ulated with the development set of TRECVID 2005. Since this set contains
images from broadcast news videos, LSCOM is particularly adapted to anno-
tate this kind of content, thus contains abstract and specific concepts (e.g. Sci-
ence Technology, Interview On Location). On the other hand, we used
1
http://www-nlpir.nist.gov/projects/tv2005/
WordNet [11] populated with the LabelMe dataset [12], referred to as the La-
belMe ontology. Contrarily to LSCOM, this ontology is very general, populated
with photographs from daily life and contains concepts such as car, computer,
person, etc. The parts of the two multimedia ontologies used in the experiments
are shown in Figure 1.
Fig. 1. The LSCOM (left) and the LabelMe (right) ontologies.
Fig. 2. The LSCOM concept Bus: a visual and a textual instance.
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) definitions of these concepts (the LSCOM definitions 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 difficult 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.
Concept A: LSCM:truck vs. LSCM:sports vs. LM:computer vs. LM:animal vs.
Concept B: LSCM:gr.vehicle LSCM:basketball LM:elec. device LM:bird
is a(A, B) 1 0.007 1 0.004
is a(B, A) 0.012 1 0.011 1
is amean (A, B) 1 0.052 1 0.062
is amean (B, A) 0.326 1 0.07 1
Base Sim. 0.848 0.959 0.915 0.390
Eucl. Sim. 0.835 0.908 0.854 0.350
SupMin Sim. 0.435 0.545 0.359 0.309
Jacc. Sim. 0.870 0.814 0.733 0.399
Cosine Sim. 0.974 0.994 0.975 0.551
Concept A: LM:gondola vs. LSCM:group vs. LSCM:truck vs. LSCM:truck vs.
Concept B: LSCM:boat ship LM:audience LM:vehicle LM:conveyance
is a(A, B) 0.016 0.006 0.022 0.022
is a(B, A) 0.009 1 0.012 0.012
is amean (A, B) 0.86 0.022 0.748 0.769
is amean (B, A) 0.167 1 0.301 0.281
Base Sim. 0.72 0.78 0.58 0.58
Eucl. Sim. 0.66 0.71 0.40 0.38
SupMin Sim. 0.069 0.082 0.22 0.22
Jacc. Sim. 0.49 0.42 0.54 0.52
Cosine Sim. 0.69 0.82 0.66 0.67
Table 1. Examples of pairs of matched intra-ontology concepts (above) and cross-
ontology concepts (below), column-wise.
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 so-
called main topic classifications (adding approx. 3 additional concepts to every
first 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 first taking the
union of all fuzzified source concepts. For every pair of concepts, we have com-
puted their Gödel subsumptional relations, as well as the degree of their similar-
ities (applying the measures from Section 5.2 and the standard cosine measure).
Apart from the classical Gödel subsumption defined 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) = avgx∈X µA0 →A (x). The results for several intra-ontology con-
cepts 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 la-
bels 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 re-
flected 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 effect 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.
Fig. 3. A fragment of the common fuzzy ontology of LSCOM (LS) and LabelMe (LM).
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 Conclusion and Open Ends
Whenever collaboration between knowledge resources is required, it is impor-
tant to provide procedures which make explicit to users the relations that hold
between different 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 different parties within a
collaborative framework.
In future work, we will investigate the impact of the choice of a reference
ontology onto the concept fuzzification 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 definition used in this work.
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