INTEGRATING FUZZY LOGIC IN ONTOLOGIES

Silvia Calegari and Davide CiucciDipartimento di Informatica, Sistemistica e Comunicazione,

Universita degli Studi di Milano Bicocca,via Bicocca degli Arcimboldi 8,

20126 Milano, Italy{calegari, ciucci}@disco.unimib.it

Keywords: Concept modifiers, fuzzy logics, fuzzy ontologies, membership modifiers, KAON, ontology editor.

Abstract: Ontologies have proved to be very useful in sharing concepts across applications in an unambiguous way.Nowadays, in ontology-based applications information is often vague and imprecise. This is a well-knownproblem especially for semantics-based applications, such as e-commerce, knowledge management, web por-tals, etc. In computer-aided reasoning, the predominant paradigm to manage vague knowledge is fuzzy settheory. This paper presents an enrichment of classical computational ontologies with fuzzy logic to createfuzzy ontologies. So, it is a step towards facing the nuances of natural languages with ontologies. Our pro-posal is developed in the KAON ontology editor, that allows to handle ontology concepts in an high-levelenvironment.

1 INTRODUCTION

An ontology is a formal conceptualization of a partic-ular domain of interest shared among heterogeneousapplications. It consists ofentities, attributes, rela-tionshipsand axioms to provide a common under-standing of the real world (Lammari and Mtais, 2004;Gruber, 1993; Guarino and Giaretta, 1995). With thesupport of ontologies, users and systems can commu-nicate with each other through an easier informationexchange and integration (Soo and Lin, 2001). On-tologies help people and machines to communicateconcisely by supporting information exchange basedon semantics rather than just syntax.

There are ontological applications where informa-tion is often vague and imprecise. For instance,the semantic-based applications of the Semantic Web(Berners-Lee et al., 2001), such as e-commerce,knowledge management, web portals, etc. Indeed, theconceptual formalism supported by a typical ontol-ogy may not be sufficient to represent uncertain infor-mation that is commonly found in many applicationdomains. For example, keywords extracted by manyqueries in the same domain may not be consideredwith the same relevance, as some keywords may bemore significant than others. Therefore, the need ofgiving a different interpretation according to the con-text emerges.

A possible solution to handle uncertain data and,hence, to tackle these problems, is to incorporatefuzzy logic into ontologies. The aim of fuzzy settheory (Klir and Yuan, 1995) introduced by L. A.Zadeh (Zadeh, 1965) is to describe vague conceptsthrough a generalized notion of set, according towhich an object may belong to a certain degree (typ-ically a real number from the interval [0,1]) to a set.For instance, the semantic content of a statement like“Cabernet is a deep red acidic wine” might have de-gree, or truth-value, of 0.6. Up to now, fuzzy sets andontologies are jointly used to resolve uncertain infor-mation problems in various areas, for example, in textretrieval (Bouquet et al., 2004; Singh et al., 2004; Ab-ulaish and Dey, 2003) or to generate a scholarly on-tology from a database in ESKIMO (Matheus, 2005)and FOGA (Quan et al., 2004) frameworks. However,in none of these examples there is a fusion of fuzzy settheory with ontologies.

The aim of this paper is to present a proposal to di-rectly integrate fuzzy logic in ontology in order to ob-tain an extension of the ontology that is more suitablefor solving uncertainty reasoning problems. It is afirst step towards the realization of a theoretical modeland of a complete framework based on ontologies thatare able to consider the nuances of natural languages.In literature, a first tentative has been made in thecontext of medical document retrieval (Parry, 2004)

by adding a degree of membership to all terms in theontology to overcome theoverloadingproblem. An-other proposal is an extension of the domain ontologywith fuzzy concept (Chang-Shing et al., 2005), how-ever only for Chinese news summarization.

This paper shows how to insert fuzzy logic duringontology creation with KAON (KAON, 2005). Thissoftware consists in a number of different modulesproviding a broad range of functionalities centeredaround creation, storage, retrieval, maintenance andapplication of ontologies. KAON allows the use ofan ontology at high-level, and the relative conceptualmodels are defined in a natural and easily understand-able way.

The rest of the paper is organized as follows: Sec-tion 2 defines a fuzzy ontology and explains how todefine and use fuzzy values in it. Section 3 presentsthe ontology editor used and it is shown how to in-tegrate it with our framework. In Section 4, we givean overview on related works and on the next steps ofour approach.

2 FUZZY LOGIC

In this section, we present a logical framework to sup-port and to reason with uncertainty. This is a focusaspect for all ontology-based applications where theuser is interested in information that often containsimprecise and vague description of concepts. For ex-ample, one may be interested in finding “avery strongflavored red wine” or in reasoning with concepts suchas “acold place”, “anexpensiveitem”, “a fastmotor-cycle”, etc.

In order to face these problems the proposed ap-proach is based on fuzzy sets theory. It has not beenchosen a particular ontology domain to explain ourtheory because our goal is to fulfil all nuances of natu-ral languages and to take into account all the differentaspects that an ontology have to consider. Our aim isto extend an ontology editor to directly handle uncer-tainty during the ontology definition, so that to enrichthe knowledge domain.

At first, let us remind the definition of a fuzzy set.Let us consider a nonempty set of objectsU , calledthe universe. A fuzzy setor generalized characteris-tic functionalis defined as a[0, 1]–valued function onU , f : U 7→ [0, 1]. Given an objectx ∈ U , f(x)represents themembership valueof x to the setf . Inthe following of this section we explain how to intro-duce fuzzy values on different objects of an ontologyand how to automatically correct them. Finally, wegive some hint on the possible applications of a fuzzyontology.

2.1 Defining a fuzzy value

The first problem to tackle is how to assign a fuzzyvalue to an entity of the ontology. The trade off is be-tween understandability and precision, since (Casillaset al., 2003)

to obtain high degree of interpretability and ac-curacy is a contradictory purpose and, in prac-tice, one of the two properties prevails over theother one. Depending on what requirement ismainly pursued, the Fuzzy Modelling field maybe divided into two different areas:

1. Linguistic fuzzy modelling – The main objec-tive is to obtain fuzzy models with a good in-terpretability

2. Precise fuzzy modelling – The main objectiveis to obtain fuzzy models with a good accu-racy.

Since our goal is to be as general as possible, boththe possibilities are given to the expert: define aprecise value or a linguistic one. In the formercase the expert, while creating the ontology, de-fines a functionf : (Concepts ∪ Instance) ×Properties 7→ Property V alue × [0, 1] with themeaning thatf(o, p) is the value that a concept oran instanceo assumes for propertyp with associateddegree. For example, in an hypothetical ontology ofcats,f (Garfield,color) = (orange,0.8) means that forthe propertycolor, the instanceGarfield, has valueor-angewith degree0.8. Or, in a wine ontology,f (wine,taste)= (full-bodied,0.4) means that the conceptwinehas afull-bodied(the value)taste(the property) withdegree0.4.

Clearly, there may exist situations in which noproperty value is necessary for a given property.For example, “Garfield has sense of humour withvalue 0.9” cannot be correctly expressed with thejust exposed formalism. In this situation, it is neces-sary to map a pair (concept/instance, property) sim-ply to [0, 1], i.e., f ′ : (Concept ∪ Instance) ×Properties 7→ [0, 1] and the above example becomesf ′(Garfield,senseof humour) = 0.9.

In order to simplify the nota-tion, we can define a unique functiong : (Concepts ∪ Instances) × (Properties ∪Prop val) 7→ [0, 1]. Thus, “Garfield has colororange with value0.8” becomesg(Garfield,orange) =0.8. Using these functiong, the expert has the chanceto choose a membership value with infinite accuracy,that is precision is preferred to interpretability.

On the other hand, the second possibility is tochoose as membership value, alabel in a given set.We have chosen the setL ={little, enough, moder-ately, quite, very, totally} which is clearly not exhaus-tive of all the possible labels, but it can intuitively bemodified as desidered.

In this case the valueg(o, p) is automatically as-signed according to Table 1.

Table 1: assignement of fuzzy value to labels.

Label Valuelittle 0.2

enough 0.4moderately 0.6

quite 0.7very 0.8

totally 1

Summing up, we give the chance to add a mem-bership value to a pair (concept/instance, property) intwo different ways: through a precise valuev ∈ [0, 1]or choosing a label in the predefined setL. Thus,through the functiong we define a new relation in theontology domain.

Another possibility is to assign a fuzzy value to anentity (concept or instance). This can be useful toovercome the problem of overloading as outlined in(Parry, 2004) and explained in Section 2.3. In thiscase the expert can define a functionh : Concepts ∪Instances 7→ [0, 1].

Let us remark that the fuzzy value assigned usingone of the two functionsg andh is a number in theunit interval[0, 1], that is, the usual support of a manyvalued logic. Hence, applications based on fuzzy on-tologies can use this value taking advantage of stan-dard and well-studied tools. For instance, in order toput together two (or more) different fuzzy values, anaggregation operator(Calvo et al., 2002) can be used.Typical examples aret–norm and t–conorms(Kle-ment et al., 2000), that is, binary mappings which givea semantic to the “OR”, “AND” operators. The mostknown are Godel norm and conorm, i.e., the min–maxoperators. Considering the above example, it may benecessary to compute the truth value of the statement“Garfield is orange AND has sense of humour”. If itis known thatf (Garfield,orange)=0.8 andf (Garfield,senseof humour) = 0.9 then [f (Garfield,orange)AND f (Garfield, senseof humour)] = min{0.8,0.9}= 0.8.

Finally, we can give the definition of fuzzy ontol-ogy.

Definition 1. A fuzzy ontologyis an ontology ex-tended with fuzzy values which are assigned throughthe two functionsg : (Concepts ∪ Instances) × (Properties ∪Prop val) 7→ [0, 1] andh : Concepts ∪ Instances 7→ [0, 1].

2.2 Updating a fuzzy value

Once an expert has created a fuzzy ontology, it is notrealistic to assume that it is perfect and that any fuzzyvalue is well-defined and suited to any environment.Thus, a mechanism to change fuzzy values in order tofit them in the best way to a specific environment or, ingeneral, to make them more correct is needed. Here,we propose a method to update fuzzy values accord-ing to results of some queries on some documents. Wedo not enter into details about how syntactically spec-ify queries, but we assume that we are able to performthem and that their results are available to us.

Let us suppose that the current fuzzy value isf andas a result of a query it must be updated tofnew. Thesimpliest possibility is to setf := fnew. However,it is reasonable to suppose that after some queries thefuzzy property has reached a stable value, hence it isnot useful to change it withfnew, losing all the his-tory of the acquired knowledge. A solution can be todiminish the importance offnew at any change:

f := f +fnew − f

Q + 1(1)

whereQ is the number of updates perfomed for thatvalue. Clearly, the valueQ must be stored in the on-tology for any defined fuzzy value.

Now, the issue is how to compute a new fuzzyvalue fnew. It is unlikely to find in a document aprecise definition of a fuzzy value, but usually a lin-guistic qualifier can be found. For example, we donot find “Cabernet has a dry taste with value0.6”, butit make sense a document such “Cabernet has a verydry taste”. So a method to make use of this kind ofinformation is needed. Here we propose an approachbased on concept modifiers (Zadeh, 1972).

A concept modifier has the effect to alter the fuzzyvalue of a property. Given a set of linguistic hedgessuch as “very”, “more or less”, “slighty”, a conceptmodifier is a chain of one or more hedges, such as“very slightly” or “very very slightly”. To any (lin-guistic) concept modifier it is necessary to associate a(numerical) membership modifier.

Definition 2. A membership modifieris a valueβ > 0which is used as an exponent to modify the value of amembership functionf asfβ .

According to their effect on a fuzzy value, a hedgecan be classified in two groups:concentrationtypeanddilation type. The effect of a concentration mod-ifier is to reduce the grade of a membership value.Thus, in this case, it must beβ > 1. For instance, tothe hedge “very”, it is usually assignedβ = 2. So,if we know thatg(Cabernet, dry taste) = 0.8, i.e.,“Cabernet has a dry taste with value0.8”, then Caber-net has a very dry taste with value0.82 = 0.64. Onthe contrary a dilation hedge has the effect to raise a

membership value, that isβ ∈ (0, 1). For instance,if to slightly it is assignedβ = 0.25 andg(Cabernet,dry taste)=0.8, then Cabernet has a slightly dry tastewith value 0.80.25 = 0.95 according to the intu-ition that if something is “dry” then it is even more“slightly dry”. Let us remark that this approach is dif-ferent from the original Zadeh’s one (Zadeh, 1972),where “slightly”, as well as other modifiers, is han-dled in a more complicated manner. This method hasthe advantage to give a uniform and simply way tomanage concept modifiers, even if, a deeper studyabout the semantic of this way of handling chain ofmodifiers is needed.

So, a concept modifier is used in literature to de-fine a new fuzzy membership given an already exist-ing one. For example, if we know the fuzzy value ofpropertyred we can infer the fuzzy membership ofpropertyvery redsimply by raising to the power 2 thevalue of red (see the above examples). However, herewe are in the opposite situation. As an example, let ussuppose to know, from an ontology, the red propertyand also, as a result of a query, that a certain objectis “very red”. Hence, from very red we need to in-fer a new red property (before “red” was fixed, hereit changes) for that object and clearly if an object isvery red it is even more red. So, if in the ontologyg(o, red) = 0.7, we must increase this value, for ex-ampleg(o, red) = 0.70.5. In conclusion, the effect ofvery is to raise the value of the property it is referredto and not to reduce its magnitude. In a schematicway, it is possible to say that in the usual case it isperformed the deduction

red→ very red

whereas in this situation:

red and very red→ red

This argument also applies to all the other conceptmodifiers. Thus, in our case what is usually consid-ered as a concentration modifier becomes a dilationone and vice versa.

Two issues need now to be faced: decide whichmodifiers we consider (and which are their values)and define a method to computeβ values for chainsof concept modifiers.About the former problem, the chosen set of hedgesis H ={very, much more, more, more or less, moder-ate, slightly}. This is only one of the possible choices.The setH can be changed according to ones needs, oncondition that it satisfies the following two properties.

• The setH is totally ordered, i.e., very< . . . <slightly and only theβ value for the smaller andgreater elements are fixed, respectively as0.5 and2.

• The two subsets of contraction hedges{more orless, moderate, slightly} and dilation hedges{very,much more, more} have the same cardinality.

These two conditions are due to the fact that we pro-pose to adopt the algorithm presented in (Khang et al.,2002) in order to compute the membership modifierof a sequence of hedges. The only difference is theorder inversion of the hedges, this is due to the use weare doing of concept modifiers which is, as explainedabove, opposite to the original approach.

Moreover, we consider also the further modifiernot which behaves as the standard involutive nega-tion on fuzzy sets:not(x) := 1 − x. As an ex-ample let us suppose thatg(Cabernet,drytaste)=0.8,then if in a document it is found that “Cabernet hasnot a dry taste” the new value isgnew(Cabernet,dry taste)= 1 − 0.8 = 0.2. This way of han-dling the not connective cannot be easily applied tochains of hedges. Using the same example, if wefind that “Cabernet has not a very dry taste”, the newvalue isgnew(Cabernet, dry taste) = 1 − √0.8 =0.11. However, “not very dry ” induces to think tosomething which is dry but not at an high degreeand this is not correctly mirrored bygnew(Cabernet,dry taste)=0.11. In (Singh et al., 2004) a solution tointegratenot in the setH of all concept modifiers ispresented. This solution does not seem to us a goodone, since it cannot be applied directly to a property,but only to another concept modifier and also in thiscase it can generate a negativeβ if the original algo-rithm of (Khang et al., 2002) is not modified. Thus,handling chains of modifiers which includenot is leftas an open problem. Pacholczyk et al. dedicated sev-eral works to the problem of linguistic negation (seefor instance (Pacholczyk, 1998; Pacholczyk et al.,2002)). So an interesting future work would be theintegration of those studies in our approach.

Another open problem is that the setH of conceptmodifiers is certainly not exhaustive of all the nuancesof natural language. Lots of elements could be addedto H and this will require new algorithms to handle it,since not all existing concept modifiers can be totallyordered or exactly split into two subgroups of samecardinality.

2.3 Examples of application

In this section we give two examples of a possible useof fuzzy ontologies. The first one is based on fuzzyvalues associated to (instance, properties) pairs andthe second one is a way to use concepts with fuzzyvalues to remedy the overloading problem.

Extending queries When performing a query on adocument, it is a usual practice to extend the set ofconcepts already present in the query with other oneswhich can be derived from an ontology. Tipically,given a concept, also its parents and children can beadded to the query and then searched in the document.

A possible use of fuzzy ontology is to extendqueries with, besides children and parents, instancesof concepts which satisfies to a certain degree thequery. Let us explain it with an example. We are givena clothes ontology and a query looking for “a verylong and black coat”. In the ontology there are twoinstances of coat:X which has property “long” withvalue0.7 andY which has property “long” with value0.3. Thus, it is natural to extend the original queryadding, not only parents and children of the concept“coat”, but also the instanceX, because“long = 0.7”can be interpreted as “very long”. On the other hand,the instanceY is not added to the extended querysince“long = 0.3” does not mean “very long”.

To make a choice on which instances have to beadded to the extended query, we have to decide howlinguistic labels are mapped to numerical values. Thesolution is again as in Section 2.1, that is only labelbelonging to setL are admitted in queries and theyare mapped to numerical values according to Table 1.If c is a concept,p is a property andl a label thenµ(c, p, l) is the value given to the labell for propertyp and conceptc. For instance in the above example,the property “a very long coat” is translated toµ(coat,long, very)= 0.8. Now, we consider all the instancesi of the conceptc and they are included in the ex-tended query if and only if :

|µ(c, p, l)− g(i, p)| ≤ ε (2)

whereε ∈ [0, 1] is a level of tolerance. Obviously,the number of instances to be added to the extendedquery depends on the value ofε, the greater isε themost are the instances. The boundary cases areε = 0,only the instances that exactly match the query areincluded, andε = 1, all the instances are included.Coming back to the example, if we fixε = 0.2, then,|µ(coat, long, very) − g(X, long)| = 0.8 − 0.7 =0.1 ≤ 0.2 = ε, whereas|µ(coat, long, very) −g(Y, long)| = 0.8− 0.3 = 0.5 ≥ 0.2 = ε. Hence,Xis included in the extended query andY is not.

Clearly, this is the simplest case where only oneproperty is present in the query. If two or more requestmust be satisfied, a generalization of equation (2) isneeded. Let us suppose that in the query there aren properties referred to the same conceptc, then in-stancei is considered iff∑n

j=1 |µ(c, pj , lj)− g(i, pj)|n

≤ ε (3)

That is we require that the mean value of the distancesbetween the values of the properties in the query andthe values of the properties in the instance is less thanthe toleranceε.

Overloading of concepts As anticipated in Sec-tion 2.1 a possible use of the fuzzy value associated toconcepts is to limit the problems due to overloading

of a concept in an ontology. The solution we are go-ing to expose has been proposed in (Parry, 2004). Inour opinion this approach is more related to statisticsthan to fuzzy logic, nevertheless it can be managed byour fuzzy ontology.

Let us suppose that a conceptc is present in dif-ferent parts of the ontology, the aim is to give an in-dication about which place is more significant withrespect to a certain domain. At a first stage, to anyconcept which is present in multiple locations is givenan equal fuzzy value such that they sum up to 1. Forinstance, ifc is present in 4 places, respectively de-noted asci then we haveh(ci) = 0.25. For anyci

the set of its local termsLi, i.e., parents and childrenin the ontology, is computed. Then, all the elementsl ∈ Li are searched in the documents under analysisand a weightwj

i is assigned to them according to therelevance they have in the document. Let us supposethat in the document under investigation, there areljoccurencies for the elementl. Then, for any conceptci and for any documentd the following function iscomputed:

µd(ci) :=∑

j

ljwji (4)

The sum over alln documents ofµd is denoted asµ:µ(ci) :=

∑nd=1 µd. Then, the new membership value

for conceptci and documentd is:

hnew(ci) :=µd(ci)µ(ci)

(5)

These values are then used to updateh(ci) accordingto Equation 1 and they are applied in order of rele-vance, so that a value due to a more relevant docu-ment is applied first and it has a greater influence thanthe following ones.

3 HOW TO USE KAON

The KAON project is a meta-project carried out at theInstitute AIFB, University of Karlsruhe and at the Re-search Center for Information Technologies (FZI).

KAON includes a comprehensive tool suite allow-ing easy creation, maintenance and management ofontologies. Furthermore, it provides a framework forbuilding ontology-based applications. An importantuser-level application supplied by KAON is an on-tology editor called OI-modeler. It allows to handleentities of an ontology in a natural way. The most im-portant features of the OI-modeler are its support formanipulation of large ontologies and the support foruser-directed evolution of ontologies. Ontologies canbe simultaneously edited by multiple users (AA.VV.,2004). The ontology navigation is more easy throughgraph-based and tree-based metaphors.

In the last years, KAON has been used in manyareas like e-commerce and b2b applications (Motiket al., 2002), autonomic and self-healing, self-configuring computational system (Stojanovic et al.,2004) and more recently, it has been applied to the Se-mantic Web (Bozsak et al., 2002; Oberle et al., 2005).

Root

Winery

Meal Course

Wine Grape Wine Region

Consumable Thing

Food Drink

Wine

body

color

flavor

grape

maker

name

sugar

full, medium, light

red, rosate, white

delicate, moderate, strong

dry, sweet, off−dry

Legend:

Concept Subconcept

instancepropertyproperty value

Figure 1: Wine ontology.

3.1 Ontologies in KAON

An ontology in KAON consists of concepts (sets ofelements), properties (specifications of how objectsmay be connected) and instances grouped in reusableunits called OI-models (ontology-instance models)(AA.VV., 2004). The conceptual model proposed al-lows to define an entity in different ways, depend-ing on the point of view of the observer. That is,an entity can be interpreted as a concept, as well asan instance. Moreover, property instantiation must bein accordance with the domain and range constraints(i.e. axioms, general rules, value-allowed) and mustobey the cardinality constraints, as specified by theproperty specifications. An OI-model may includeother OI-models, and have immediate access to alldefinitions from the included model.

Figure 1 is an example of an ontology in KAON:it represents only a partial ontology definition aboutwine. We have chosen such ontology because it islargely widespread and known, and so it is moresimply to understand the reasoning approach used inKAON.

In KAON language, it is possible to define well-known symmetric, transitive and inverse proper-

ties, with the addition of modularization and meta-modeling (AA.VV., 2004). Obviously, each of thesefeatures allow to manage two types of implicit knowl-edge: axioms and general rules. The formers are astandard set of rules, such as the transitive properties,the latters are general rules to combine and to adaptinformation defined in an ontology domain.

Moreover, KAON language allows to specify so-called lexical entries (i.e. labels, synonyms, stems, ortextual documentation) which reflect various nuancesof natural languages. For example, the same lexicalentry may be associated with several elements: thelabel BEAR may be associated with an instance rep-resenting a bear as an animal or as a puppet. Further-more, the instances can be defined in different lan-guages, namely English, German, French, Spanish,Portuguese, Arabic and Chinese.

3.2 Fuzzy ontologies in KAON

Our aim is to enrich KAON language adding the pro-posed fuzzy-sets approach. In the following, it isshowed how we have integrated our framework in theKAON project.

Figure 2: New KAON ontology overview

Figure 2 represents a “Fuzzy Inspector” developedto create in KAON the fuzzy ontology. The new panelhas been called “Fuzzy Logic” (see right lower cornerin Figure 2) and it allows the expert an easy fuzzylogic integration. The Fuzzy Inspector is composedby a table representing fuzzy entity, membership de-gree and number of updatesQ.

The domain expert can choose his fuzzy entity(concept or instance) simply clicking up through twotypes of interface proposed by KAON, namely graph-based and tree-based metaphors, partially showed inthe screenshot. Moreover, the expert can select thelink between an instance and a property adding itsfuzzy value.

Thus, the expert can create real fuzzy ontology do-main selecting the entity and directly inserting fuzzylogic through the Fuzzy Inspector Panel purposely de-veloped. In Figure 2, we propose two ways to use theFuzzy Logic Panel. In the first row the expert types

the membership degree according to his point of view.In the second row he can choose the apposite value bya list. The selected element in the list will be referredto an a-priori defined numerical value as explained inSection 2.1. In the definition phase of the ontologythe number of the updates is zero. This value willbe changed during the queries in accordance with thefunctions defined in previous sections.

KAON’s ontology language is based on RDFS(RDFS, 2004) with proprietary extensions for al-gebraic property characteristics (symmetric, transi-tive and inverse), cardinality, modularization, meta-modelling and explicit representation of lexical infor-mation.

In literature, all the limits about the RDFS arewell-known. Thus, it has been developed KAON2(KAON2, 2005) that is a successor of the KAONproject. The main difference as to previous KAONversion is the supported ontology language, namelyKAON used a proprietary extension of RDFS,whereas KAON2 is based on OWL DL (OWL, 2005).

OWL DL is a sublanguage of OWL (OWL, 2005)and it supports those users who want the maximumexpressiveness without losing computational com-pleteness (all conclusions are guaranteed to be com-puted) and decidability of reasoning systems (all com-putations will finish in finite time).

In more details, KAON2’s language is based on acombination of the OWL DL and OWL Lite sublan-guages (KAON2, 2005) of the OWL Web OntologyLanguage (OWL, 2005).

Recently, some proposals to integrate fuzzy logicin OWL have been presented (Straccia, 2005; Stoiloset al., 2005; Pan et al., 2005).

The more complete and suitable to our studyseems to be the extension ofSHOIN (D) presentedin (Straccia, 2005).

Thus, the next and last step to the integration offuzzy ontology in KAON2 is a deeper analysis of allthese approaches and the adaptation of one of them toour situation. This is out of the scope of the presentpaper and will be proposed in a forthcoming work.

4 CONCLUSION

In this paper, we have introduced fuzzy logic directlyin the ontology during the domain definition, enrich-ing the actual features proposed by other ontology ed-itors. The proposed solution allows to represent andto reason with uncertain information. This is a deli-cate problem for all those areas where the applicationsare based on ontology.

The domain expert has two possibilities to add amembership value in an ontology domain: through apair ({concept,instance},property)or through an en-

tity {concept, instance}. In both solutions, he/she canassign this degree through a precise valuev ∈ [0, 1]or choosing a label in the predefined setL.

We have also proposed a method, based on con-cept modifiers, to automatically update the member-ship degree during queries, useful, as example, for theextraction of more relevant documents.

Furthermore, we have presented two possible ex-amples of application: to extend a query and to over-come the problem of overloading.

KAON has been the ontology editor chosen to in-troduce fuzzy logic during the ontology definition.We have integrated fuzzy logic in KAON, developinga suitable Fuzzy Inspector Panel.

In the future, it is necessary to develop fuzzy-OWLin KAON2 and to test all the proposed framework.Furthermore, we plan to enrich our theory consideringlinguistic negation. This is another crucial topic inorder to handle all the uncertainty situations proposedby natural languages within the ontological domain.

REFERENCES

AA.VV. (2004). Developer’s Guide for KAON 1.2.7. Tech-nical report, FZI Research Center for Information andWBS Knowledge Management Group.

Abulaish, M. and Dey, L. (2003). Ontology Based FuzzyDeductive System to Handle Imprecise Knowledge.In In Proceedings of the 4th International Conferenceon Intelligent Technologies (InTech 2003), pages 271–278.

Berners-Lee, T., Hendler, T., and Lassila, J. (2001). Thesemantic web.Scientific American, 284:34–43.

Bouquet, P., Euzenat, J., Franconi, E., Serafini, L., Stamou,G., and Tessaris, S. (2004). Specification of a commonframework for characterizing alignment.IST Knowl-edge web NoE, 2.2.1.

Bozsak, E., Ehrig, M., Handschuh, S., Hotho, A., Maedche,A., Motik, B., Oberle, D., Schmitz, C., Staab, S., Sto-janovic, L., Stojanovic, N., Studer, R., Stumme, G.,Sure, Y., Tane, J., Volz, R., and Zacharias, V. (2002).KAON - Towards a large scale Semantic Web. InE-Commerce and Web Technologies, Third InternationalConference, EC-Web 2002, proceedings, volume 2455of LNCS, pages 304–313. Springer-Verlag.

Calvo, T., Mayor, G., and Mesiar, R., editors (2002).Ag-gregation Operators. Physica–Verlag, Heidelberg.

Casillas, J., Cordon, O., Herrera, F., and Magdalena, L.(2003). Accuracy improvements to find the balanceinterpretability–accuracy in linguistic fuzzy model-ing:an overview. InAccuracy Improvements in Lin-guistic Fuzzy Modeling, pages 3–24. Physica-Verlag,Heidelberg.

Chang-Shing, L., Zhi-Wei, J., and Lin-Kai, H. (2005). Afuzzy ontology and its application to news summa-

rization. IEEE Transactions on Systems, Man, andCybernetics-Part B: Cybernetics, 35:859–880.

Gruber, T. (1993). A Translation Approach to Portable On-tology Specifications.Knowledge Acquisition, 5:199–220.

Guarino, N. and Giaretta, P. (1995). Ontologies and Knowl-edge Bases: Towards a Terminological Clarification.In Mars, N., editor,Towards Very Large KnowledgeBases: Knowledge Building and Knowledge Sharing,pages 25–32. IOS Press, Amsterdam.

KAON (2005). Karlsruhe Ontology and Semantic WebTool Suite (KAON). http://kaon.semanticweb.org.

KAON2 (2005). Karlsruhe Ontology and Se-mantic Web Tool Suite 2 (KAON2).http://kaon2.semanticweb.org.

Khang, T. D., Storr, H., and Holldobler, S. (2002). Afuzzy description logic with hedges as concept modi-fiers. InThird International Conference on IntelligentTechnologies and Third Vietnam-Japan Symposium onFuzzy Systems and Applications, pages 25–34.

Klement, E. P., Mesiar, R., and Pap, E. (2000).TriangularNorms. Kluwer Academic, Dordrecht.

Klir, G. and Yuan, B. (1995).Fuzzy Sets and Fuzzy Logic:Theory and Applications. Prentice Hall.

Lammari, N. and Mtais, E. (2004). Building and maintain-ing ontologies: a set of algorithms.Data and Knowl-edge Engineering, 48:155–176.

Matheus, C. (2005). Using Ontology-based Rules for Situ-ation Awareness and Information Fusion. InPositionPaper presented at the W3C Workshop on Rule Lan-guages for Interoperability.

Motik, B., Maedche, A., and Volz, R. (2002). A ConceptualModeling Approach for Semantics-Driven EnterpriseApplications. InProceedings of the First Interna-tional Conference on Ontologies, Databases and Ap-plication of Semantics (ODBASE-2002), volume 2519of LNCS, pages 1082–1099. Springer-Verlag.

Oberle, D., Staab, S., Studer, R., and Volz, R. (2005). Sup-porting application development in the semantic web.ACM Trans. Inter. Tech., 5:328–358.

OWL (2005). Ontology Web Language (OWL).http://www.w3.org/2004/OWL/.

Pacholczyk, D. (1998). A new approach to linguisticnegation based upon compatibility level and tolerancethreshold. In Polkowski, L. and Skowron, A., edi-tors,Proceedings of RSCTC98, volume 1424 ofLNAI,pages 416–423.

Pacholczyk, D., Quafafou, M., and Garcia, L. (2002). Opti-mistic vs. pessimistic interpretation of linguistic nega-tion. In Proceedings of AIMSA02, volume 2443 ofLNAI, pages 132–141.

Pan, J., Stamou, G., Tzouvaras, V., and Horrocks, I. (2005).f-swrl: A Fuzzy Extension of SWRL. InICANN 2005,volume 3697 ofLNCS, pages 829–834. Springer-Verlag.

Parry, D. (2004). A fuzzy ontology for medical documentretrieval. InProceedings of The Australian Workshopon DataMining and Web Intelligence (DMWI2004),pages 121–126, Dunedin.

Quan, T., Hui, S., and Cao, T. (2004). FOGA: A Fuzzy On-tology Generation Framework for Scholarly SemanticWeb. InKnowledge Discovery and Ontologies (KDO-2004). Workshop at ECML/PKDD.

RDFS (2004). Resource Description Framework Schema(RDFS). http://www.w3.org/TR/PR-rdf-schema.

Singh, S., Dey, L., and Abulaish, M. (2004). A Frameworkfor Extending Fuzzy Description Logic to Ontologybased Document Processing. InProceedings of AWIC2004, volume 3034 ofLNAI, pages 95–104. Springer-Verlag.

Soo, V. W. and Lin, C. Y. (2001). Ontology-based infor-mation retrieval in a multi-agent system for digital li-brary. In6th Conference on Artificial Intelligence andApplications, pages 241–246.

Stoilos, G., Stamou, G., Tzouvaras, V., Pan, J. Z., and Hor-rocks, I. (2005). Fuzzy OWL: Uncertainty and theSemantic Web. InInternational Workshop of OWL:Experiences and Directions (OWL-ED2005), Galway,Ireland.

Stojanovic, L., Schneider, J., Maedche, A., Libischer, S.,Studer, R., Lumpp, T., Abecker, A., Breiter, G., andDinger, J. (2004). The role of ontologies in autonomiccomputing systems.IBM Systems Journal, 43:598–616.

Straccia, U. (2005). Towards a Fuzzy Description Logicfor the Semantic Web Preliminary Report. InESWC 2005, volume 3532 ofLNCS, pages 167–181.Springer-Verlag.

Zadeh, L. A. (1965). Fuzzy sets.Inform. and Control,8:338–353.

Zadeh, L. A. (1972). A fuzzy-set-theoretic interpretation oflinguistic hedges.Journal of Cybernetics, 2:4–34.