fuzzy computational ontologies in contexts || modeling uncertainty in knowledge representation

Chapter 4 Modeling Uncertainty in Knowledge

Representation

The classical view in cognitive psychology holds that an object is either aninstance of a concept or it is not. In terms of mathematics, every conceptis a crisp set. However, as we have discussed above, many concepts do nothave clear boundaries or definitions. Different objects have different degrees ofmembership or typicality with respect to a certain concept. In this section, wegive a review of studies that investigate how graded membership, vaguenessand uncertainty are modeled. Several extensions to existing ontology modelsor description logics involves fuzzy sets, therefore we will start by brieflyreviewing the basic notions of fuzzy set theory.

4.1 Fuzzy Set Theory

Fuzzy set theory is a mathematical theory first formalized by Zadeh [1] tohandle uncertainty and imprecision in information systems. Fuzzy set theorycan be considered as an extension of the classical (non-fuzzy) set theory,which is also generally called crisp sets in the literature [2]. Classical settheory allows the membership of the elements in a set in binary terms, abivalent condition, i.e., an element either belongs or does not belong to theset. However, there exists much knowledge that are fuzzy; knowledge that isvague, imprecise, uncertain, ambiguous, inexact, or probabilistic in nature.The reason is that the thinking and reasoning of human usually involve fuzzyconcepts, originating from inherently inexact human concepts. For example,there is no particular quantitative value when defines the term ‘young’. Fromthe perspectives of some people, age 35 is young, but maybe from others’perspectives, age 35 is not young. The concept young has no clean boundary.For most people, age 1 is definitely young and age 100 is definitely not young,which age 35 has some degree of being young and usually depends on thecontexts.

Classical set theory represents a set A by explicitly or implicitly enumer-

Y. Cai et al., Fuzzy Computational Ontologies in Contexts© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

38 Chapter 4 Modeling Uncertainty in Knowledge Representation

ating all its elements as follows:

A = {a1, a2, a3, a4, · · · , an}.

In fuzzy set theory, membership μA(x) of elements x in a set A is no longerlimited to 0 (non-member) and 1 (member). Instead, each fuzzy set has acharacteristic function, which assigns a value within a specific range (usuallyfrom 0 to 1) to each element in the universal set to indicate the membershipgrade of that element. A characteristic function is therefore also called amembership function [2]. Formally, for a universal set X , the membershipfunction of a fuzzy set A is denoted by

μA : X → [0, 1].

Fuzzy sets can be used to model concepts which do not have clear-cutboundaries. An example is the concept ‘hot.’ There is no particular tem-perature above which one may consider something as ‘hot.’ The concept‘hot’ can be modeled as a fuzzy set, and the membership function can assignmembership grade to different temperature values, where higher temperaturevalues receive higher membership grades. Consider another example, whenwe classify students into ‘tall’ and ‘not-tall’ students, we face such a question:if students who are taller than 1.8m tall are to be considered as ‘tall’, thenshould we exclude a student whose height is 1.79m from ‘tall students’? Fig-ures 4.1 and 4.2 show the membership degree distribution of ‘tall students’using crisp set theory and fuzzy set theory, respectively.

Fig. 4.1 Membership degree distribution of crisp set.

Fig. 4.2 Membership degree distribution of fuzzy set.

4.2 Uncertainty in Ontologies and Description Logics 39

Operations on crisp sets have their counterparts in fuzzy sets, which aregeneralized version of the original operations. The three basic operations oncrisp sets include complement, intersection and union. These three operationscan be generalized to fuzzy sets in more than one way, as long as they satisfycertain axioms [2]. In particular, Zadeh [1] proposes several operations whichare generally regarded as the standard operations. The following equationsare commonly used to determine the membership function of the resultantset under standard complement, intersection and union respectively.

μA(x) = 1− μA(x),μA∩B(x) = min[μA(x), μB(x)],μA∪B(x) = max[μA(x), μB(x)].

Fuzzy sets allow systems to model uncertainty and imprecision by intro-ducing graded membership degrees in sets. It also gives inception to otheruseful theories such as fuzzy logic and possibility theory. Fuzzy logic is basedon fuzzy set theory, and it can deal with reasoning that is approximate ratherthan precisely deduced from classical predicate logic. It allows in linguisticform the set membership values to imprecise concepts like ‘heavily’, ‘quite’and ‘very’. These theories find applications in many different domains. Forexample, fuzzy set theory and fuzzy logic are used in controllers (e.g., Refs.[3, 4]), databases and information retrieval systems (e.g., Refs. [5 – 7]) andexpert systems (e.g., Refs. [8, 9]).

4.2 Uncertainty in Ontologies and Description Logics

Concepts in classical ontologies and description logics are interpreted as crispsets. There are various proposals of extending ontologies with probabilistictheory or fuzzy set theory. For example, Parry in Ref. [10] uses fuzzy settheory to model fuzzy concepts and assign membership degrees to instances.In particular, Parry [10] proposes a fuzzy ontology for retrieval of medicaldocuments. It makes use of fuzzy membership value to indicate how likelyan ‘overloaded’ term (a term with several different meanings) is located in aparticular location in the ontology.

Ding and Peng [11] propose a method to extend the ontology languageOWL with Bayesian networks to represent uncertainty in ontologies. Firstly,they augment OWL with probabilistic markups, so that conditional proba-bilistic information can be encoded in ontologies. Secondly, a set of transla-tion rules is defined to convert the ontology into a Bayesian network, whichis used in reasoning tasks. As a result, the ontology supports both commonreasoning tasks as well as probabilistic reasoning. A similar extension on thelanguage OWL is described in Ref. [12], in which fuzzy set theory is used toallow vague and imprecise concepts, such as ‘hot’ and ‘fast’, to be representedin OWL.


Dubois et al. [13] propose a frame-based object-centered representation(O.C.R.), which incorporates fuzzy set theory to model classes (concepts) ina domain of interest. O.C.R. models several different common phenomenain human thinking, such as typicality, uncertainty and vagueness. Classesare intensionally described in terms of attributes (properties), of which thevalues are classified into two types, namely allowed values and typical values,where the ranges of these values are described by fuzzy sets. O.C.R. modelstypicality by employing the notion of typical range of attributes. The typicalrange T (a, C) of an attribute a of the class C is the set of typical values thatan instance of C can take for a. This range is represented by a fuzzy set,where typical values have higher membership grades than those less typical.For example, in the class of ‘Birds’, the attribute ‘way of locomotion’ hasa typical value ‘fly’, but other less typical values, such as ‘walk’ and ‘swim’also exist in the set of range, although they are assigned smaller values ofmembership grade.

In Ref. [14], Tamma and Bench-Capon present an extended ontologyknowledge model that allows more explicit representation of semantic in-formation about concepts. In this model, there are three characterizationsof properties: (1) attribute behavior over time, (2) modality, and (3) pro-totypical and exceptional properties. The model is a frame-based knowledgemodel-based on classes, slots and facets. Classes are collections of objectssharing the same properties. Slots, also known as attributes, are used todescribed concepts, and are themselves described by a set of additional con-straints called facets. One of the facets is value prototypes, which specifiesthe prototypical values of the slot.

There are also a number of research works that propose extensions tostandard description logics to handle fuzziness and uncertainty in conceptsand categorization. For example, Koller et al. [15] propose a probabilisticextension to Description Logics. Straccia [16] combines fuzzy set theory andDescription Logics and introduces fuzzy A L C , in which concepts are in-terpreted as fuzzy sets. Stoilos et al. present a fuzzy S H I N in Ref. [17].Different fuzzy Description Logics vary in their expressive power, complexityand reasoning capabilities.

In addition, Steffen et al. [18] further extend the expressiveness of fuzzyDescription Logics by introducing fuzzy hedges. Fuzzy hedges are terms thatmodify the extent to which an adjective or a concept is used to describecertain situation. ‘Very’, ‘more or less’, ‘quite’ are examples of hedges. Hedgescan be modeled by modifying the membership function of a fuzzy set. Forexample, the membership function of the concept ‘very hot’, constructed byadding the hedge ‘very’ to the concept ‘hot’ can be obtained by raising theoriginal membership function to a higher power: μvery hot(a) = μhot(a)2.This work extends this idea and proposes a framework of fuzzy DescriptionLogics with hedges as concept modifiers.

Besides, Straccia [16] proposes a fuzzy A L C and a fuzzy S H OI N (D)[19]. Stoilos et al. [17] present a fuzzy S H I N . These fuzzy DLs vary in

4.2 Uncertainty in Ontologies and Description Logics 41

possessing different expressive power, complexity and reasoning capabilities.Some fuzzy ontologies are constructed based on fuzzy DLs or fuzzy logic [20,21]. Some works apply fuzzy ontologies for some applications. For instance,Cross and Voss [22] explore the potential that fuzzy mathematics and ontolo-gies have for improving performance in multilingual document exploitation.Parry [23] uses fuzzy ontology to improve medical document retrieval. Theseworks can represent membership degrees of different objects in concepts. Nev-ertheless, in these models, object memberships are given by users manuallyor obtained by fuzzy functions defined by users. These works lack a formalmechanism to obtain the membership degrees of objects in concepts auto-matically based on the defining properties of concepts and properties whichobjects possess. Besides, there is no consideration of how people representingconcepts in their mind.

On the other hand, Giordano et al. [24, 25] focus specifically on reasoningabout typicality in Description Logics. They propose adding a ‘typicality’operator to A L C . The operator is intended to select the ‘most normal’ or‘most typical’ instances of a concept. As a result, in addition to ABoxes andTBoxes, a knowledge base may contain subsumption relations that involvestypicality.

Recently, we also propose a fuzzy description logic named fom-DL forSemantic Web environment [26]. In fom-DL , there is an alphabet of distinctconcepts (C), roles (R), objects (I) and properties (P). It adopts the uniquename assumption in fom-DL . The syntax of fom-DL is as follows.

Role. Each role name RN is a fuzzy role in fom-DL . A valid role R isdefined by the abstract syntax: R := RN |R−. The inverse relation of roles issymmetric, and to avoid considering roles such as R−−, we defined a functionInv, which returns the inverse of a role, more precisely Inv(R) := RN− ifR = RN , and Inv(R) = RN if R = RN−. Roles are organized in a hierarchy.

Concept. Each concept name CN ∈ C is a fuzzy concept in fom-DL .We denote a primitive concept by A, then concepts C and D are formed outas follows.

C, D −→ �|⊥|A|C �D|C D|¬C|∀R.C|∃R.C| �n R.C|�n R.C|$R.C|∀R1, . . . , Rn.C|∃R1, . . . , Rn.C.

Object. Each object name IN ∈ I is an object in fom-DL .The semantics of fuzzy DL are provided by a fuzzy interpretation which

is a pair I = 〈ΔI , ·I 〉 where the domain ΔI is a non-empty set of objectsand ·I is a fuzzy interpretation function, which maps• an object name a to elements of aI ∈ ΔI ;• a concept name C to a membership function CI : ΔI → [0, 1], and we

consider the object membership of an object ai in a concept C is denotedby μC(ai) and μC(ai) = CI (ai). Thus, a concept C is considered as afuzzy set of objects C = {au1

1 , au22 , · · · , aun

n }, where ai is an object in ΔI

and ui is the membership of ai in concept C, i.e., CI (ai);• a role name R to a membership function RI : ΔI ×ΔI → [0, 1]. A role


R is actually considered as a fuzzy set of object pairs R = {< a1, b1 >w1 ,< a2, b2 >w2 , ..., < an, bn >wn}, where < ai, bi > is a role instance (i.e., apair of objects) and wi is the membership of the role instance < ai, bi >in R, i.e., RI (ai, bi);

• a property name P to a membership function PI : RI × CI → [0, 1].For a property, it is interpreted as a fuzzy set of pairs roles and concepts,i.e., P = {(< ai, bi >, bi)vi | < ai, bi >wi∈ R, bui

i ∈ C}. If an objectai has a fuzzy role < ai, bi > with object bi, RI (ai, bi) = wi > 0 andCI (bi) = ui > 0, then we say ai possesses a property member (< ai, bi >,bi) of property P = R.C to a degree PI (< ai, bi >, bi) = vi wherePI (< ai, bi >, bi) = min(RI (ai, bi), CI (bi)).Figure 4.3 shows the semantics of fuzzy DL .

Fig. 4.3 Semantics of the fuzzy DL .

4.3 Semantic Similarity

One important task in knowledge representation is to determine the degreeof semantic similarity between concepts [27]. With a measure of similarity,a system will be able to obtain concepts that are similar or closely relatedto each other based on certain properties. This in fact has a wide rangeof application. For example, due to the distributive nature of the SemanticWeb, there must be more than one ontology that describes similar conceptsin a particular domain. When software agents using different ontologies wantto communicate with one another, they have to match concepts in differentontologies [28, 29]. In this case, they must judge whether two terms refer tothe same concept or two closely related concepts with the help of a measureof similarity. As in the case of information retrieval, determining semanticsimilarity between concepts is also an important task [30, 31], as it allowsthe retrieval system to identify similar concepts and provide the most relevant

4.3 Semantic Similarity 43

information to the users.There are in fact quite a number of similarity measures used to assess

the similarity between terms, concepts and ontologies, depending on the rep-resentation model used. For example, similarity between two terms can bedetermined by using a simple substring matching algorithm [32]. If conceptsor objects are represented as vectorial data, with each dimension representinga distinctive feature, distance functions can be used to calculate the distancebetween two objects [33]. For example, two most commonly used distancefunctions are the Euclidian and weighted Euclidian distance functions (seeTable 4.1).

Table 4.1 Two commonly used distance measures

Name of Function Distance Function

Euclidian d(x, y) =qPp

i=1(xi − yi)2

Weighted Euclidian d(x, y) =qPp

i=1 αi(xi − yi)2

In instance-based ontology matching, similarity between two concepts isusually determined by the number of instances they share. For example, theJaccard’s coefficient [34] is used in the GLUE ontology matching system [28]:

Sim(A, B) =|x ∈ (A ∩B)||x ∈ (A ∪B)| .

This function compares the number of instances that belong to both conceptA and concept B to the number of instances that belong to A or B only. Thesimilarity between A and B will be higher as the number of instances sharedby them increases.

In Ref. [27], Cross reviews and presents two types of semantic similar-ity measures, namely network distance models and information theoreticalmodels. For network distance models, similarity is determined by the dis-tance between the nodes in the ontology than corresponds to the concepts inquestion [35]. In order to reflect the edge distances, weights have been addedto the edges between nodes in the ontologies to provide better assessmentof similarity [36, 37]. Information theoretical models determine similarity byusing information theory. This is based on the idea that similarity betweentwo concepts can be judged by the degree to which they share information[27]. For example, the information shared by two concepts c1 and c2 can beapproximated by the information content of the lowest super-concept c3 thatsubsumes them in the hierarchy [38]:

Sim(c1, c2) = − log p(c3).

Although these measures are based on different approaches, it has been noted[27] that they can be viewed as variation of Tversky’s [39] parameterized


ration model of similarity:

Sim(X, Y ) =f(X ∩ Y )

f(X ∩ Y ) + α× f(X − Y ) + β × f(Y −X),

where f(·) is a function which compares the properties or shared instancesof the concepts X and Y .

4.4 Contextual Reasoning

McCarthy [40] was the first to propose formalizing context in intelligent sys-tems. He introduces the notation ist(c,p) to denote the assertion that a propo-sition p is true in context c. Giunchiglia [41] uses context as a means of for-malizing the idea of localization, which takes ‘context to be a set of facts usedlocally to prove a given goal plus the inference routines used to reason aboutthem.’ Some subsequent efforts in formalizing context in logical languagesinclude [42, 43]. These works focus on how context can be formally repre-sented in a knowledge representation system, and how reasoning processescan accommodate changes in context.

As research and development of the Semantic Web proceed, an increas-ingly important issue in the use of ontologies is how context can be modeled.In particular, Obrst and Nichols [44] mention two issues. Firstly we have toinvestigate how concepts, properties, and judgement of membership of indi-vidual objects are interpreted differently when there is a change in the contextof the reasoning tasks. Secondly, there is the problem of how ontologies canbe designed to represent contexts formally.

Grossi et al. [45, 46] propose a theoretical framework to handle context inDescription Logics. The framework is intended for modeling situations suchas ‘concept A is a kind of concept B in context C.’ The framework involvesa contextual taxonomy model in which a set of models represents a set ofdifferent contexts. Subsumption relations between concepts only hold in spe-cific contexts. It provides a formal semantics for contextualized subsumptionexpressions as well as the possibility of describing operations (such as com-bination or abstraction) on contexts.

On the other hand, a logical extension called Context Description Frame-work [47] to the existing Resource Description Framework (RDF) has alsobeen proposed. Context Description Framework defines context of a state-ment (a triple in RDF) as a set of other statements, which describe a certaincondition of an environment. To accommodate this change, a contextual rangefor a property is added to a statement predicate. Thus, a statement in CDFbecomes a quadruple.

4.5 Summary 45

4.5 Summary

In this chapter, we have gone through a brief review of Semantic Web, on-tologies, and Description Logics. We have also reviewed some theories incognitive psychology about human reasoning, concepts and categorization.While there are already quite a number of proposals of extending ontologiesand Description Logics to handle fuzziness and typicality, we are not awareof any work that directly address the differences between membership andtypicality. We believe addressing this issue is important when using ontolo-gies to model real world concepts. In the following chapters, we will describein details our proposal and its applications in recommendation systems.

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