A MODEL FOR REPRESENTING VAGUE LINGUISTIC TERMS

AND FUZZY RULES FOR CLASSIFICATION IN ONTOLOGIES

Cristiane A. Yaguinuma, Vinícius R. T. Ferraz, Marilde T. P. Santos, Heloisa A. Camargo

Department of Computer Science, Federal University of São Carlos, Rod. Washington Luis Km 235, São Carlos, SP, Brazil

Tatiane M. Nogueira

Institute of Mathematics and Computer Sciences, University of São Paulo

Av. Trabalhador São-carlense, 400, São Carlos, SP, Brazil

Keywords:

Knowledge representation, Fuzzy set theory, Fuzzy ontology, Fuzzy reasoning, Classiﬁcation.

Abstract:

Ontologies have been successfully employed in applications that require semantic information processing.

However, traditional ontologies are not able to express fuzzy or vague information, which often occurs in

human vocabulary as well as in several application domains. In order to deal with such restriction, concepts

of fuzzy set theory should be incorporated into ontologies so that it is possible to represent and reason over

fuzzy or vague knowledge. In this context, this paper proposes a model for representing fuzzy ontologies

covering fuzzy properties and fuzzy rules, and we also implement fuzzy reasoning methods such as classical

and general fuzzy reasoning, aiming to support classiﬁcation of new instances based on fuzzy rules.

1 INTRODUCTION

Ontologies have been widely used in applications re-

garding knowledge representation, such as the Se-

mantic Web, which investigates the semantics of con-

tent and services over the Web. In the context of com-

puter and information sciences, an ontology deﬁnes a

set of representational primitives with which to model

a domain of knowledge or discourse (Gruber, 2009).

As ontologies model semantic elements and support

reasoning, they have been applied to improve seman-

tic information processing among humans and com-

putational systems.

Although expressive, the background formalism

of traditional ontologies is not able to represent im-

precise or vague information, which often occurs in

human language and in several application domains.

For example, it is difﬁcult to model concepts like

young, dark, hot and large, for which a clear and pre-

cise deﬁnition is not possible (Straccia, 2006). Fuzzy

sets (Zadeh, 1965) provide a meaningful and powerful

representation of vague concepts expressed in natural

language. Several fuzzy sets representing linguistic

concepts, such as low, medium and high, are often em-

ployed to deﬁne states of a variable (fuzzy variable)

(Klir and Yuan, 1995). Such variables are often used

in production rules, which support knowledge infe-

rence based on fuzzy reasoning methods, e.g. classi-

cal and general methods (Cordón et al., 1999). There-

fore, such rules can be used to infer the classiﬁcation

of elements into speciﬁc categories according to the

values of fuzzy variables.

In this sense, it is important that ontologies repre-

sent vague or fuzzy information and support approxi-

mate reasoning mechanisms. Hence, fuzzy reasoning

methods should be associated to fuzzy ontologies, im-

proving the expressiveness of concepts and relation-

ships modelled. Speciﬁcally, classiﬁcation methods

based on fuzzy rules are useful to several applica-

tions, including Semantic Web and Text Mining, by

classifying contents (Web pages and documents) into

concepts of a domain depending on fuzzy properties

associated to linguistic terms.

In order to face these issues, we propose a model

for representing fuzzy ontologies that considers the

classic and general fuzzy reasoning methods for clas-

siﬁcation based on rules containing fuzzy properties.

This paper is organized as follows. Section 2 dis-

cusses related work on fuzzy ontology representa-

tion. Next, Section 3 describes the proposed model

for fuzzy ontology representation, followed by a case

study presented on Section 4. Finally, Section 5 con-

cludes this paper and points future works.

438

A. Yaguinuma C., R. T. Ferraz V., T. P. Santos M., A. Camargo H. and M. Nogueira T. (2010).

A MODEL FOR REPRESENTING VAGUE LINGUISTIC TERMS AND FUZZY RULES FOR CLASSIFICATION IN ONTOLOGIES.

In Proceedings of the 12th International Conference on Enterprise Information Systems - Artiﬁcial Intelligence and Decision Support Systems, pages

438-442

DOI: 10.5220/0002976204380442

 SciTePress

2 RELATED WORK ON FUZZY

ONTOLOGY

Among the approaches for fuzzy ontology representa-

tion, some researches consider ontologies composed

of fuzzy classes and fuzzy relationships to represent

the semantics of domains. In this case, fuzzy class and

fuzzy relationship respectively correspond to fuzzy

set and fuzzy relation of the fuzzy set theory (Zadeh,

1965). Fuzzy OWL (Stoilos et al., 2006) and f-OWL

DL (Pan et al., 2008) extend OWL language elements

in order to represent such features in ontologies.

Recently, fuzzy ontology approaches are mov-

ing forward to express fuzzy properties, membership

functions and linguistic hedges (Straccia, 2006; Cale-

gari and Ciucci, 2007). Fuzzy properties or fuzzy

datatypes extend attributes of traditional ontologies,

corresponding to fuzzy variables whose values can be

vague linguistic terms deﬁned as fuzzy sets. For ex-

ample, the age attribute can assume fuzzy linguistic

values such as young, adult or old that correspond

to fuzzy sets. Linguistic hedges are special linguis-

tic terms by which other terms are modiﬁed (Klir and

Yuan, 1995). Terms like very, more or less, fairly are

classic examples of hedges. Any linguistic hedge may

be interpreted as a unary operation on the unit interval

[0, 1]. By representing these fuzzy semantic elements,

ontologies can model a semantics closely related to

the vagueness of human vocabulary.

Some researches have also pointed to fuzzy rule

representation. For example, Damásio et al. (Damá-

sio et al., 2008) extend the RuleML language to ex-

press degrees of truth assigned to propositons and

rules, however their approach do not support fuzzy

properties and membership functions. Fuzzy DL sys-

tem (Bobillo and Straccia, 2008) has achieved better

expressiveness, as it represents fuzzy rules contain-

ing linguistic terms and implements reasoning mech-

anisms based on Mamdani method.

As the review of the literature shows, there is a

growing trend to increase expressiveness in fuzzy on-

tologies. Following this direction, it is interesting that

fuzzy ontologies support reasoning mechanisms to-

wards classiﬁcation based on rules containing fuzzy

properties. Aiming to support these features, it is fun-

damental to represent fuzzy properties, membership

functions and rules in a suitable way. In this sense,

Section 3 describes the proposed model for represent-

ing fuzzy properties and classiﬁcation rules for further

fuzzy reasoning in ontologies.

3 MODEL FOR FUZZY

PROPERTIES AND RULES IN

ONTOLOGIES

In order to represent fuzzy class, fuzzy property and

fuzzy rule, we present some deﬁnitions to compre-

hend how we have incorporated them in ontologies:

Deﬁnition 1. Fuzzy class, which corresponds to a

fuzzy set, is a class whose individuals can belong to it

with a membership degree between the interval [0, 1].

When a fuzzy class corresponds to a fuzzy linguis-

tic term associated to a continuous attribute, it can

be deﬁned by a parameterized membership function,

such as triangular and trapezoidal functions (Klir and

Yuan, 1995).

Deﬁnition 2. Fuzzy property is deﬁned as a property

or attribute that can assume vague linguistic values

represented by fuzzy classes. This deﬁnition is based

on the fuzzy variable deﬁnition, which corresponds to

a variable whose values are fuzzy linguistic terms.

Deﬁnition 3. Fuzzy rule is deﬁned as an if-then rule

whose antecedent contains fuzzy properties and the

consequent has a class deﬁned in the ontology (clas-

siﬁcation rule). Fuzzy reasoning methods can be ap-

plied over these rules to derive the class of an individ-

ual based on the values of its fuzzy properties.

From these deﬁnitions, we propose the model il-

lustrated as a graph in Figure 1. In this model, the

Fuzzy Variable element is responsible for connecting

a Fuzzy Property to its linguistic terms represented as

Membership Function Fuzzy Class. Such connection

is respectively made by hasFuzzyProperty and has-

FuzzyValue relationships. The Membership Function

Fuzzy Class is a fuzzy class that is deﬁned by a pa-

rameterized membership function. This class can rep-

resent a linguistic term, whose label is described by

linguisticLabel property of type String. The Trape-

zoidal and Triangular subclasses represent two possi-

ble parameterized membership functions that can be

used to model a fuzzy class. To deﬁne more param-

eterized functions, subclasses can be added as well.

Finally, the parameters of membership functions are

represented by leftZeroParameter, leftOneParameter,

rightZeroParameter, rightOneParameter properties of

type ﬂoat. These parameters determine which values

of the fuzzy property that the membership function

assumes minumum (zero) and maximum (one) val-

ues, both considering left and right sides of the trapez-

ium. In case of triangular functions, only three param-

eters are required, with leftOneParameter correspon-

A MODEL FOR REPRESENTING VAGUE LINGUISTIC TERMS AND FUZZY RULES FOR CLASSIFICATION IN

ONTOLOGIES

439

Figure 1: Model for representing fuzzy properties and their linguistic terms as parameterized fuzzy classes.

ding to the center of the triangle.

An important feature of the proposed model is that

it is an abstract representation, thus independent of

ontology language syntax. So, it can be instantiated

using any traditional ontology language that models

representational primitives such as classes, instances,

attributes and relationships. This is a relevant contri-

bution in comparison to related work, since existing

approaches usually extend ontology language syntax

to incorporate fuzzy concepts, consequently losing

compatibility with reasoners and applications.

Regarding fuzzy rules, we have used Jena Frame-

work (Carroll et al., 2004) to implement classical and

general fuzzy reasoning methods, considering fuzzy

properties modelled in OWL and rules deﬁned ac-

cording to Jena rule syntax. The choice for OWL

and Jena Framework was just a matter of implemen-

tation, other languages and technologies could also be

used. Next section depicts a case study involving text

mining, illustrating a real application of the proposed

model and implemented fuzzy reasoning methods.

4 CASE STUDY:

CLASSIFICATION OF TEXT

DOCUMENTS

The conducted tests involved the classiﬁcation of sci-

entiﬁc documents considering Artiﬁcial Intelligence

research area and its subareas. Fifty documents were

used, whose attributes ﬁrstly corresponded to the fre-

quency of keywords that appear on them, obtained by

applying Text Mining techniques (study reported in

(Nogueira et al., 2009)). However, as the generated

document-attribute matrix was too sparse and high

dimensional, we have clustered the documents with

fuzzy c-means algorithm (Bezdeck, 1981), thus re-

ducing the dimension of the matrix. In this case, at-

tributes represent the relationship between documents

and identiﬁed clusters, with a degree corresponding

to the strength of such relation. At the end of this

pre-processing step, we generated a document-cluster

matrix (snippet in Table 1) that represents the fuzzy

properties of documents. Last column contains the

document class assigned by domain experts, where

FL stands for Fuzzy Logics, M for Mining and ML for

Machine Learning. All these categories correspond to

classes deﬁned in the ontology.

Table 1: Document-cluster matrix generated by fuzzy c-

means.

ID C0 C1 C2 C3 C4 C5 Class

5 0,173 0,0378 0,143 0,271 0,179 0,193 FL

22 0,059 0,222 0,262 0,144 0,163 0,145 M

35 0,335 0,485 0,037 0,036 0,022 0,082 ML

Next step is modelling fuzzy properties in the on-

tology. In this experiment, we consider that fuzzy

properties correspond to relationships between doc-

uments and identiﬁed clusters, named from C0 to

C5, which can assume low, medium and high linguis-

tic values, deﬁned by uniformly distributed triangular

membership functions. Listing 1 illustrates how C0

property and its low linguistic value are modelled ac-

cording to the proposed model, instantiated in OWL.

<fuz:FuzzyVariable rdf:ID="c0_fuzzy_variable">

<fuz:hasFuzzyProperty rdf:resource="#c0"/>

<fuz:hasFuzzyValue>

<fuz:TriangularFuzzyClass rdf:ID="c0_low">

<fuz:linguisticLabel rdf:datatype="&xsd;string">low

</fuz:linguisticLabel>

<fuz:leftZeroParameter rdf:datatype="&xsd;ﬂoat">3.13E−8

</fuz:leftZeroParameter>

<fuz:leftOneParameter rdf:datatype="&xsd;ﬂoat">3.13E−8

</fuz:leftOneParameter>

<fuz:rightZeroParameter rdf:datatype="&xsd;ﬂoat">0.489177

</fuz:rightZeroParameter>

</fuz:TriangularFuzzyClass>

</fuz:hasFuzzyValue>

. . .

</fuz:FuzzyVariable>

Listing 1: C0 property and its low linguistic value.

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Once we have modelled all fuzzy properties, they

can be used in fuzzy rules for classiﬁcation of sci-

entiﬁc documents. In this experiment, fuzzy rules

should contain C0 to C5 properties in the antecedent

part, and a class in the consequent that corresponds to

the inferred class of the ontology. In order to auto-

matically learn these rules, we have applied the Wang

and Mendel method (Wang and Mendel, 1992) over

the document-cluster matrix. As a result, we have ob-

tained nineteen rules in total, which were modelled

using Jena rule syntax. Due to space limitations, List-

ing 2 shows only two of them, just to illustrate how

they are speciﬁed according to the proposed model.

[rule1: (?x c0 ’low’), (?x c1 ’low’), (?x c2 ’high’), (?x c3 ’low’),

(?x c4 ’low’), (?x c5 ’low’) −> (?x rdf:type Machine_Learning)]

[rule2: (?x c0 ’low’), (?x c1 ’medium’), (?x c2 ’low’), (?x c3 ’low’),

(?x c4 ’medium’), (?x c5 ’low’) −> (?x rdf:type Fuzzy_Logics)]

Listing 2: Fuzzy rules generated by Wang and Mendel.

At this stage, fuzzy reasoning methods can be ap-

plied to classify scientiﬁc documents regarding Ar-

tiﬁcial Intelligence subareas. When a new docu-

ment needs to be classiﬁed, it passes through the pre-

processing step in order to obtain its fuzzy property

values. After that, the user chooses a fuzzy reasoning

method (classical or general) to obtain the inferred

class. In our tests, general method performed better

than classical one, as the former considers all ﬁred

rules by combining their association degree. How-

ever, this research does not intend to evaluate the

best method, our goal is making them available for

ontology-based applications that require the represen-

tation of fuzzy properties and classiﬁcation rules.

Concluding this case study, we have observed

that the proposed model contributed to represent

the vagueness present in textual information content.

Moreover, fuzzy reasoning methods can automati-

cally infer the classes of new documents, an informa-

tion that can be analyzed by document retrieval sys-

tems for improving query results.

5 CONCLUSIONS AND FUTURE

WORK

We have proposed a model for representing fuzzy

properties and fuzzy rules in ontologies, which can

be instantiated using any traditional ontology lan-

guage that models basic representational primitives.

Furthermore, fuzzy reasoning methods (classical and

general) were implemented in order to support clas-

siﬁcation of new instances according to their fuzzy

property values. The results obtained from the study

case demonstrate that the proposed model contributed

to manage vagueness on text documents, representing

a good approach not only for classiﬁcation but also

for organization of the text documents.

Finally, we plan to incorporate more fuzzy set

concepts, such as linguistic hedges, fuzzy relations

etc. We intend to support other types of rules and their

fuzzy reasoning methods (e.g. Mamdani and Larsen

methods), as well as defuzziﬁcation methods.

ACKNOWLEDGEMENTS

We thank CAPES and INEP agencies for supporting

this research, inside the scope of WebPIDE project.

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