EXTENDING OWL BY FUZZY DESCRIPTION LOGIC

Abstract:

Technology about ontology plays an important role in all kinds of technologies of Semantic Web. OWL, the standard web ontology language is proposed by W3C. However, vocabularies in OWL only represent certain and complete concepts and roles. In order to represent fuzzy knowledge, this paper extends existing OWL language with some new vocabularies by fuzzy constructors、axioms and constraints. The paper maps semantics of these new vocabularies to fuzzy description logic and directly reasons fuzzy knowledge using constraint propagation calculus. The paper presents a new method of representing and reasoning fuzzy ontology.

Keywords:

OWL; fuzzy logic; fuzzy OWL;

1.         Introduction

With the development of the Semantic Web1, research about ontology is playing a central role as a source of shared and exchanged information over Semantic Web. OWL5, the standard web ontology language proposed by W3C recently, is intended to represent terms and their interrelationships in applications. However, vocabularies in OWL only represent certain and complete knowledge. There are so much uncertain and incomplete knowledge in real world, especially in the World Wide Web, which is huge and can be partially known.

In the last decades, impressive technologies about fuzzy and uncertain knowledge have been developed in literature [3.4.6.7.8]. It is an interesting and appealing idea to extend OWL using these technologies. The literature [4] extends OWL using probabilistic knowledge. In fact, uncertain knowledge or vague concepts is as important as probabilistic knowledge in real world. In the paper, We extend OWL by fuzzy constructors、axioms and constraints (denoted FOWL). Moreover, we map semantics of new fuzzy terms to fuzzy description logic and directly reason fuzzy knowledge by constraint propagation calculus

In the following section we recall fuzzy description logic and constraint propagation calculus from literature [7]. Section 3 elaborates fuzzy constructors、axioms and constraints in FOWL. Section 4 gives a reasoning example. Section 5 is summary and further research work..

2.         Fuzzy description logic

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2.1.         Syntax

Fuzzy description logic can present vague concepts and roles as well as interoperates in these concepts and these roles. From a fuzzy set⁹ point of view, vague concepts or roles can be seen as fuzzy sets in universe U or universe . A fuzzy set S with respect to a universe U is characterized by a membership function . If A、B are fuzzy concepts and R is a atomic fuzzy role，then  are fuzzy concepts. The membership functions of these concepts satisfy following restrictions: for all  

A fuzzy knowledge base (FK) consists of a set  of fuzzy terminological axioms, and a set  of fuzzy assertions. A fuzzy terminological axiom knowledge base  consists of a set of axioms of the form or . A fuzzy assertion knowledge base  consists of a set of three fuzzy constrains of the form , where a、b are objects ( individuals and variables) in domain U; . A fuzzy constraint is an expression having one of the forms and   , where  are objects ( individuals and variables) in domain U; . Though three fuzzy constrains are only involved in fuzzy assertions knowledge base, other forms will be used in reason procedure.

2.2.   Semantics

A fuzzy interpretation is a pair  where is the domain whereas I is an interpretation function mapping different individuals in universe U into different element, a fuzzy concept A into a membership function , a fuzzy role R into a membership function .Moreover, the interpretation function I satisfies the following equations: for all  

A fuzzy interpretation  satisfies a fuzzy terminological axiom (  or ) iff for  or . A fuzzy interpretation   satisfies a fuzzy terminological axiom knowledge base  ( ) iff the fuzzy interpretation satisfies every terminological axiom in . Similarly, a fuzzy interpretation satisfies fuzzy assertions, iff for (A fuzzy interpretation is extended to variables by mapping these variables into elements of the fuzzy interpretation domain). A fuzzy interpretation satisfies a fuzzy assertion knowledge base  ( ) iff the fuzzy interpretation satisfies every assertion in . A fuzzy interpretation  satisfies a fuzzy knowledge base or is a model of a fuzzy knowledge base ( ) iff it satisfies every fuzzy terminological axiom or assertion in FK.

 

2.3.   Reason tasks and constraint propagation calculus

Given a fuzzy knowledge base FK, the most important reason task is deciding satisfiability of FK. If FK is satisfied, for given fuzzy assertion or fuzzy terminological axiom  another important reason task is deciding whether  or  is a fuzzy entailment of the fuzzy knowledge base FK. The fuzzy entailment problem can be reduced to the satisfiability problem.

If fuzzy knowledge base can be transformed into pure fuzzy assertion knowledge base, the satisfiability problem can be resolved using constraints propagation rules. The transformation work from the literature [6] can be extended as follows:

·         Elimination of concept subsumption: each concept subsumption  is replaced with a concept equation , where  is a new primitive concept,  is new fuzzy terminological knowledge base.

·         Expansion of : every defined concept (the lift argument of a concept equation) which occurs in the defining term of another concept equation (the right argument of a concept equation) is substituted by its defining term to acquire . The process is iterated until there remain only undefined concepts in the right arguments of concept equation.

·         Expansion of : every primitive concept occurring in  is substituted by its defining term in . This yields .

The above transformation has the nice property that  iff , where is obtained by replacing every primitive concept occurring in with its defining term in .

All problems can be reduced to determine whether a finite set S of fuzzy constraints is satisfiable or not. The problem is based on a set of constraints propagation rules transforming a set S of fuzzy constraints into “simpler” satisfiability preserving sets  until either all  contain a clash or some is completed and clash-free.

A set of fuzzy constraints S contains a clash iff it contains either one of the constraints in Table 1or S contains a conjugated pair (Under the condition the row-column pair of fuzzy constraints in table 2 is a conjugated pair.).

 Table 1 Clashes             Table 2 Conjugated Pairs            

Concerning the constraints propagation rules, for 5 connective  and 4 relations , there are 20 rules. The rules are as follow:

 If x is a new variable and there is no such that both ( ) are already in the constraint set.

Because rules about  are same as the rules about   ,  they are omitted.

3.         Extending fuzzy terms for OWL

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3.1    Extending fuzzy constructors and axioms

From the point of fuzzy set¹⁴, common concepts are special fuzzy concepts, that is, for all individual in domain U membership function either is 1 or is 0. So class descriptions in OWL can naturally be extended to class descriptions of fuzzy concepts. If  “fdl” (fuzzy description logic) is namespace of fuzzy knowledge, all  class descriptions (intersection, union, complement) and concept axioms (subsumption, equation) can use RDF/XML syntax same as OWL. Moreover, prefix of these terms is “fdl”. Particular formats are in table 3.

Table 3 Fuzzy OWL

Fuzzy constructor / FDL  syntax	Examples for FOWL
fdl:Class / A、B	<fdl:Class rdf:ID="A"/> <fdl:Class rdf:ID="B"/>
fdl:intersection -Of /	<fdl:Class>  <fdl:intersectionOf fdf:parseType="Collection"> <fdl:Class rdf:about="#A"/> <fdl:Class rdf:about="#B"/> </fdl:intersectionOf> </fdl:Class>
fdl:unionOf /	<fdl:Class> <fdl:unionOf rdf:parseType="Collection"> <fdl:Class rdf:about="#A"/> <fdl:Class rdf:about="#B"/> </fdl:unionOf> </fdl:Class>
fdl:complement-Of /	<fdl:Class> <fdl:complementOf> <fdl:Class rdf:about="#A"/> </fdl:complementOf> </fdl:Class>>
fdl:someValues -From /	<fdl:Restriction> <fdl:onProperty rdf:resource="#R" /> <fdl:someValuesFrom rdf:resource="#A" /> </fdl:Restriction>
fdl:allValues -From /	<fdl:Restriction> <fdl:onProperty rdf:resource="#R" /> <fdl:allValuesFrom rdf:resource="#A"/> </fdl:Restriction〉
fdl:subClassOf /	<fdl:Class rdf:ID="A">   <fdl:subClassOf rdf:resource="#B" /> </fdl:Class>
fdl:equivalent -Class /	<fdl:Class rdf:about="#A"> <fdl:equivalentClass rdf:resource="#B"/> </fdl:Class>

 

3.2    Extending fuzzy constraints

Fuzzy constraints are also important formats in fuzzy knowledge base. There are not matching components of fuzzy constraints in OWL. Fuzzy constraints can be seen as range of individual belonging to fuzzy concepts with respect to the domain. So we may represent fuzzy constraints with range of membership function of these concepts. We define term “fdl: individual” and  “fdl: individuals” representing singular individual and dual individuals respectively. The term has attribute “fdl:name” whose value restricts individual’s name. If the term represents dual individuals, its attribute term come forth twice. We also define term “fdl:membershipOf” which represent degree of individual belonging to the fuzzy concept. Four degree term matching the term are “fdl:moreOrEquivalent”、“fdl:lessOrEquivalent”、“fdl:moreThan”、“ fdl:lessThan”. They represent constraints such as more or equivalent, less or equivalent, more, and less in order. Table 4 shows in detail.

Table 4   Fuzzy constraints in FOWL

Fuzzy constraints	Examples for FOWL
	<fdl: individual fdl:name="a">   <fdl:membershipOf rdf:resource="#A"/>   <fdl:moreOrEquivalent fdl:value=n/> </fdl:individual>
	<fdl: individual fdl:name="a">   <fdl:membershipOf rdf:resource="#A"/>   <fdl:lessOrEquivalent fdl:value=n/> </fdl:individual>
	<fdl: individuals fdl:name="a" fdl:name=”b”>  <fdl:membershipOf rdf:resource="#R"/>   <fdl:moreOrEquivalent fdl:value=n/> </fdl:individuals>
	<fdl: individual fdl:name="a">   <fdl:membershipOf rdf:resource="#A"/>   <fdl:moreThan fdl:value=n/> </fdl:individual>
	<fdl: individual fdl:name="a">   <fdl:membershipOf rdf:resource="#A"/>   <fdl:lessThan fdl:value=n/> </fdl:individual>
	<fdl: individuals fdl:name="a" fdl:name=”b”>   <fdl:membershipOf rdf:resource="#R"/>   <fdl:lessOrEquivalent fdl:value=n/> </fdl:individuals>
	<fdl: individuals fdl:name="a" fdl:name=”b”>   <fdl:membershipOf rdf:resource="#R"/>   <fdl:moreThan fdl:value=n/> </fdl:individuals>
	<fdl: individuals fdl:name="a" fdl:name=”b”>   <fdl:membershipOf rdf:resource="#R"/>   <fdl:lessThan fdl:value=n/> </fdl:individuals>

 

 

4.         An example

Example 1: Suppose we have two images i1 and i2. Their contents are driving cars in the street. An underlying image analysis tool recognizes, among all the recognized objects, there is a Ferrari in image i1, while there is a Porsche in image i2. Furthermore, a semantic image indexing tool established that, image i1 is about a Ferrari to some degree 0.6, whereas image i2 is about a Porsche to some degree 0.8. The FOWL snippet is as figure 2. We want to know if image i1 is about a car to some degree 0.6. This is a typical problem of deciding if the assertion is fuzzy entailment of the fuzzy OWL. The question can be represented as figure 3.

Figure 1 FOWL snippet

Figure 2 Question

Reason process is as follow:

1.Elimination of concept subsumption

2. Expansion of terminological axiom knowledge base（The less format of the question merges into conversing assertion knowledge base as figure 4 ）

4.          Finish reasoning task with the constraints propagation rules. Detailed process represented by fuzzy description logic is in the appendix 1. There is a clash in the above FOWL, so the question is fuzzy entailment of the FOWL.

5.         Summary and future work

This paper extends OWL language by fuzzy constructors 、axioms and constraints. Moreover, the paper maps semantics of new vocabularies to fuzzy description logic and directly reasons fuzzy knowledge using constraint propagation calculus.

We are actively working on resolving how to implement parser and reasoning machine for FOWL. The parser can parse fuzzy knowledge base represented by FOWL and the reasoning machine can automatically finish related reason tasks of fuzzy ontology.

 

Acknowledgements

The work is (Partially) supported by the NSFC major research program: "Basic Theory and Core Techniques of Non-Canonical Knowledge" (60496322) and Open Foundation of Beijing Municipal Key Laboratory of Multimedia and Intelligent Software Technology.

References

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