C Web

Extension of OWL with Dynamic Fuzzy Logic

Zhiming Cui, Wei Fang, Xuefeng Xian, Shukui Zhang, and Pengpeng Zhao

Jiangsu Provincial Key Laboratory of Computer Information Processing Technology,

Soochow University, Suzhou

The Institute of Intelligent Information Processing and Application,

Soochow University, Suzhou, 215006, P.R. China

************@****.***.**

Abstract. In recent years, ontology has played a major role in knowl-

edge representation. Ontology languages are based on description logics.

Though they are expressive enough, they cannot express and reason with

fuzzy and dynamic knowledge on the Semantic Web. To deal with uncer-

tain and dynamic knowledge on the Semantic Web and its applications,

a new fuzzy extension of description logics,OWL and Ontology based

on Dynamic fuzzy logic called the dynamic Description logics(DFDL),

dynamic fuzzy Ontology(DFO) and dynamic fuzzy OWL (DFOWL) are

presented. The syntax and semantics of DFDL, DFO and DFOWL are

formally de ned, and the forms of axioms and assertions are speci ed.

The research indicates the DFOWL provides more expressive power

for the Semantic Web, and overcomes the insu ciency of OWL as the

ontology language for the Semantic Web.

Keywords: dynamic fuzzy logic; semantic web; ontology; OWL.

1 Introduction

In recent years, Ontology has played a major role in knowledge representation for

the Semantic Web. Ontology is a conceptualization of a domain into a human

understandable, and machine-readable or machine-processable format consist-

ing of entities, attributes, relationships, and axioms[1].The OWL(Web Ontology

Language) is designed for use by applications that need to process the content

of information instead of just presenting information to humans. And the OWL

is intended to provide a language that can be used to describe the classes and

relations between them that are inherent in Web documents and applications.

OWL facilitates greater machine interpretability of Web content than that sup-

ported by XML, RDF, and RDFS by providing additional vocabulary along with

a formal semantics[2].

The Semantic Web[3] is a vision for the future of the Web in which information

is given explicit meaning, making it easier for machines to automatically process

and integrate information available on the Web. Description logics (DLs)[4] are

widely used on the semantic web. Fuzzy extensions of description logics import

the fuzzy theory to enable the capability of dealing with fuzzy knowledge[3].

The fuzzy knowledge plays an important role in many domains that face a huge

L. Chen et al. (Eds.): APWeb and WAIM 2009, LNCS 5731, pp. 67 76, 2009.

c Springer-Verlag Berlin Heidelberg 2009

68 Z. Cui et al.

amount of imprecise and vague knowledge and information, such as text mining,

machine learning, information integration and natural language processing[5].

On the Semantic Web, the knowledge expression is a very key problem, but a

lot of knowledge has a dynamic and fuzzy characters, traditional logical method

or fuzzy logic or some other knowledge expression models are very di cult to

express them accurately and e ectively, such as, She is a girl who becomes more

and more beautiful. Here become, beautiful have embodied dynamic

character and fuzzy character su ciently. If we use the existing approaches

to resolve these problems having dynamic and fuzzy characters will be very

di cult, they can only represent static knowledge.

In addition, SHOIN(D) is the theoretical counterpart of the OWL Descrip-

tion Logic[6]. Thus, in the paper, we de ne a dynamic fuzzy extension of the

OWL language considering fuzzy SHOIN(D). We have extended the syntax and

semantic of fuzzy SHOIN(D) with the possibility and dynamic to add a concept

modi er to a relationship and introducing a novel constructor which enables us

to de ne a subset of concepts with a membership value greater or lower that a

xed value. The main contribution of this paper is the description of how we

transfer a classical DL, ontology and OWL to dynamic fuzzy description logic

(DFDL), dynamic fuzzy ontology (DFO) and dynamic fuzzy OWL (DFOWL)

which have better representation and inference ability for fuzzy and dynamic

knowledge on the Semantic Web.

2 Dynamic Fuzzy Description Logics

A lot of information on the Semantic Web[7][3] are uncertain, imprecise and

dynamic. And, the traditional Description Logics cannot express and inference

these dynamic knowledge e ciently. Then, to deal with these uncertain and

dynamic knowledge on the Semantic Web, we have extended the Fuzzy Set and

Description Logic based on Dynamic Fuzzy Logic(DFL)[8][9].

2.1 Dynamic Fuzzy Logic

De nition 1. A statement having character of dynamic fuzzy is called dynamic

fuzzy proposition that is usually symbolized by capital letters A, B, C... . E.g.1

Here is a DF proposition: It will be getting hot soon.

De nition 2. A dynamic fuzzy number [0,1],which is used to mea-

aa

sure a dynamic fuzzy proposition s true or false degree, is called dynamic fuzzy

proposition s true or false. It is usually symbolized by, ( b, b ), . . .,

aa cc, ) = or, min =,max =, the same are as

where ( a a a a aa a aa a

follows.

De nition 3. A dynamic fuzzy proposition can be regarded as a variable whose

value is in the interval [0,1] [, ]. The variable is called dynamic fuzzy propo-

sition variable that is usually symbolized by small letter.

Extension of OWL with Dynamic Fuzzy Logic 69

Operation rules of any dynamic fuzzy variable [0,1] are pre-

xx yy

scribed as follows:

1 Negation

The negation of variable is presented by, and =((1

xx xx xx, 1 ))

-x x

2 Disjunction =max x

x y y

xx yy

3 Conjunction =min xx yy xx yy

4 Condition =max xx yy xx yy xx yy

5 bi-direction =min(max,max xx yy xx yy xx yy

De nition 4. Dynamic fuzzy calculus formations can be de ned as follows:

(1) A simple dynamic fuzzy variable itself is a well-formed formula.

(2) If P is a well-formed formula, P is a well-formed formula,

xx xx

too.

(3) If P and Q, are well-formed formulas, P Q,

xx yy xx yy, )P Q, P Q, P Q are also well-

(x x yy xx yy xx yy

formed formulas.

(4) A string of symbols including proposition variable connective and brackets

is well-formed formula if and only if the strings can be obtained in a nite of

steps, each of which only applies the earlier rules (1),(2) and (3).

The main formulas can be found in reference[8][9].

2.2 Dynamic Fuzzy Description Logics

The fuzzy description logic(FDL or FALC)[13][14][15] interpret concepts or roles

as fuzzy sets of individuals or individual pairs. Such concepts and roles are called

fuzzy concepts and fuzzy roles. But the Fuzzy Description Logics (FDL) cannot

express the dynamic knowledge. Then FDL is extended with dynamic fuzzy ca-

pabilities to yield DFDL (Dynamic Fuzzy Description Logics) in which concepts

are interpreted as dynamic fuzzy sets. For example, in DFDL, a concept C is in-

terpreted as a dynamic fuzzy set and a statement like a is C has a truth-value

in [0,1] [, ]. In this case, I (s) is an interpretation function mapping C into

a membership function C I (s),C I (s) : I (s) [0, 1] [, ]. S is state of DFDL.

Acting on concepts, the crisp operations of conjunction, disjunction, negation

and quanti cation are normally extended to their dynamic fuzzy counterparts.

By using the dynamic fuzzy logic, we can say it is becoming hotter and hotter

to a degree of 0.17, another day is hot to a degree of 0.19.

DFDL are a set Nc of concept names and a set NR of role names. The syntax

of DFDL is the extending syntax combined FDL with DFL.

Concepts and relationship C,D of DFDL are de ned with the following syntax

rules:

70 Z. Cui et al.

(1) C, D C C D C D R.C R.C C D

(2) C,C I (s) C C I (s)

(3) R r R

Where Ci Nc, R NR, is an action [0,1] [, ], r is an atom relationship.

De nition 5. Recursive formula de nition of DFDL predicate formation is as

follows:

(1) An atom ( rst order logical symbols) is a formula.

(2) Assertion formula and general formula are both formula.

(3) If G and H are formulas, T is a dynamic fuzzy truth value of assignment, is a free variable in DFDL, G,G H,G H,G H, G H, G,

xx xx, )G, G are all formulas.

( x x xx

(4) Any string of symbol is a formula of DFDL, if and only if the string can

be obtained in a nite of steps, each of which only applies the earlier rules (1),

(2) and (3).

The assertion formula of DFDL is as follows:

Dynamic Fuzzy TBox DFTB. A dynamic fuzzy TBox DFTB consists of a

nite set of dynamic fuzzy concept inclusion axioms of the form,,n>,, and, where c is a concept including axiom, and

is action.

Dynamic Fuzzy ABox DFAB. Adynamic fuzzy ABox DFAB consist of a

nite set of dynamic fuzzy concept and fuzzy role assertion axioms of the form,, R(a,b),C(a), and C(a, b), Where a, b is a

concept or role assertion.

Dynamic Fuzzy Knowledge Base DFKB. A dynamic fuzzy knowledge base

DFKB= consists of a dynamic TBox DFTB, and a dynamic

fuzzy ABox DFAB.

De nition 6. If the knowledge base DFKB= of DFDL C is

satis ed with the equation: W =DFTB, and W =DFAB, then the DFKB is

satis able, which denoted by W =DFKB.

2.3 Semantic Interpretation of DFDL

In DFDL, s is a state of DFDL. The fuzzy interpretation of s is de ned as

mapping I =, where I (s) is a nonempty set as the domain.

Mapping function I (s) makes a concept map to a dynamic fuzzy subset of I (s),

and let relationship map a subset of I (s) I (s) . The semantics of DFDL

is extended. The main idea is that concepts and roles are interpreted as fuzzy

subsets of an interpretation domain. Therefore, axioms, rather being satis ed

(true) or unsatis ed (false) in an interpretation, become a degree of truth in

[0,1] [, ].

Extension of OWL with Dynamic Fuzzy Logic 71

In general, according to construction operators and description logic construct

the complex concept and simple concepts relations. Then DFDL at least contain

the following operators: Conjunction, Disjunction, Not, Existential quan-

ti cation, and Value restriction . DFDL combined with the basis of the time

constraints operator [, ] and FDL will extend to DFDL. So, the syntax

and semantic of DFDL are shown in table 1 and table 2.

Table 1. Interpretation of concepts in DFDL

DFDL Syntax Semantic

AI (s) I (s)

A

Concept

RI (s) I (s) I (s)

R

Role name

C I (s) DI (s)

C D

Conjunction

C I (s) DI (s)

C D

Disjunction

{x I (s) y (x,y) RI (s) y CI (s) }

R.C

Value restriction

{x I (s) y (x,y) RI (s) }

R.C

Existential quanti cation

I (s)

Top

Bottom

I (s) C I (s)

C

Negation

Table 2. Interpretation of axioms in DFDL

Abstract Syntax Syntax Semantic

I (s) I (s)

AI (s) C1 . . . Cn

A C1 . . . Cn

Class(A partial C1,. . .,Cn )

I (s) I (s

AI (s) = C 1 . . . Cn

A= C1 . . . Cn

Class(A complete C1,. . .,Cn )

I (s) I (s)

C1 C2 C1 C2

SubClassOf(C1,C2 )

I (s)

= (RI (s) )+

Transitive(R) Tr(R) R

I (s) I (s)

P 1 P2 P1 P2

SubPropertyOf(P1,P2 )

I ( s ) I ( s) I (s),oi ) Ri

Valued(R1,o1 ). . . value(Rn,on ) (o,oi ):Ri (o

I (s) I ( s) I ( s)

SameIndividual(o1,. . ., on ) o1 =o2 =. . . =on o1 =o2 =. . . =on

I (s) I (s)

oi =oi,,i=j

Di erntIndividuals(o1,. . ., on ) oi =oi, i=j

The assertions of DFDL is:, I = i C I (s) (aI (s) )

n, and the terminological axioms is A=C or A CI =A C i x I (s) AI (s)

(x) C I (s) .

De nition 7. A rule of Dynamic Fuzzy Production can be de ned as follows:

P Q, CDF, I . Its right hand side I is a group of conditions, left hand

side is some actions. Premise Q and conclusion P both may be DF. CDF is

( 0, 0 ) CDF ( 1, 1 ), be called Degree of Con dence.

When Q = a1 (u1 )a2 (u2 ). . . an un,if Q1 = a1 (v1 ) a1 (v2 ). . . a1 (vn ). t1 (herein

n

1 2

t :0, where [, ] is symbolized by dynamic fuzzy oper-

ator. acts on the current state and the action waiting for

executing is s. In our case s is denoted by state, is symbolized by state set,

s is indicated by the value or content of the variable in the state

x

x

xx, ),s> stands for the variable waiting for evaluating in the state

s,

In general, according to construction operators the description logic construct the

complex concept and relation on simple concepts and relations. Description logic

is usually at least contain the following operators: Conjunction, Disjunction, Not, Existential quanti cation, and Value restriction . The basic OWL

is added on the basis of the time constraints operator, will form DFOWL.

x

x

A dynamic fuzzy interpretation I with respect to a concrete domain D is a

pair I =( I (s), I (s) ) consisting of a non empty set I (s) (called the domain at

the state s),disjoint from D,where I (s) is the domain of interpretation, as

in the classical case, and I (s) is an interpretation function which maps con-

cepts (roles) to a membership function I (s) [0,1] [, ]( I (s) I (s) [0,1]

[, ]), which de nes the dynamic fuzzy subset C I (s) (RI (s) ). To each c Ic

an element in D, to each T Rc a subset of and to each n-array concrete

predicate d the interpretation dD n . D

A dynamic fuzzy concept C is satis able i there exists some dynamic fuzzy

interpretation I for which there is some a I (s) such that C I (s) (a) = n,

and n [0,1] [, ], A dynamic fuzzy interpretation I satis es a TBOX T i

a I (s) RI (s) DI (s), for each R C, and a I (s) RI (s) = DI (s), for

each R C .

5 Related Work

Nowadays, on the Semantic Web quite a lot of ontology languages exist,like the

OWL and DAML+OIL. Both these languages, use Description logic(Dls) as their

underlying formal for representation of knowledge as well as for performing tasks.

Fuzzy sets theory, introduced by L. A. Zadeh [10], allows to deal with imprecise

and vague data, so that a possible solution is to incorporate fuzzy logic into on-

tologies. In On the Expressiveness of the Languages for the Semantic Web -

Making a Case for A Little More, Ch. Thomas and A. Sheth introduce the

need for fuzzy probabilistic formalisms on the Semantic Web, in particular within

OWL. In Fuzzy ontologies for information retrieval on the WWW, D. Parry

uses fuzzy ontologies, and presents a broad survey of relevant techniques, leading

up to the notions of fuzzy search and fuzzy ontologies. Handling faceted or vague

information is an open issue in many research areas, for example as in object-

oriented databases systems[15]. Also the conceptual formalism supported by a

typical ontology[9] may not be su cient to represent uncertain information that is

commonly found in many application domains. In addition, the way in which con-

cepts and relations are usually expressed can be inadequate to handle the nuances

Extension of OWL with Dynamic Fuzzy Logic 75

of natural languages used by human to describe and to understand the context in

which they live. The fuzzy set theory [4], originally introduced by L. A. Zadeh

[10], allows one to denote non-crisp concepts. Fuzzy sets and ontologies have been

jointly used to resolve uncertain information problems in various areas, for exam-

ple, in text retrieval [17][1]or to generate a scholarly ontology from a database in

the ESKIMO [2] and FOGA [6] frameworks. However, there is not a complete fu-

sion of fuzzy set theory with ontologies in any of these examples. Kang et al.[15]

presented a description logics for fuzzy ontologies on semantic web.

6 Conclusion

The success of the deployment of the Semantic Web will largely depend on

whether useful ontologies will emerge, allowing shared agreements about vocab-

ularies for knowledge representation. [14] In this paper we have extended the DL

language, ontology and OWL with dynamic fuzzy logic theory. We introduce the

DFL theory like semantic interpretation and reasoning rules The combination of

transitive and inverse roles allow us encode and reason with fuzzy and dynamic

knowledge on the Semantic Web. Furthermore,the incorporation of DFL allows

us to encode and reason with imprecise and dynamic knowledge. More repre-

sentation features and e cient reasoning and analysis algorithms with DFL on

semantic web are the future work.

Acknowledgments. This research was partially funded by the grants from the

Natural Science Foundation of China under grant No.60673092 and No.60873116;

the 2008 Jiangsu Key Project of science support and self-innovation under grant

No.BE2008044; the Higher Education Graduate Research Innovation Program

of Jiangsu Province in 2008 under grant No.CX08B-099Z.and the Project of

Jiangsu Key Laboratory of Computer Information Processing Technology under

grant No.KJS0820.

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