Exploiting Shared Ontology with Default Information for Web Agents

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EXPLOITINGSHARED ONTOLOGYWITH DEFAULT INFORMATION FOR WEB

AGENTS

YINGLONG MA

∗

, BEIHONG JIN

†

, ANDMINGQUAN ZHOU

‡

Abstrat. Whendierentagentsommuniatewitheahother,thereneedstobesomewaytoensurethatthemeaningofwhat

oneagentembodiesisauratelyonveyedtoanotheragent. Ithasbeenarguedthatontologiesplayakeyroleinommuniation

amongdierent agents. However, insome situations, beause thereexist terminologial heterogeneities and inompletenessof

pieesofinformationamongontologiesusedbydierentagents,ommuniationamongagentswillbeveryomplexanddiultto

takle. Inthispaper,weproposedasolutiontotheproblemforthesesituations. Weuseddistributeddesriptionlogitomodelthe

mappingsbetweenontologiesusedbydierentagentsandfurthermakeadefaultextensiontotheDDLfordefaultreasoning.Then,

baseonthedefaultextensionoftheDDLmodel,aompleteinformationqueryanbereduedtohekingdefaultsatisabilityof

theomplexoneptorrespondingtothequery.

Keywords. Ontology,DesriptionLogi,Multi-agentSystem,Satisability,Defaultreasoning.

1. Introdution. Agents often utilize the servies of other agents to perform some given tasks within

multi-agentsystems[1℄. Whendierentagentsommuniatewith eahother, there needstobesomewaysto

ensurethatthemeaningofwhatoneagentembodiesisauratelyonveyedtotheotheragent. Ontologiesplay

akeyroleinommuniationamongdierentagentsbeausetheyprovideanddeneasharedvoabularyabouta

denitionoftheworldandtermsusedinagentommuniation. Inreal-lifesenarios,agentssuhasWebagents

[2℄ need to interat in a muh wider world. The future generation Web, alled Semanti Web [3℄ originates

from the form of deentralized voabularies - ontologies, whih are entral to the vision of Semanti Web's

multi-layerarhiteture[4℄. InthebakgroundofthefutureSemantiWebintelligene,thereareterminologial

knowledge bases (ontologies), reasoning engines, and also standards that make possible reasoning with the

markedoneptsontheWeb. ItnowseemslearthatSemantiWebwillnotberealizedbyagreeingonasingle

global ontology, but rather by weaving together a large olletion of partial ontologies that are distributed

arosstheWeb [5℄. In thissituation, theassumption that dierentagentsompletely sharedavoabulary is

unfeasible andevenimpossible. Infats,agentswilloften useprivate ontologiesthat dene termsin dierent

waysmaking itimpossiblefor theother agentto understand theontentsof amessage[6℄. Ushold identies

somebarriersforagentommuniation,whihanbelassiedintolanguageheterogeneityandterminologial

heterogeneity [7℄. In this paper, we will fous on terminologyheterogeneityand not onsider the problem of

languageheterogeneity.

Tooverometheseheterogeneityproblems,thereisaneedto alignontologiesusedbydierentagents,the

mostoftendisussedare mergingandmappingof ontologies[8, 9℄. However,it seemsthat theeortsarenot

enough. For ommuniationamong agentswith heterogeneous ontologies, there are still someproblems that

requiretobesolved. In somesituations,onlyinompleteinformationanbegot. These happensometimeas

unavailabilityofpieesofinformation,sometimeassemantiheterogeneities(here,terminologialheterogeneities

are foused on) among ontologies from dierent soures. Another problem is that there always exist some

exeptionalfats,whihonit withommonsenseinformation. Forexample,ommonlybirdany,penguin

belongstobird,butpenguinouldn'ty. Inthesesituations,ommuniationamongagentswillbemoreomplex

and diult to takle. We must onsider not only the alignment of ontologies used by dierent agents, but

also the impliit default information hidden among these ontologies. Then, information reasoning for query

shouldbebasedonboththese expliitlyrepresentedontologiesandimpliitdefaultinformation. Thisform of

reasoningisalled default reasoning,whih isnon-monotoni. Little attention,however,hasbeenpaidto the

problemofendowingtheselogisabovewithdefault reasoningapabilities.

For a long time, representation and reasoning in desription logi (DL) [10℄ have been used in a wide

rangeof appliations, whih areusually givenaformal, logi-based semantis. Anotherdistinguishedfeature

istheemphasisonreasoningasaentral servie. Desriptionlogiisveryusefulfordening,integrating,and

∗

ComputerSienesandTehnologyDepartment,NorthChinaEletriPowerUniversity,Beijing102206,P.R.China,

m_y_longotaix.isas.a . n,

†

TehnologyCenterofSoftwareEngineering,InstituteofSoftware,ChineseAademyofSienes,P.O.Box8718,Beijing100080,

P.R.China,jbhotaix.isas.a.n,

‡

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maintainingontologies, whih providetheSemantiWeb withaommonunderstanding ofthebasisemanti

oneptsusedtoannotateWebpages. Theyshouldbeidealandidatesforontologylanguages[11℄. DAML+OIL

[12, 15℄ and OWL [13℄ are learexamplesof Desription Logis. Reently, Borgidaand Serani proposed an

extensionof theformalframeworkofdesriptionLogi todistributed knowledgemodels[14℄, whiharealled

distributeddesriptionlogi(DDL).ADDLonsistsofasetofterminologialknowledgebases(ontologies)and

aset ofso-alledbridgerules betweenoneptdenitions from dierentontologies. Twokindsof bridgerules

areonsideredin DDL.AnotherimportantfeatureofDDL istheabilityto transformadistributed knowledge

baseinto aglobalone. Inotherwords,theexisting desriptionlogireasonersanbeappliedforderivingnew

knowledge.

We adopt the view of [6℄ that the mappings between ontologies will mostly be established by individual

agentsthat use dierentavailableontologies in order to proess agiventask. Inouropinion, itis asolution

to model the mappings between ontologies used by dierent agentsusing aDDL and further makea default

extensiontotheDDLfordefaultreasoning. Then,baseonthedefaultextensionoftheDDLmodel,aomplete

information queryanbe reduedto hek default satisabilityof the omplexonept orresponding to the

query.

Thispaperisorganizedasfollows. Setion2presentsourmotivationinmakingdefaultextensiontoDDLfor

ommuniationamong multiple agents. Setion 3introduesrepresentationandreasoningrelatedtoontology.

Distributeddesriptionlogiareintroduedpartiularly. InSetion4,weprovideaformalframeworkfordefault

extensionto desriptionlogi. DefaultreasoningbasedonanEDDTis disussedin Setion5. Meanwhile,an

algorithmisproposed forhekingthedefaultsatisabilityofagivenoneptoraterminologialsubsumption

assertion. Setion 6andSetion7arerelatedworkandonlusion,respetively.

2. Motivation. Inorder to proessagiven taskin multi-agent systems,itis importantandessentialto

ommuniate witheah otheramong dierent agents. However,there oftenexist someterminologial

hetero-geneitiesandinompletenessofpieesofinformationamongontologiesusedbydierentagents,whihmakean

agentnotompletelyunderstandtermsusedbyanotheragent. Inthesituations,itisdiultandeven

impos-sibletorealizetheommuniationamongagents. Weproposeasolutiontothisproblemforthesituation. We

modeltheknowledgerepresentationofmulti-agentsusingdistributed desriptionlogi. Theinternalmappings

betweenontologiesusedbydierentagentsaredenedusingtheso-alledbridgerulesofdistributeddesription

logi. Then,bydefaultextensiontotheDDLmodel,weanexpressexpliitlysomedefaultinformationhidden

amongtheseontologies. BasedontheextensiontoDDLmodel,aqueryanbereduedtohekingthedefault

satisabilityof aoneptoranassertionorrespondingto thequery. Morepreisely,anadapted algorithm is

proposed forhekingdefaultsatisability.

Fig.2.1. Thesituationofommuniationproblembetweentwoagents

Inorder toexpress theproblem toberesolvedlearer,we makesomeassumptionfor simpliity. Weonly

onsider ommuniationbetween twoagents, whose ontologies are enoded onthe samelanguage. Then, we

assumethatontologiesusedbythetwoagentshavesuientoverlapsuhthatinternalmappingsbetweenthem

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Inmulti-agentsystems,ontologiesareusedastheexpliitrepresentationofdomainofinterest. Toproessa

giventask,anagentperhapsusesmultipleontologies,whihusuallysupplementeahotherandformaomplete

model. However,in themodel,thedefaultinformationamongthese ontologiesisnotonsidered. Forexample,

weperhapsestablishtheinternalmappingspeifyingthatBIRDisasublassofNON_SPEAKING_ANIMAL.

Throughthe agentusing ontology 1, we take the following queryBIRD

∧

NON_SPEAKING_ANIMAL. The

foundquestionisthat theagentusing ontology1doesn'tknowthemeaningoftheterm

NON_SPEAKING_ANIMAL

whihonlyanbeunderstoodbytheagentusingontology2. Togetompleteandorretresultsofquery,the

twoagentsmust oordinate eah other. Another questionis that the query resultsof theagent will inlude

SPARROW and PARROT. We will nd the results are partially orret beause the lass of PARROT an

speak like man. The reason getting partially orret results is that wehave not onsidered the default fat:

in mostases, birdsannot speak; parrots belong to the lass of birds but they an speak. In our opinion,

defaultinformationshouldbeonsideredandaddedintothemodelwithmultipleontologies,whihwillforma

suientlyompletelymodel. Then,availablereasoningsupport forontologylanguagesis basedonthemodel

withdefaultinformation.

3. Representationand ReasoningRelated to Ontologies. Aformalandwell-foundedontology

lan-guageisthebasisforknowledgerepresentationandreasoningabouttheontologiesinvolved. DesriptionLogi

isaformalismforknowledgerepresentationandreasoning. Desriptionlogiisveryusefulfordening,

integrat-ing, andmaintainingontologies, whih provide theSemantiWeb withaommon understandingof thebasi

semantioneptsused to annotate Webpages. It should beidealandidatesfor ontologylanguages. One of

theimportantproposalsthathavebeenmadeforwell-foundedontologylanguagesfortheWebisDAML+OIL.

Reently, desription logihas heavily inuenedthe developmentof the semantiWeb language. For

exam-ple, DAML+OILontologylanguageis just analternativesyntax forveryexpressivedesriptionlogi[12℄. So

in the followingsetions, weuse syntax and semantirepresentations of desription logiinvolved instead of

DAML+OIL. Desription Logis is equipped with a formal, logi-based semantis. Its another distinguished

featureistheemphasisonreasoningasaentralservie.

3.1. DesriptionLogi. ThebasinotationsinDLarethenotationofoneptsembraingsome

individ-ualsonadomainofindividuals,androlesrepresentingbinaryrelationsonthedomainofindividuals. Aspei

DLprovidesaspeisetofonstrutorsforbuildingmoreomplexoneptsandroles. Forexamples:

thesymbol

⊤

isaoneptdesriptionwhihdenotesthetoponept,whilethesymbol

⊥

standsforthe

inonsistentoneptwhihisalled bottomonept.

thesymbol

⊓

denotesoneptonjuntion,e.g.,thedesriptionPerson

⊓

Maledenotesthelassofman.

the symbol

∀

R.C denotes the universal roles quantiation (also alled value restrition), e. g., the

desription

∀

hasChild.Maledenotesthesetofindividualwhosehildrenareallmale.

thenumberrestritiononstrutor (

≥

nR.C) and(

≤

nR.C),e.g., thedesription(

≥

1hasChild.Dotor)

denotesthelass ofparentswhohaveat leastonehildrenandallthehildrenaredotors.

Thevariousdesriptionlogisdierfrom onetoanotherbasedonthesetofonstrutorstheyallow. Here,

weshowthesyntaxandsemantisof

ALCN

[16℄,whih arelistedasFigure3.1.

Thenweanmakeseveralkindsofassertionsusingthesedesriptions. Thereexist twokindsofassertions:

subsumptionassertionsof theform

C

⊑

D

andassertionsaboutindividualsoftheform

C

(

a

)

or

p

(

a, b

)

,where

C

and

D

denoteConepts,

p

denotesrole, and

a

and

b

areindividual,respetively. Forexamples,theassertion

P arent

⊑

P erson

denotes the fat the lass of parents is subsumed by thelass of person. The desription

P erson

(

a

)

denotes that the individual

a

is a person while the desription

hasChild

(

a, b

)

denotes

a

has a hild whois

b

. Theolletion ofsubsumption assertionsis alled Tbox, whih speies theterminologyused to desribe some appliation domains. A Tbox an be regarded as a terminologial knowledge base of the

desriptionlogi.

AninterpretationforDL

I

= (∆

I

_,

_•

I

₎

,where

∆

I

isadomainofobjetsand

•

I

theinterpretationfuntion.

Theinterpretation funtion maps roles into subsetsof

∆

I

_×

_∆

I

, onepts intosubsets of

∆

I

andindividuals

intoelementsof

∆

I

. Satisfationsandentailmentsin DLTboxwillbedesribedusingfollowingnotations:

I |

=

C

⊑

D

i

C

I

_⊑

_D

I

I |

=

T

,iforall

C

⊑

D

in

T

,

I |

=

C

⊑

D

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DL Syntax DL Semanti

¬C

L

\C

I

C

⊓

D

C

I

_∩

_D

I

C

⊔

D

C

I

_∪

_D

I

∃P.C

{x|∃y.

(

x, y

)

∈

P

I

_∧

_y

_∈

_C

I

_}

∀R.C

{x|∀y.

(

x, y

)

∈

P

I

_→

_y

_∈

_C

I

_}

C

⊑

D

C

I

_⊆

_D

I

P

⊑

R

P

I

_⊆

_R

I

C

⊑ ¬D

C

I

_∩

_D

I

₌

_∅

≥

nP.C

{

x

∈

L

| k

{

y

∈

L

|

(

x, y

)

∈

P

I

_∧

_y

_∈

_C

I

}

k ≥

n

}

≤

nP.C

{

x

∈

L

| k

{

y

∈

L

|

(

x, y

)

∈

P

I

_∧

_y

_∈

_C

I

}

k ≤

n

}

C

(

a

)

a

∈

C

I

P

(

a, b

)

(

a, b

)

∈

P

I

Fig.3.1. Syntaxandsemantisofontologyrepresentation

T

|

=

C

⊑

D

,iforallinterpretations

I

,

I |

=

C

⊑

D

suhthat

I |

=

T

3.2. Distributed DesriptionLogi. ADDL isomposed ofaolletionof distributed"DLs,eah of

whih representsasubsystem of thewhole system. All ofDLs in DDL are notompletely independentfrom

oneanotherasthesamepiee ofknowledgemightbepresentedfrom dierentpointsofviewin dierentDLs.

Eah DL autonomously represents and reasonsabout a ertain subset of the whole knowledge. Distributed

desription logi(DDL) an better present heterogeneousdistributed systemsby modeling relations between

objetsandrelationsbetweenoneptsontainedindierentheterogeneousontologies.

ADDL onsistsofaolletionof DLs,whihis written{

DL

i

}

i

∈

I

, everyloalDLin DDLisdistinguished bydierentsubsripts. TheonstraintrelationsbetweendierentDLsaredesribedbyusingso-alledbridge

rules" in an impliit manner, while the onstraints between the orresponding domains of dierent DLs are

desribed by introduing the so-alled semantis binary relations". In order to support diretionality, the

bridgerules from

DL

i

to

DL

j

will beviewedasdesribing ow of information"from

DL

i

to

DL

j

from the pointof viewof

DL

j

. InDDL,

i

:

C

denotestheonept

C

in

DL

i

,

i

:

C

⊑

D

denotessubsumptionassertion

C

⊑

D

in

DL

i

. A bridgerule from

i

to

j

is desribed aordingto following twoforms:

i

:

C

⊑

−

→

j

:

D

and

i

:

C

−

⊒

→

j

:

D

. Theformerisalled into-bridge rule,and thelatteralled onto-bridgerule. A DDLembraes aset ofsubsumptionassertions,whiharealledDTB.A distributedTbox(DTB)isdenedbasedonTboxes

in allof loal DLsand bridge rules between these Tboxes. A DTB

DT

= (

{T

i

}

i

∈

I

, B

)

, where

T

i

is Tboxin

DL

i

,and forevery

i

6 =

j

∈

I

,

B

=

{B

ij

}

, where

B

ij

is aset ofbridgerulesfrom

DL

i

to

DL

j

. A DTBanbe regardedasadistributed terminologialknowledgebase forthedistributeddesriptionlogis.

ThesemantisfordistributeddesriptionlogisareprovidedbyusingloalinterpretationforindividualDL

andonnetingtheirdomainsusingsemantisbinaryrelations

r

ij

. Adistributedinterpretation

J

=

(

{I

i

}

i

∈

I

, r

)

of DT onsists ofinterpretations

I

i

for

DL

i

overdomain

∆

I

i

, and afuntion

r

assoiatingto eah

i, j

∈

I

a binaryrelation

r

ij

⊆

∆

I

i

×

_∆

I

j

.

r

ij

(

d

) =

{d

′

_∈

_∆

I

j

|

₍

_{d, d}

′

₎

∈

_r

ij

}

,andforany

D

∈

∆

I

j

,

r

ij

(

D

) =

∪

d

∈

D

r

ij(

d

)

. Notethatsemantirelation

r

mustbeboldeverywhere.

Adistributed interpretation

J

d-satises(written

|

=d

)theelementsof DTB

DT

= (

{T

i

}

i

∈

I

, B

)

aording tofollowinglauses: Forevery

i, j

∈

I

J |

=d

i

:

C

⊑

−

→

j

:

D

if

r

ij

(

C

I

i

₎

⊆

_D

I

j

J |

=d

i

:

C

⊒

−

→

j

:

D

if

r

ij

(

C

I

i

₎

⊇

_D

I

j

J |

=d

i

:

C

⊑

D

if

I

i

|

=

C

⊑

D

J |

=d

T

i

,ifforall

C

⊑

D

in

T

i

suhthat

I

i

|

=

C

⊑

D

J |

=d

DT

,ifforevery

i, j

∈

I

,

I

i

|

=d

T

i

and

I

i

|

=d

b

,forevery

b

∈ ∪B

ij

DT

|

=d

i

:

C

⊑

D

,ifforeverydistributed interpretation

J

,

J |

=d

DT

implies

J |

=d

i

:

C

⊑

D

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DT={{

T

1

={PARROT

⊑

BIRD,SPARROW

⊑

BIRD},

T

2

={PARROT

⊑

FLYING_ANIMAL, GOAT

⊑ ¬

SPEAKING_ANIMAL}

B

={1:PARROT

⊑

2:PARROT}}

DF={BIRD(

x

):PARROT(

x

)/

¬

SPEAKING_ANIMAL(

x

)} Fig.4.1.DTandDoftheDDT

∆

I

1

={parrot1,parrot2,sparrow,swan},PARROT

I

1

={parrot1,parrot2}

SWAN

I

1

={swan},SPARROW

I

1

={sparrow}

BIRD

I

₁

={parrot1,parrot2,sparrow,swan}

∆

I

2

={parrot,goat,buttery},PARROT

I

2

={parrot}

GOAT

I

2

={goat},FLYING_ANIMAL

I

2

={parrot,buttery}

¬

SPEAKING_ANIMAL

I

₂

={goat},

r

12

={(parrot1,parrot),(parrot2,parrot)}

Fig.4.2.ThedistributedinterpretationoftheDDT

betweenontologiesusedbydierentagentsaredenedusingtheso-alledbridgerulesofdistributeddesription

logi. As mentioned in Setion 2,however,DDL model is not suient for modeling ommuniationamong

multiple agents with heterogeneousontologiesbeause the default information among these ontologies is not

onsidered.Inthissituation,querybasedonmulti-agentsystemswillbepossibletogetpartiallyorretresults.

Toonstrutasuientlyompletelymodel,defaultinformationshouldbeonsideredandaddedintotheDDL

modelwithmultiple ontologies. Inthefollowing,wedisuss theproblemof defaultextensiontoDDL.

Ourdefaultextensionapproahisoperatedonadistributed terminologialknowledgebase. Adistributed

terminologialknowledge base originally embraes only somestrit information(i. e., the informationhaving

beenexpressedexpliitlyindistributedterminologialknowledgebase). Defaultinformationisusedforgetting

ompleteandorretinformationfrom multipledistributed ontologies. Weshouldonsiderawayto expliitly

inludeandexpressthedefaultinformationinadistributed terminologialknowledgebaseforreasoningbased

on these distributed ontologies. Tobe ableto inlude default information in distributed knowledge base, we

rstlyintroduethenotationdesriptionofadefault rule.

Definition 4.1. A defaultrule isof the form

P

(

x

) :

J

1

(

x

)

, J

2

(

x

)

,

· · ·

, J

n(

x

)

/C

(

x

)

,where

P, C

and

J

i

are onept names (

1 ≤

i

≤

n

), and

x

isa variable.

P

(

x

)

is alledthe prerequisite of the default, all of

J

i(

x

)

are alledthe justiationsof the default, and

C

(

x

)

isalled theonsequentof the default. The meaningof default rule

P

(

x

) :

J

1

(

x

)

, J

2

(

x

)

,

· · ·

, J

n(

x

)

/C

(

x

)

anbeexpressedasfollows:

If there exists an interpretation

I

suh that

I

satises

P

(

x

)

and doesn't satisfy every

J

i(

x

)

(

1 ≤

i

≤

n

), then

I

satises

C

(

x

)

. Otherwise, if

I

satisesevery

J

i(

x

)

(

1 ≤

i

≤

n

),then

I

satises

C

(

x

)

.

Forexample,to statethatapersonanspeakexeptifs/he isadummy,weanusethedefaultrule

Person(x):Dummy(x)/CanSpeak(x).

IfthereisanindividualnamedJohninadomainofindividuals,thentheloseddefaultruleis

Person(John):Dummy(John)/CanSpeak(John).

To dealwith strit taxonomies informationas well asdefault information in distributed knowledge base,

thedenitionofdistributedknowledgebaseshouldbeextendedforinludingasetofdefaultrules. Weallthe

distributed terminologialknowledge base withexpliit default informationdefault distributed terminologial

knowledgebase,whihisdenoted asDDT.

Definition4.2. AdefaultdistributedterminologialknowledgebaseDDT=(DT,D),whereDTistheDTB

ofdistributeddesription logi,andD isasetof default rules.

An example of a DDT is shown in gure 4.1. The DT of the DDT is based on twoloal terminologial

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The satisfationproblem ofDDT should be disussed forqueriesbased onit. The satisfationsymbolis

denoted as

|

=dd

. The kindof satisabilityof these elements in DDT meansthat theyshould satisfy notonly

DT,butalsothesetDofdefaultrules. SoweallsatisabilityofelementsinDDTdefaultsatisability. Default

satisability servesasaomplementof satisability denition in adistributed terminologialknowledge base

withdefault rules. InqueriesbasedonDDT,thedenition willbeusedto detetsatisabilityofaonept or

assertion.

Definition4.3. Adistributedinterpretation

J

dd-satises(written

|

=dd

)theelementsof

DDT

= (

DT, D

)

, aording tofollowing lauses: For every default rule

δ

inD,

δ

=

P

(

x

) :

J

1

(

x

)

, J

2

(

x

)

, . . . , J

n(

x

)

/C

(

x

)

, for every

i, j

∈

I

J |

=dd

DDT

,if

J |

=d

DT

and

J |

=d

δ

J |

=dd

DT

,if

J |

=d

DT

and

J |

=d

δ

J |

=d

δ

,if

J |

=d

P

⊑

C

implies

J

2

d

J

k

⊑ ¬C

forall

k

(

1 ≤

k

≤

n

)

J |

=d

P

⊑

C

,if

i

6 =

j

,suhthat

J |

=d

i

:

P

⊑

C

or

J |

=d

i

:

P

⊑

−

→

j

:

C

or

J |

=d

j

:

C

−

⊒

→

i

:

P

DDT

|

=dd

DT

,ifforalldistributed interpretation

J

,

J |

=dd

DDT

implies

J |

=dd

DT

Inadistributed knowledgebase,defaultinformationmayhavebeenused duringreasoning,but aDDT is

notreallyhelpfulforreasoningwithdefaultinformationindistributedknowledge. Someadditionalinformation

with respet to default rules should be inludedexpliitly into DT. A losed default rule of the form

P

(

x

) :

J

1

(

x

)

, J

2

(

x

)

,

· · ·

, J

n(

x

)

/C

(

x

)

anbedividedintotwoparts:

P

(

x

)

→

C

(

x

)

and

J

i(

x

)

→

C

(

x

)

,(

1 ≤

i

≤

n

). We allthe rstpartfullled rule, and theseond exeptionalrules. A rule ofthe form

A

(

x

)

→

B

(

x

)

means for every(distributed)interpretation

I

,

x

∈

A

I

,then

x

∈

B

I

,i.e.

A

⊑

B

,where

A

and

B

areoneptnames,and

x

denotesanindividual.

Definition 4.4. An extendeddistributedknowledge baseEDDT isonstrutedbasedona DDT=(DT,D),

aording tothe following lauses: Foreverydefault rule

δ

inD,

δ

=

P

(

x

) :

J

1

(

x

)

, J

2

(

x

)

, . . . , J

n(

x

)

/C

(

x

)

, 1) Dividing into two parts whih embraefullled rules and exeptionalrules, respetively. Thefullled

rule denotes that it holds in most ases until the exeption fats appear, while the exeptional rules denote

someexeptionalfats.

2)Adding

P

⊑

C

and

J

i

⊑

C

into DT(

1 ≤

i

≤

n

),whiharetheassertionsorrespondingtofullled rule andexeptionalrules,respetively

3) Setting the priorities ofdierentrules forseleting appropriaterulesduring reasoning. The assertions

orrespondingtoexeptionalruleshavethehighestpriority,whileoriginalstritinformationhasnormalpriority.

Theassertionsorrespondingtofullled rulesaregiventhelowestpriority.

IntheourseofonstrutinganEDDT,defaultinformationhasbeenaddedintodistributedknowledgebase

for default reasoning, beause these default information may havebeen used during reasoning. Exeptional

information has been assigned the highest priority to avoid oniting with some strit information, while

fullled rules would beused onlyin thesituation that no other stritinformation anbeused, its priority is

least. Asimplied viewof theEDDT basedonthe DDT and itsinterpretation (shownin gure 4.1and 4.2)

anbefoundin gure4.3. Thedefaultrule

BIRD

(

x

) :

P ARROT

(

x

)

/SP EAKIN G

_

AN IM AL

(

x

)

isdivided intoonefullledruleand oneexeptionalrule,thefullledrule

BIRD

⊑ ¬SP EAKIN G

_

AN IM AL

andthe exeptionalrule

P ARROT

⊑

SP EAKIN G

_

AN IM AL

has been added into EDDT. Infat, an EDDTan be reognized as a olletion of integrated ontologies with default information expressed expliitly. Default

reasoning an be performed based on an EDDT. In the following setion, we will fous on how the default

reasoning basedon EDDT will be realized. Meanwhile, an adapted algorithm will bedisussed for heking

defaultsatisabilityofomplexoneptsand subsumptionassertions.

5. Reasoning with Default Information. Reasoning with default information provides agents using

dierent ontologies with stronger query apability. In our opinion, a query based on DDT an boil down

to heking default satisability of omplex onept in aord with the query. Based on desription logis,

satisability of aomplex onept is deided in polynomial time aording to Tableau algorithm for

ALCN

[10,16℄. AnimportantresultofDDListheabilitytotransformadistributedknowledgebaseintoaglobalone.

Sotheexisting desriptionlogireasonersanbeappliedforderiving newknowledge. Thiswouldallowus to

(7)

Fig.4.3. AnexampleofEDDT

knowledgebase ofaDDL willbetransformedtothereasoningproblem ofterminologialknowledgebase ofa

globalDLorrespondingtotheDDL.Soinouropinion,detetingdefaultsatisabilityofaDDLisjustdeteting

thedefaultsatisabilityoftheglobalDLinaordwiththeDDL.AdefaultextensiontoTableaualgorithmfor

ALCN

DLanbeusedfordetetingdefault satisabilityof

ALCN

oneptsbasedonanEDDT.

Definition 5.1. A onstraint set S onsists of onstraints of the form C(x), p(x,y), where C and p are

onept nameandrolename, respetively. Bothxandy arevariables.

An

I

-assignmentmaps avariable

x

intoaelementof

∆

I

. If

x

I

_∈

_C

I

, the

I

-assignmentsatises

C

(

x

)

. If

(

x

I

_{, y}

I

₎

_∈

_p

I

,the

I

-assignmentsatises

p

(

x, y

)

. Ifthe

I

-assignmentsatiseseveryelementinonstraintset

S

, itsati ses

S

. Ifthere exist aninterpretation

I

and an

I

-assignmentsuh that the

I

-assignmentsatises the onstraintset

S

,

S

issatisable.

S

issatisable ialltheonstraintsin

S

aresatisable.

ItwillbeonvenienttoassumethatalloneptdesriptionsinEDDTareinnegationnormalform (NNF).

Usingde-Morgan'srulesandtheusualrulesforquantiers,any

ALCN

oneptdesriptionanbetransformed

into an equivalent desriptionin NNF in linear time. Forexample, theassertion desription

SP ARROW

⊑

BIRD

anbetransformedtheform

¬SP ARROW

⊔

BIRD

. Toheksatisabilityofonept

C

,ourextended algorithmstarts withonstraintset

S

=

{C

(

x

)

}

, andapplies transformationrules in anextended distributed knowledgebase. Theonept

C

issatisableitheonstraintset

S

isunsatisable. Inapplyingtransformation rules,ifthereexistallobviousonits(lashes)in

S

,

S

isunsatisable,whihmeanstheonept

C

issatisable. Otherwise,

S

isunsatisable. Thetransformationrulesare derivedfrom oneptsand assertionsin EDDT. If theonstraintset

S

before the ationis satisable,

S

after theation is also satisable. The transformation rulesofdefaultextensiontosatisabilityalgorithmareshownasFigure5.1.

Whentheadaptedalgorithmisusedfordetetingdefaultsatisabilityof

ALCN

onepts,everyationmust

preservesatisability. Beauseifanation don'tpreservesatisability, weannotensuretheonditionthat if

the onstraint set before the ationis satisable then theset after theation is satisable. In the extension

algorithm,wemustprovetheationspreservesatisability.

Theorem 5.2. The ationof the appliedtransformationrulespreserves satisability.

Proof. Beause aDDL an be regardedas aglobal DL, forsimpliation, we useinterpretation

I

of the

globalDLfordistributed interpretation

J

oftheDDL.

Intheextensionalgorithm,everystepmayinvolvetheationsofsometransformationrulesthatareapplied.

sowe must proveall of these ations in these stepspreserve satisability. Beause the ationsin theseond

stepareoriginallyderivedfromthelassialTableaualgorithm,wehaveknowntheypreservesatisability[10℄.

Theremainderoftheproofwillonlyonsidertheationsintherststepandthethirdstep.

1)Intherststep,theationonditionisthat foranydefaultruleoftheform

P(

x

) : J

1

(

x

)

,

J

2

(

x

)

, . . . ,

Jn(

x

)

/

C(

x

)

(8)

Exeptionalrules:(UsedforStep1)

Condition:

Foranydefault ruleofthe formP(

x

):J

1

(

x

), J

2

(

x

),...,J

n

(

x

)/C(

x

), thereexistsJ

i

(

x

)(1

6 i

6

n)isontained inS,butSdoesn't ontain

¬

C(

x

).

Ation:

S=S

∪

{

¬

C(

x

)}

Stritrules:(UsedforStep2)

⊓−

rule:

Condition:

{(C

⊓

D)(

x

)}

⊆

S,butSdoesn'tontainbothC(

x

)andD(

x

). Ation:

S=S

∪

{C(

x

),D(

x

)}

⊔−

rule:

Condition:

{(C

⊔

D)(

x

)}

⊆

Sbut{C(

x

),D(

x

)}

∩

S=

∅

. Ation:

S=S

∪

{C(

x

)}orS=S

∪

{D(

x

)}

∃−

rule:

Condition:

{(

∃

R.C)(

x

)}

⊆

S,butthereisnoindividualname

y

suhthatSontainsC(

x

)andR(

x

,

y

). Ation:

S=S

∪

{C(

y

),R(

x

,

y

)}

∀−

rule:

Condition:

{(

∀

R.C)(

x

),R(

x

,

y

)}

⊆

S,butSdoesn'tontainC(

y

). Ation:

S=S

∪

{C(

y

)}

≥

n-rule:

Condition:

(

≥

nR)(x)S,theredoesn'texistindividualnamesy

1

,y

2

,

· · ·

,y

n

suhthatR(x,y

i

)andy

i

6 =

y

j

areinS,(1

≤

i

≤

j

≤

n). Ation:

S=S

∪

R(x,y

i

)

∪

y

i6

=

y

j

,(1

≤

i

≤

j

≤

n),wherey

1

,

· · ·

,y

n

aredistintindividualnamesnotourringinS.

≤

n-rule:

Condition:

distintindividualnamesy

1

,

· · ·

,y

n

+1

areontained inSsuhthat(

≤

nR)(x)and R(x,y

1

),

· · ·

,R(x,y

n

+1

)areinS,and y

i6

=

y

j

is notinSforsomei,j,

1

≤

i

≤

j

≤

n+1. Ation:

for eahpairy

i

and y

j

,suhthat 1

≤

i

≤

j

≤

n+1and yi

=

6

yjisnot inS,the S

i,j

:=[y

i

/y

j

℄SisobtainedfromSbyreplaing eah ourreneofy

i

byy

j

.

Fullledrule:(UsedforStep3)

Condition:

noothertransformationrulesisappliable,andforanydefaultruleoftheformP(

x

):J

1

(

x

),J

2

(

x

),...,J

n

(

x

)/C(

x

),{P(

x

)}

⊆

S,but alloftheJ

i

(

x

)(1

6 i

6

n)andC(

x

)arenotontainedinS.

Ation:

S=S

∪

{C(

x

)}

Fig.5.1.TheadaptedTableaurulesusedfordetetingdefaultsatisabilityof

ALCN

onepts

2) In thethird step, the ation onditionis that {P(

x

)}

⊆

S, S doesn't ontainall of theJ

i

(

x

) (1

6 i

6

n) and noothertransformation rulesanbe applied. If theonstraintset Sbefore theationis satisable, then

there exists an interpretation

I

suh that

I

satises all of elements ofS. Beause {P(

x

)}

⊆

S, then

I

satises P(

x

). Furthermore, we know that

I

doesn't satisfy any J

i

(

x

) (1

6 i

6

n), otherwise, there would exist other exeptionalruleswhih anbeapplied. Beause

I

satisesP(

x

)beforetheation. SofromDenition 4.1, we get

I

satisesC(

x

). Beause

I

satisesbothSandC(

x

),weget

I

satisesS

∪

{C(

x

)}.

From above proofs, we an onlude that every ation in the applied transform rules, in the extension

algorithm,preservessatisability.

AsmentionedinDenition4.4,anEDDTembraesthreetypesoftransformationrules: stritinformation,

fullledinformationandexeptionalinformation. Thesedierenttypesofinformationaregivendierentlevels

ofpriority. Here,weusethesymbolSR to denotetheset ofstritfats in anEDDT,FRto denotetheset of

(9)

Algorithm:hekingdefault satisabilityofCbasedon theEDDT

Require:AnEDDTwhihembraesSR,FRandER.

Ensure:thedesriptionsofSR,FRandERinNNF.

1.S

0

=C(x),i=1;

2.applystritrulesandtransformS

0

intoS

i

; 3.foreahr

∈

ERdo //Step1 4. ifS

i

meetstheonditionofr

5. applyrtoS

i

andresultofation: S

i

+1

←

S

i

; 6. i=i+1;

7. ifthereexistlashesinS

i

8. returnCissatisable";

9. endif

10. endif

11. endfor

12. foreahr

∈

SRdo //Step2

13. ifS

i

meetstheonditionofrandS

i

isn'tlabeledClash" 14. applyrtoS

i

andresultofation: S

i

+1

←

S

i

;

15. i=i+1;

i

17. S

i

islabeledClash"; 18. endif

19. endif

20. endfor

21. foreahr

∈

FRdo //Step3

22. ifS

i

meetstheonditionofrandS

i

isn'tlabeledClash" 23. applyrtoS

i

andresultofation: S

i

+1

←

S

i

;

24. i=i+1;

i

26. S

i

islabeledClash"; 27. endif

28. endif

29. endfor

30. iftheleafnodesofallpossiblebranhesintheonstrutedtree-likemodelarelabeledClash"

31. returnCissatisable";

32. elsereturnCisunsatisable";

33. endif

SR

=

{¬P ARROT

⊔

BIRD,

¬SP ARROW

⊔

BIRD,

¬P ARROT

⊔

F LY IN G

_

AN IM AL,

¬GOAT

⊔ ¬SP EAKIN G

_

AN IM AL}

F R

=

{¬BIRD

⊔ ¬SP EAKIN G

_

AN IM AL}

ER

=

{¬P ARROT

⊔

SP EAKIN G

_

AN IM AL}.

Thesubsumption assertionsto be hekedshould betransformed into their negationdesriptionin NNF

aordingto the theorem [10℄:

A

⊑

B

is satisable i

A

⊓ ¬B

is unsatisable, where

A

and

B

are onept desriptions,respetively. Forexample,thesubsumptionassertion

SP ARROW

⊑ ¬SP EAKIN G

_

AN IM AL

will be transformed into the onept desription with negation

SP ARROW

⊓

SEAKIN G

_

AN IM AL

. In thefollowing,wewilldesribepartiularlytheextensionalgorithmforhekingdefaultsatisabilityofagiven

onept. Thedefaultextensionalgorithmanbedividedintothreesteps. Intherststep,weapplyexeptional

rulestoonstraintset beausetheyhavethehighestpriority. Ifexeptionalrulesanbeusedforthedeteted

onept, strit rules will not be used. Otherwise, ifno exeptionalrules an be used, the strit rules anbe

applied to onstraint set (step 2). The reason why we do like this is to avoid oniting with some strit

information. Anotherreasonisto savereasoningtime. Instepthree,onlyin thesituationthatnootherstrit

informationanbeused, ouldfullled rulesbeused. Thedefaultextensionalgorithmeitherstops beauseall

ationsfail withobviousonits,oritstops withoutfurther usedrules.

ThefollowingexampleshowninFigure5.2demonstratesthealgorithmwithatree-likediagram. Wewantto

(10)

P ARROT

(

x

)

/SP EAKIN G

_

AN IM AL

(

x

)

, weknowthat

P ARROT

(

x

)

isn't ontained in S

0

, Then, in the rst step, the exeptional rule

¬P ARROT

(

x

)

⊔

SP EAKIN G

_

AN IM AL

(

x

)

an not be applied to S

0

. In thefollowingsteps,weapplystrit rules,thereasoningontinuesuntil itstopswithobviousonits. Finally,

the leaf node of every branh in this tree-like diagram is notated using Clash" tag. So we know the

on-straint

SP ARROW

⊓

SP EAKIN G

_

AN IM AL

arenotsatisable. Thatistosay,thesubsumptionassertion

SP ARROW

⊑ ¬SP EAKIN G

_

AN IM AL

issatisable.

S

0

=

{

(SPARROW

⊓

SPEAKING

_

ANIMAL)(

x

)

}

⊓

S

1

=

S

0

∪ {

SPARROW(

x

)

,

¬

SPEAKING

_

ANIMAL(

x

)

}

¬

SPARROW

⊔

BIRD(

x

)

S

2

=

S

1

∪ {¬

SPARROW(

x

)

}

S

2

=

S

1

∪ {

BIRD(

x

)

}

//Clash

¬

BIRD(

x

)

⊔ ¬

SPEAKING

_

ANIMAL(

x

)

S

3

=

S

2

∪ {

BIRD(

x

)

}

S

3

=

S

2

∪ {¬

SPEAKING

_

ANIMAL(

x

)

}

//Clash //Clash

Fig.5.2. Detetingdefaultsatisabilityofomplexonept

Please note that the extension algorithm an takle both general subsumption assertions and assertions

aboutexeptionalfats. Inanotherexample shownin Figure 5.3,wewanttohekwhether thesubsumption

assertion

P ARROT

⊑

SP EAKIN G

_

AN IM AL

issatisable,thatistosay,wehekthedefaultsatisability oftheonept

P ARROT

(

x

)

⊓ ¬SP EAKIN G

_

AN IM AL

(

x

)

,whihtransformedinto aonstrainset. Inthe rststep,whentheexeptionalrule

¬P ARROT

(

x

)

⊔

SP EAKIN G

_

AN IM AL

(

x

)

isappliedtoonstraintset, the omplete onits our. So weknow theonept

P ARROT

(

x

)

⊓ ¬SP EAKIN G

_

AN IM AL

(

x

)

is not satisable,whihmeansthat thesubsumptionassertion

P ARROT

⊑

SP EAKIN G

_

AN IM AL

is satisable. Thenreasoningproessstops withoutapplyingother transformationrules. Thisanbeservedasanexample

ofreasoningforanexeptionalfat.

S

0

=

{

(PARROT

⊓ ¬

SPEAKING

_

ANIMAL)(

x

)

}

⊓

S

1

=

S

0

∪ {

PARROT(

x

)

,

¬

SPEAKING

_

ANIMAL(

x

)

}

¬

PARROT

⊔

SPEAKING

_

ANIMAL(

x

)

S

2

=

S

1

∪ {¬

PARROT(

x

)

}

S

2

=

S

1

∪ {

SPEAKING

_

ANIMAL(

x

)

}

//Clash

Clash

Fig.5.3.Anexampleof detetingexeptionalfat

Inthefollowing,wegiveabriefofdisussionofomplexityissuesaboutthedefaultsatisabilityalgorithm.

(11)

Proof. From [16℄, we know that satisability of

ALCN

-onept desriptions is PSPACE-omplete. As

mentionedabove,ourdefaultsatisabilityalgorithmfor

ALCN

-oneptdesriptionsanbedividedintothree

steps. Infat, everystepis just thesatisabilityalgorithm for

ALCN

. Thenthe sequeneof thethree steps isalsoessentiallythesatisabilityalgorithm for

ALCN

. Sowegetthe onlusionthat defaultsatisabilityof

ALCN

-oneptdesriptionsisPSPACE-omplete.

6. Related work and Disussions. In the desription logis ommunity, a number of approahes to

extenddesriptionlogis with default reasoninghavebeenproposed. Baaderand Hollunder [17℄ investigated

theproblemsaboutopendefault indetailanddened apreferene relation. Theapproah isnotrestritedto

simplenormal default. Twokindsof default rules were introdued by Straia[18℄. Therst kindis similar

to the fulled rules in our approah. The seond kind of rules allows for expressing default information of

llersof roles. Lambrix[19℄ presentedadefault extension todesriptionlogisfor usein anintelligentsearh

engine, Dwebi. Besides the standardinferenes, Lambrix added anewkind of infereneto desription logi

framework to desribe whether an individual belongs to a onept from a knowledge base. Calvanese [20℄

proposedaformalframeworktospeifythemappingbetweentheglobalandtheloalontologies. Maedhe[21℄

also proposed a framework for managing and integratingmultiple distributed ontologies. Stukenshmidt [6℄

exploitedpartialsharedontologiesinmulti-agentommuniationusinganapproximationapproahofrewriting

onepts. However, default information wasnot onsidered in these dierent frameworks and systems. An

importantfeatureofourformalframeworkdistinguishedfromotherworkisthatourdefaultextensionapproah

is basedon DDL. Toour best knowledge, little work hasbeendone to pay attention to default extension to

DDLforommuniationamongagents.

There is an alternative proposal for dealing with the problem of the example shown in Figure 2.1. For

example,ifthetermSPARROWinsteadofBIRDin ontology1ismappedintotheterm

NON_SPEAKING_ANIMAL

inontology2,andthetermPARROTinontology1isnotmappedintothetermNON_SPEAKING_ANIMAL,

then there is no default informationto be onsidered. It seemsthat wehaveavoided theproblem of default

information betweenthe twoontologies using the inter-ontologymapping. However,in fat,this approah is

exhausted and unsalable. If there are a lot of terms belonging to the sublasses of BIRD to be added into

ontology1,wehaveto mapeveryoneofthese addedtermsinto NON_SPEAKING_ANIMALin ontology2.

In thesituation, we will nd the alternativeapproah is muh exhausted and unsalable. In ontrast to the

alternativeapproah,ourdefaultextensionapproahto DDLonsiderstheinter-ontologymappingeortsand

thesalabilityofontologiesusedbydierentagentsaskeyfeatures.

Regarding to the omplexity issue of the proposed default satisability algorithm, we will nd that the

algorithminreasenomoreomplexitythansatisabilityalgorithmfor

ALCN

. Itmeansthatweanperform

reasoningwith strit information aswell asdefault information in thesame timeand spae omplexity. The

future work inludes aexible mehanismfor parsing exhangedmessagesamong agents. ACLs are used to

onstrutandparseexhangedmessagesrequiredbybothpartiipants. Then,oneptsdened inDAML+OIL

ontologylanguageanbereadilyombinedwiththemehanism,thusinreasingtheexibilityofmessages,and

heneaessibilityandinteroperabilityofservieswithinopenenvironments.

7. Conlusion. Inthispaper,anapproahisproposedtoenablesagentsusingdierentontologiesonthe

Web to exhangesemantiinformation solely relyingoninternally provided mappingbetween theontologies.

Beause of the semanti heterogeneity among these ontolgies, it is diult for an agent to understand the

terminologyofanotheragent. Togetompleteandorretsemantiinformationfrommultipleontologiesused

bydierentagents,default information among these ontologies should beonsidered. Ourapproahis based

ondefaultextensiontoDDL.Thedistributedterminologialknowledgebaseisoriginallyusedtopresentstrit

information. To perform default reasoningbased on DDL, strit aswell asdefault informationis takeninto

aount. Then, all of default information above is added into anextended default distributed terminologial

knowledgebase(EDDT),whihisonstrutedfromadefaultdistributedterminologialknowledgebase(DDT).

ThedefaultTableaualgorithmisusedonEDDTwheredierentruleshavedierentpriority: exeptionalrules

havethehighestpriority,andfullledrulestheleast. Reasoningwithdefaultinformationprovidesagentsusing

(12)

OurapproahenablesagentsusingdierentontologiesontheWebtoexhangesemantiinformationsolely

relyingoninternallyprovidedmappingbetweentheontologies. Butsofar,ourapproahisonsideredasabasi

mehanismforfailitatingagentommuniation. Toapplyitinpratie,thereisstill alot ofworkto bedone

[23℄. For example, moresophistiated agent ommuniationprotools, similar to KQML[22℄ and FIPA [24℄,

haveto be developed forgettingomplete andorret informationthroughagents. Usingthe ommuniation

protools, onepts dened in DAML+OIL ontology languageanbe readilyombined with the mehanism,

thusinreasing theexibility ofmessages,and heneaessibilityand interoperabilityof servieswithin open

environments.

Aknowledgments. This researh is partially supported by theNational Grand Fundamental Researh

ProgramofChinaunderGrantNo.TG1999035805,2002CB312005,theChineseNational863"High-Teh

Pro-gramunder GrantNo.2001AA113010.

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Editedby: ShahramRahimi,RaheelAhmad

Reeived: Otober10,2005