OverviewoftheTutorialHistoryandBasicsSyntaxSemanticsABoxesTboxesInferenceProblemsandtheirinterrelationshipandRelationshipwithotherlogicalformalismsApplicationsofDLsERdiagramswithicomdemo ID: 116926
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DescriptionLogics|Basics,Applications,andMoreIanHorrocksInformationManagementGroupUniversityofManchester,UKUlrikeSattlerTeachingandResearchAreaforTheoreticalComputerScienceRWTHAachen,GermanyRWTHAachenGermany1 OverviewoftheTutorialHistoryandBasics:Syntax,Semantics,ABoxes,Tboxes,InferenceProblemsandtheirinterrelationship,andRelationshipwithother(logical)formalismsApplicationsofDLs:ER-diagramswithi.comdemo,ontologies,etc.includingsystemdemonstrationReasoningProcedures:simpletableauxandwhytheyworkReasoningProceduresII:morecomplextableaux,non-standardinferenceprob-lemsComplexityissuesImplementing/OptimisingDLsystemsRWTHAachenGermany2 DescriptionLogicsfamilyoflogic-basedknowledgerepresentationformalismswell-suitedfortherepresentationofandreasoningaboutàterminologicalknowledgeàcon gurationsàontologiesàdatabaseschemata{schemadesign,evolution,andqueryoptimisation{sourceintegrationinheterogeneousdatabases/datawarehouses{conceptualmodellingofmultidimensionalaggregationà:::descendentsofsemanticsnetworks,frame-basedsystems,andKL-ONEakaterminologicalKRsystems,conceptlanguages,etc.RWTHAachenGermany3 ArchitectureofaStandardDLSystem.ConcreteSituationTerminologyFather=Manu9haschild.�...Human=MammaluBiped.John:HumanuFatherJohnhaschildBillKnowledgeBaseIIDescriptionRWTHAachenGermany4 IntroductiontoDLIADescriptionLogic-mainlycharacterisedbyasetofconstructorsthatallowtobuildcomplexconceptsandrolesfromatomicones,conceptscorrespondtoclasses/areinterpretedassetsofobjects,rolescorrespondtorelations/areinterpretedasbinaryrelationsonobjects,Example:HappyFatherintheDLALC Manu(9has-child:Blue)u(9has-child:Green)u(8has-child:HappytRich)RWTHAachenGermany5 IntroductiontoDL:SyntaxandSemanticsofALCSemanticsgivenbymeansofaninterpretationI=(I;I):ConstructorSyntaxExampleSemanticsatomicconceptAHumanAIIatomicroleRlikesRIIIForC;DconceptsandRarolenameconjunctionCuDHumanuMaleCI\DIdisjunctionCtDNicetRichCI[DInegation:C:MeatInCIexistsrestrict.9R:C9has-child.Humanfxj9y:hx;yi2RI^y2CIgvaluerestrict.8R:C8has-child.Blondfxj8y:hx;yi2RI)y2CIgRWTHAachenGermany6 IntroductiontoDL:OtherDLConstructorsConstructorSyntaxExampleSemanticsnumberrestriction(nR)(7has-child)fxjjfy:hx;yi2RIgjng(;ALCN)(nR)(1has-mother)fxjjfy:hx;yi2RIgjnginverseroleR has-child fhx;yijhy;xi2RIgtrans.roleRhas-child(RI)concretedomainu1;:::;un:Ph-fatherage,age.�fxjhuI(x);:::;uIn(x)i2Pgetc.Manyotherdi erentDLs/DLconstructorshavebeeninvestigatedRWTHAachenGermany7 IntroductiontoDL:KnowledgeBases:TBoxesForterminologicalknowledge:TBoxcontainsConceptde nitionsA_=C(Aaconceptname,Cacomplexconcept)Father_=Manu9has-child.HumanHuman_=Mammalu8has-child :Human;introducemacros/namesforconcepts,canbe(a)cyclicAxiomsC1vC2(Cicomplexconcepts)9favourite:Breweryv9drinks:Beer;restrictyourmodelsAninterpretationIsatis esaconceptde nitionA:=Ci AI=CIanaxiomC1vC2i CI1CI2aTBoxTi Isatis esallde nitionsandaxiomsinT;IisamodelofTRWTHAachenGermany8 IntroductiontoDL:KnowledgeBases:ABoxesForassertionalknowledge:ABoxcontainsConceptassertionsa:C(aanindividualname,Cacomplexconcept)John:Manu8has-child.(MaleuHappy)Roleassertionsha1;a2i:R(aiindividualnames,Rarole)hJohn;Billi:has-childAninterpretationIsatis esaconceptassertiona:Ci aI2CIaroleassertionha1;a2i:Ri haI;aI2i2RIanABoxAi Isatis esallassertionsinA;IisamodelofARWTHAachenGermany9 IntroductiontoDL:BasicInferenceProblemsSubsumption:CvDIsCIDIinallinterpretationsI?w.r.t.TBoxT:CvTDIsCIDIinallmodelsIofT?;structureyourknowledge,computetaxonomyConsistency:IsCconsistentw.r.t.T?IsthereamodelIofTwithCI6=;?ofABoxA:IsAconsistent?IsthereamodelofA?ofKB(T,A):Is(T,A)consistent?IsthereamodelofbothTandA?InferenceProblemsarecloselyrelated:CvTDi Cu:Disinconsistentw.r.t.T,(nomodelofIhasaninstanceofCu:D)Cisconsistentw.r.t.Ti notCvTAu:A;DecisionProcduresforconsistency(w.r.t.TBoxes)suceRWTHAachenGermany10 IntroductiontoDL:BasicInferenceProblemsIIFormostDLs,thebasicinferenceproblemsaredecidable,withcomplexitiesbetweenPandExpTime.Whyisdecidabilityimportant?Whydoessemi-decidabilitynotsuce?Ifsubsumption(andhenceconsistency)isundecidable,andàsubsumptionissemi-decidable,thenconsistencyisnotsemi-decidableàconsistencyissemi-decidable,thensubsumptionisnotsemi-decidableàQuestfora\highlyexpressive"DLwithdecidable/\practicable"inferenceproblemswhereexpressivenessdependsontheapplicationpracticabilitychangedoverthetimeRWTHAachenGermany11 IntroductiontoDL:HistoryComplexityofInferencesprovidedbyDLsystemsoverthetimelateearly'90smid'90slateUndecidableExpTimePSpaceNPPTimeInvestigation of Complexity of Inference Problems/AlgorithmsstartsCrack, KrisClassic (AT&T)LoomKL-ONEFact, DLP, RaceRWTHAachenGermany12 IntroductiontoDL:State-of-the-implementation-artInthelast5years,DL-basedsystemswerebuiltthat4canhandleDLsfarmoreexpressivethanALC(closerelativesofconverse-DPDL)Numberrestrictions:\peoplehavingatmost2children"Complexroles:inverse(\has-child"|\child-of"),transitiveclosure(\o spring"|\has-child"),roleinclusion(\has-daughter"|\has-child"),etc.4implementprovablysoundandcompleteinferencealgorithms(forExpTime-completeproblems)4canhandlelargeknowledgebases(e.g.,Galenmedicalterminologyontology:2,740concepts,413roles,1,214axioms)4arehighlyoptimisedversionsoftableau-basedalgorithms4perform(surprisinglywell)onbenchmarksformodallogicreasoners(Tableaux'98,Tableaux'99)RWTHAachenGermany13 RelationshipwithOtherLogicalFormalisms:FirstOrderPredicateLogicMostDLsaredecidablefragmentsofFOL:Introduceaunarypredicate AforaconceptnameAabinaryrelationRforarolenameRTranslatecomplexconceptsC;Dasfollows:tx(A)= A(x);ty(A)= A(y);tx(CuD)=tx(C)^tx(D);ty(CuD)=ty(C)^ty(D);tx(CtD)=tx(C)_tx(D);ty(CtD)=ty(C)_ty(D);tx(9R:C)=9y:R(x;y)^ty(C);ty(9R:C)=9x:R(y;x)^tx(C);tx(8R:C)=8y:R(x;y))ty(C);ty(8R:C)=8x:R(y;x))tx(C):ATBoxT=fCivDigistranslatedasT=8x:^1intx(Ci))tx(Di)RWTHAachenGermany14 RelationshipwithOtherLogicalFormalisms:FirstOrderPredicateLogicIICisconsistenti itstranslationtx(C)issatis able,Cisconsistentw.r.t.Ti itstranslationtx(C)^Tissatis able,CvDi tx(C))tx(D)isvalidCvTDi t)8x:(tx(C))tx(D))isvalid.;ALCisafragmentofFOLwith2variables(L2),knowntobedecidable;ALCwithinverserolesandBooleanoperatorsonrolesisafragmentofL2;furtheraddingnumberrestrictionsyieldsafragmentofC2(L2with\countingquanti ers"),knowntobedecidableFincontrasttomostDLs,addingtransitiveroles/transitiveclosureoperatortoL2leadstoundecidabilityFmanyDLs(likemanymodallogics)arefragmentsoftheGuardedFragmentFmostDLsarelesscomplexthanL2:L2isNExpTime-complete,mostDLsareinExpTimeRWTHAachenGermany15 RelationshipwithOtherLogicalFormalisms:ModalLogicsDLsandModalLogicsarecloselyrelated:ALC multi-modalK:CuD C^D;CtD C_D:C :C;9R:C hRiC;8R:C [R]Ctransitiveroles_ transitiveframes(e.g.,inK4)regularexpressionsonroles_ regularexpressionsonprograms(e.g.,inPDL)inverseroles_ converseprograms(e.g.,inC-PDL)numberrestrictions_ deterministicprograms(e.g.,inD-PDL)ënoTBoxesavailableinmodallogics;\internalise"axiomsusingauniversalroleu:C:=D [u](C,D)ënoABoxavailableinmodallogics;usenominalsRWTHAachenGermany16