62 isacontradictoryof denotedby if2 2 RRsRdisasetofstrictRsanddefeasibleRdinferencerulesoftheform1nand1nrespectivelywhereiaremetavariablesrangingoverw ID: 163455
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ofpreferencesmayimplyfurtherpreferencesandtherebymakeresolutionsbasedonconictingpreferencesimpossible.Finally,resolutionsareimpossibleifsomeattackssucceedirrespectiveofpreferences(e.g.,attacksonnegationasfailureassumptions).Inouropinion,suchsubtletiescanonlybefullyappreciatedinasettingwherethestructureofargumentsandthenatureofattackandtheuseofpreferencetodenedefeatsismadeexplicit.TothisendwestudyresolutionsintheASPIC+framework[16,17,18],ageneralframeworkforargumentationwithpreferencesthatintegratesandfurtherde-velopsAImodelsofstructuredargumentation.In[16,17,18]conditionshavebeeniden-tiedunderwhicharangeofpossibleinstantiationsofASPIC+satisfy[9]'srationalitypostulates,whileanumberofexistingapproachestostructuredargumentationhavebeenshowntobeaninstanceorspecialcaseofASPIC+.Therefore,studyingresolutionsinthisframeworkarguablymakesthestudyasgeneralaspossible.Section2reviewsASPIC+,afterwhichSection3denesresolutionsofASPIC+argumentationframeworks,denedundertheassumptionthattheseareinducedbytheacquisitionoffurtherpreferenceinformation.Section4thenevaluatesthegrounded,pre-ferredandstablesemanticsagainst[14]'sproperties.Wealsoshowthatwhileingeneralthepreferredandstablesemanticsfailtheseproperties,onecanidentifyspecicinstanti-ationsofASPIC+thatsatisfythem.Sections3and4alsoillustratetheabovementionedlimitationsofconsideringonlyresolutionsofsymmetricattacks,andpointtosomelimitsofabstractmodelsofargumentation.Inparticular,ifresolutionsaremodelledwithoutspecifyingthestructureofarguments,thenitiseasytooverlookassumptionsmadeattheabstractlevelthatdonotholdforallreasonableinstantiationsoftheabstractframework.2.TheASPIC+frameworkWereviewtheASPIC+frameworkasdenedin[16,17].ASPIC+assumesanunspec-iedlogicallanguageL,anddenesargumentsasinferencetreesformedbyapplyingstrictordefeasibleinferencerulestopremisesthatarewellformedformulae(wff)inL.Astrictrulemeansthatifoneacceptstheantecedents,thenonemustaccepttheconsequentnomatterwhat.Adefeasiblerulemeansthatifoneacceptsallantecedents,thenonemustaccepttheconsequentifthereisinsufcientreasontorejectit.Todeneattacks,minimalassumptionsonLaremade;namelythatcertainwffareacontraryorcontradictoryofcertainotherwff,andthatdefeasibleinferencerulescanbenamedinthelanguageLthroughtheuseofanamingconventionn.Apartfromthistheframe-workisstillabstract:itappliestoanysetofstrictanddefeasibleinferencerules,andtoanylogicallanguagewithadenedcontraryrelation.ThebasicnotionofASPIC+isanargumentationsystem.Argumentsarethenconstructedw.r.taknowledgebasethatisassumedtocontainthreekindsofformulas.Denition1AnASPIC+argumentationsystemisatupleAS=(L;�;R;n;)whereLisalogicallanguageand�isacontrarinessfunctionfromLto2L,suchthat:'isacontraryof if'2 , 62 ';'isacontradictoryof (denotedby`'=� '),if'2 , 2 '.R=Rs[Rdisasetofstrict(Rs)anddefeasible(Rd)inferencerulesoftheform'1,...,'n!'and'1,...,'n)'respectively(where'i;'aremeta-variablesrangingoverwffinL),andsuchthatRs\Rd=;. Denition4AattacksBiffAundercuts,rebutsorunderminesB,where:AundercutsargumentB(onB0)iffConc(A)2 n(r)forsomeB02Sub(B)suchthatB0'stoprulerisdefeasible.ArebutsargumentB(onB0)iffConc(A)2 'forsomeB02Sub(B)oftheformB001;:::;B00n)'.InsuchacaseAcontrary-rebutsBiffConc(A)isacontraryof'.ArgumentAunderminesB(onB0)iffConc(A)2 'forsomeB0=','2Prem(B)nKn.InsuchacaseAcontrary-underminesBiffConc(A)isacontraryof'orif'2Ka.Anundercut,contrary-rebut,orcontrary-undermineattackissaidtobepreference-independent,otherwiseanattackispreference-dependent.Then,AdefeatsB(denotedA!B)iffAattacksB(denotedA*B)onB0,andeither:A*B0ispreference-independent,or;A*B0ispreference-dependentandAB0.Thesuccessofrebuttingandunderminingattacksthusinvolvescomparingthecon-ictingargumentsatthepointswheretheyconict.Thedenitionofsuccessfulunder-miningexploitsthefactthatanargumentpremiseisalsoasubargument.Addinganargumentordering(whichmayormaynotbedenedonthebasisofthepartialpreordersondefeasiblerulesandnon-axiompremises)toanargumentationtheoryconsistingofanargumentationsystemandaknowledgebase,yieldsastructuredargumentationframework:Denition5LetATbeanargumentationtheory(AS;KB).Astructuredargumen-tationframework(SAF)denedbyAT,isatriplehA,C,iwhereAisthesetofall,oronlyc-consistent,argumentsconstructedfromKBinAS,isapartialpreorderonA,and(X;Y)2CiffXattacksY.ThejustiedargumentsunderDungsemantics[11]canthenbedened.Torecap,aDungframeworkconsistsofabinaryrelationBoverasetofargumentsA.Then:SAisconictfreeiff8X;Y2S,(X;Y)=2B;X2Aisacceptablew.r.t.someSAiff8Ys.t.(Y;X)2Bimplies9Z2Ss.t.(Z;Y)2B.Then,aconictfreesetSis:anadmissibleextensioniffX2SimpliesXisacceptablew.r.t.S;acompleteextensioniffX2SwheneverXisacceptablew.r.t.S;apreferredextensioniffitisasetinclusionmaximalcompleteextension;thegroundedextensioniffitisthesetinclusionminimalcompleteextension;astableextensioniffitispreferredand8Y=2S,9X2Ss.t.(X;Y)2B.Fors2fcomplete,preferred,grounded,stableg,XisscepticallyorcredulouslyjustiedunderthessemanticsifXbelongstoall,respectivelyatleastone,sextension.If=hA,C,iisaSAF,andDthedefeatrelationobtainedfromCandthepreferenceordering,thenlettingDbethebinaryrelationB,thejustiedargumentsofarethejustiedargumentsoftheDungframework(A;D).In[18]itisshownthatundersomeintuitiveassumptionsonthestrictinferencerules,axiompremisesandthepreferencerelation,instantiationsofASPIC+satisfy[9]'srationalitypostulatesforargumentation. Denition6Letbeapartialpreorderoveraset�.Then0extendsiff0and8X;Y2�,X YimpliesX 0Y.Let=(A;C;)beaSAF.Then0=(A;C;0)preference-extendsiff0extends.Tomotivatethedenitionofextends,recallthatisapartialpreorder.ThusitdoesnotsufcetodeneextendsintermsoftheconditionX YimpliesX 0Yalone.Toseewhy,supposeXYandYX,whichimpliesXtY;thatistheyareeffectivelyassignedthesamestrength.Hence,itmightbethat0preservesthestrictpreferencesin,butXYandYX.ButwecertainlywanttopreservetheassignmentofequalstrengthtoXandY.Ontheotherhand,itdoesnotsufcetodeneextendsintermsofthecondition0alone.ThisisbecausegivenonlyXYandsoX Y,wewantthatthisstrictpreferencebepreservedintheextendedargumentordering.However,ifX0YandY0X,thenthisstrictpreferencewouldnotbepreserved.Itisstraightforwardtothenshowthatif(A;C;0)preference-extends(A;C;),andD0andDarethedefeatrelationsrespectivelydenedby0and,thenD0D.Furthermore:3Proposition1Let=(A;C;)bedenedby(K;1)in(L;�;R;2),anddenedonthebasisof1and2accordingtotheweakestorlastprinciples(asdenedinSec-tion5.1in[17]).Forany01thatextends1,andany02thatextends2,theSAF0=(A;C;0)denedby(K;01)in(L;�;R;02),preference-extends.Wecannowdenethenotionofapreference-basedresolution:Denition7Let0=(A;C;0)beaSAFthatpreference-extends=(A;C;),andletD0andDbedefeatrelationsrespectivelydenedby0and.Then0isapreference-basedresolutionofiffD0D.Theresolutionsdenedheredifferfromtheresolutionsin[3,14],whichonlyre-solvesymmetricrelationsbetweenarguments.Firstly,preferencesmayresultindenyingthedialecticalsuccessofanasymmetricattackasadefeat,bothinabstractapproachestoargumentation(e.g.,ValueBasedFrameworks[6]),butalsoinstructuredapproaches.Forexample,inASPIC+anargumentwithastricttoprulecanasymmetricallyattackanargumentwithadefeasibletoprulewhilebeingweakerthanitstarget,sothattheattackdoesnotresultinadefeat.Also,classicallogicapproachestoargumentationwithpref-erences[2](andtheirformulationinASPIC+[16,17])deneonlyasymmetricattacksfromtheconclusionofanattackingargumenttothepremiseoftheattackedargument.Secondly,someresolutionsmaynotbepossible.IfXandYinareequallystrong(XtY),andXandYattackandsodefeateachother,thenanyfurtherpreferences,andthusanyfurtherresolution,preservestheassignmentofequalstrengthandthusthesymmetricdefeat.Forexample,iftwocontradictingwitnessesorexpertsweredeemedtobeequallycredible,thennofurtherpreferencescanchangethis.Likewise,iftwocon-ictinglawswereregardedofequalhierarchicalstatusthennofurtherpreferencescanchangethis.Thirdlybothattacksinasymmetricattackmayfailtosucceedasdefeats,asillustratedbythefollowingexample4. 3Spacelimitationsprecludeinclusionofproofsforallbutthekeyresultsinthispaper.Allresultsnotformallyproveninthispaper,canbefoundinSection9,[17]4Resultsof[16,17,18]implythat[9]'sconsistencypostulatesholdforthisandallourfurtherexamples. respectively.Figures1-b)and1-c)showthetwopossiblepreference-basedresolutions,obtainedrespectivelybyextending0toinclude:q0:p(andsoP P0,P Q,Q0 Q)and:p0:q(andsoP0 P,Q P,Q Q0).ArgumentSisinthegroundedextensionofbothresolutions,butnotinthegroundedextensionofFigure1-a). Figure1.b)andc)arethetwopreference-basedresolutionsofa)However,onecanproveLefttoRightScepticalfornitaryframeworks:Theorem6IfXisinthegroundedextensionof=(A;C;),thenXisinthegroundedextensionofeverypreference-basedresolution0of.PROOF.LetDandD0bethedefeatrelationdenedbyand0respectively.LetSni=1Fibethegroundedextensionobtainedbyiterativeapplicationofthecharac-teristicfunctionFto(A;D)(i.e.,F1=F(;),Fi=F(Fi�1)).LetSmi=1Gibethegroundedextensionobtainedbyiterativeapplicationofthecharac-teristicfunctionFto(A;D0).Weshowbyinductiononi,thatX2FiimpliesX2Gi:Basecase(i=1):F1=fXj:9Y;(Y;X)2Dg.Hence,sinceD0D,G1F1.InductiveHypothesis:Forji,X2FjimpliesX2Gj.GeneralCase:SupposeX2Fi,Y!DX.Then9Z2Fi�1,Z!DY.ByinductivehypothesisZ2Gi�1.SupposeY!D0X,Z9D0Y(sinceZ 0Y).ByLemma34in[17],either9Y02Sub(Y)s.t.Y0!D0Z,or9Y+Z0s.t.Y+Z0!D0Zonsomedefeasiblesub-argumentZ0ofZ(Y+Z0isanargumentstrictlyextendingthedefeasiblesub-argumentsofYandallthedefeasibleargumentsofZexceptZ0).SinceZ2Gi�1,then9W2Gj;ji,W!D0Y.QEDThepreferredandstablesemanticsfailLefttoRightSceptical.Toillustrate,considertheargumentationsystemwhereLandthecontraryrelationsaredenedasinExample2,andthearguments(builtfromassumptionpremises)anddefeatsareshowninFigure2-a).Weassumenoorderingonthedefeasiblerulesandnon-axiompremises,andso=(recallthatattacksonassumptionpremisesarepreferenceindependent).fD;Bgisthesinglepreferred/stableextensionandsosetofscepticallyjustiedarguments.Now,considerthepreference-basedresolutioninFigure2-b)obtainedbyaddingtheordering e):ac)a,andsoD A(underthelastlinkprincipleasdenedinSection5.1[17]).Thesinglepreferred/stableextensionofthisresolutionis;. Figure2.b)isapreference-basedresolutionofa)In[14]andsubsequently[3],itisshownthatRighttoLeftScepticalholdsforthepreferredsemantics,forresolutionsasdenedin[3,14].Wenowshowthatonceweaccountforpreference-basedresolutions,RighttoLeftScepticalfails,eveninthecasewheresuchresolutionsonlyresolvesymmetricattacks.Example7ConsidertheargumentationsystemwhereLandthecontraryrelationsaredenedasinExample2,andwhere:Rs=;,Rd=fb;c)a,a)xg.TheknowledgebaseconsistsofKn=;,Kp=f:b;b;:c;cg,Ka=fb;c;ag,and:b0c,:c0b.Basedoneithertheweakestorlastlinkprinciples,D C,E B.WeobtaintheargumentsanddefeatsshowninFigure3-a).fD;E;Agisoneofthepreferred/stableextensions,andsoXisnotscepticallyjustied.Wenowenumerateallpossiblewaysofextendingtheordering0ontheordinarypremises,andthus(byProposition1)(denedundereithertheweakestorlastlinkprinciples),andtheresultantresolutions.NotethatextendingtheorderingondefeasibleruleswillmakenodifferenceasonlytheattacksbetweenBandD,andEandCarepreferencedependent): Figure3.b)isapreference-basedresolutionofa) 1.Extendingwith:b0byieldsD BandtheresolutioninFigure3-b).Thepreferred/stableextensionsarefB;E;XgandfB;C;Xg.2.Extendingwith:c0cyieldsE CandtheresolutioninFigure3-c).Thepreferred/stableextensionsarefC;D;XgandfB;C;Xg.3.Extendingwith:b0band:c0cyieldsD B,E C,andtheresolution(notshown)withpreferredextensionfB;C;Xg4.Extendingwithb0:b,thenbytransitivity:c0c,yieldingB D,E C,andtheresolutioninFigure3-d).Thepreferred/stableextensionisfC;D;Xg.5.Extendingwithc0:c,thenbytransitivity:b0b,yieldingD BandC E,andtheresolutioninFigure3-e).Thepreferred/stableextensionisfB;E;Xg6.Extendingwith:b0:c,thenbytransitivity:b0b,andweareincase1.7.Extendingwith:c0:b,thenbytransitivity,:c0c,andweareincase2.8.Extendingwithb0c,thenbytransitivity:c0c,andweareincase2.9.Extendingwithc0b,thenbytransitivity:b0b,andweareincase1.Thecounter-examplethusshowsthatforallpreference-basedresolutions,Xisascepticallyjustiedargument.HoweverXisnotascepticallyjustiedargumentof.Example7illustratestheimpossibilityofconstructingaresolution0withDasym-metricallydefeatingBandEasymmetricallydefeatingC,whichwouldyieldapre-ferred/stableextensionfD;E;AgthatexcludesX,andthuswouldpreserveRighttoLeftSceptical.TheonlyreasonRighttoLeftScepticalholdsforpreferredsemanticsin[3,14]isthatintheabstractsetupallresolutionsarepossible,including0.HowevertheASPIC+instantiation(cases4and5)illustratesthatgiventheexistingpremiseordering,anyextensionmakingB D(andsoD!B)thenimpliesE C(andsoC!E),andanyextensionmakingC E(E!C)thenimpliesD B(B!D).Theresultsforpreferred/stablesemanticsarenegative.However,thequestionnatu-rallyarisesastowhetherparticularASPIC+instantiationssatisfythedesiredproperties,andunderwhatrestrictions.InwhatfollowsweshowthatthepropertiesaresatisedbyparticularclassicallogicinstantiationsofASPIC+.Consideranargumentationsystem(L;�;R;n;)whereLisastandardproposi-tionalorrst-orderlanguage,�isdenedasclassicalnegation,RconsistsonlyofstrictinferencerulesRswhichconsistsofallvalidrst-orderinferencesoverL,and=.Let(K;0)beanyknowledgebasewithKn=Ka=;,Kp=�,�L,and0atotalpreorderoverKp(�).Let=(A;C;)whereAisthesetofc-consistentargumentsanddenedundertheweakestorlastlinklinkprinciple.Wewrite(�;0)todenotesuchaSAF.Then,forthestablesemantics,LefttoRightScepticalandRighttoLeftScep-ticalcanbeshownbyexploitinganequivalence(Theorem32in[17])betweentheaboveclassicallogicinstantiationofASPIC+andBrewka'spreferredsubtheories[8]:Denition8Adefaulttheoryisatuple(�;),where�isasetofclassicalrstorderformulae,isatotalpre-orderand(�1;:::;�n)theinducedpartitionintoequiva-lenceclasses,suchthat8 ;2�, iff 2�i;2�j,i&#x-278;j.Apreferredsubtheoryisaset=1[:::[nsuchthatfori=1:::n,1[:::[iisamaximal(undersetinclusion)consistentsubsetof�1;:::;�i.HenceforthwewritePS((�;))todenotethesetf1;:::;ngofallpreferredsubtheoriesof�.Intuitively,apreferredsubtheoryisobtainedbytakingamaximalundersetinclusionconsistentsubsetof�1,extendingthiswithamaximalconsistentsubsetof�2,extendingthiswithamaximalconsistentsubsetof�3,andsoon. Allthisworkimplicitlyassumesthatsuchadditionsordeletionsareindependentofeachother,anassumptionthatmaynotholdforinstantiations.Weconcludebypointingtofuturework.TheinstanceofASPIC+thatisshowninSection4tosatisfybothproperties,suggestsinvestigatingotherconditionsunderwhichpropertiesaresatised.Also,theintuitionsunderlyingtheproperties,suggestotherprop-ertiesbywhichsemanticscouldbeevaluated.Forexample,`XisacredulouslyjustiedargumentofiffXisascepticallyjustiedargumentofsomepreference-basedreso-lution0',andtheweakerpostulate`XisacredulouslyjustiedargumentofiffXisacredulouslyjustiedargumentofsomepreference-basedresolution0'.References[1]L.AmgoudandP.Besnard.Aformalanalysisoflogic-basedargumentationsystems.InProc.4thInternationalConferenceonScalableUncertainty(SUM'10),pages42–55,2010.[2]L.AmgoudandC.Cayrol.Areasoningmodelbasedontheproductionofacceptablearguments.AnnalsofMathematicsandArticialIntelligence,34(1-3):197–215,2002.[3]P.BaroniandM.Giacomin.Onprinciple-basedevaluationofextension-basedargumentationsemantics.ArticialIntelligence,171(1015):675–700,2007.[4]R.BaumanandG.Brewka.ExpandingArgumentationFrameworks:EnforcingandMonotonicityRe-sults.InProc.2010ConferenceonComputationalModelsofArgument(COMMA2010),75–86,2010.[5]P.Baroni,P.E.Dunne,andM.Giacomin.Ontheresolution-basedfamilyofabstractargumentationsemanticsanditsgroundedinstance.ArticialIntelligence,175(3-4):791–813,2011.[6]T.J.M.Bench-Capon.Persuasioninpracticalargumentusingvalue-basedargumentationframeworks.JournalofLogicandComputation,13(3):429–448,2003.[7]A.Bondarenko,P.M.Dung,R.A.Kowalski,andF.Toni.Anabstract,argumentation-theoreticapproachtodefaultreasoning.ArticialIntelligence,93:63–101,1997.[8]G.Brewka.Preferredsubtheories:anextendedlogicalframeworkfordefaultreasoning.InProc.11thInternationalJointConferenceonArticialintelligence,pages1043–1048,1989.[9]M.CaminadaandL.Amgoud.Ontheevaluationofargumentationformalisms.ArticialIntelligence,171(5-6):286–310,2007.[10]C.Cayrol,F.DupindeSaint-CyrandM-C.Lagasquie-SchiexRevisionofanArgumentationSystem.InProc.KnowledgeRepresentation(KR'08),124-134,2008.[11]P.M.Dung.Ontheacceptabilityofargumentsanditsfundamentalroleinnonmonotonicreasoning,logicprogrammingandn-persongames.ArticialIntelligence,77(2):321–358,1995.[12]T.F.Gordon,H.Prakken,andD.N.Walton.TheCarneadesmodelofargumentandburdenofproof.ArticialIntelligence,171:875–896,2007.[13]N.GorogiannisandA.Hunter.Instantiatingabstractargumentationwithclassicallogicarguments:Postulatesandproperties.ArticialIntelligence,175:1479–1497,2011.[14]S.Modgil.Hierarchicalargumentation.InProc.10thEuropeanConferenceonLogicsinArticialIntelligence(JELIA),pages319–332,2006.[15]S.Modgil.Reasoningaboutpreferencesinargumentationframeworks.ArticialIntelligence,173(9-10):901–934,2009.[16]S.ModgilandH.Prakken.Revisitingpreferencesandargumentation.InIJCAI2011,2011.[17]S.ModgilandH.Prakken.Ageneralaccountofargumentationandpreferences.Technicalreport,www.dcs.kcl.ac.uk/staff/smodgil/GAP.pdf,2012.[18]H.Prakken.Anabstractframeworkforargumentationwithstructuredarguments.ArgumentandCom-putation,1(2):93–124,2010.[19]H.Prakken.Somereectionsontwocurrenttrendsinformalargumentation.InA.Artikis,et.al(eds.)LogicPrograms,NormsandAction.EssaysinHonourofMarekJ.SergotontheOccasionofhis60thBirthday,pp.249–272.Springer,2012.[20]B.vanGijzelandH.Prakken.RelatingCarneadeswithabstractargumentation.InProceedingsofthe22ndInternationalJointConferenceonArticialIntelligence(IJCAI-11),pages1113–1119,2011.