ALattice-basedApproachtoComputingWarrantedBeliefsinSkepticalArgumentat - PDF document

ALattice-basedApproachtoComputingWarrantedBeliefsinSkepticalArgumentationFrameworksCarlosIvanChesnevarGuillermoRicardoSimariDept.ofComputerScienceandEng.…UniversidadNacionaldelSur…8000BahšaBlanca,RGENTINATel.+54-291-459-5135…Fax+54-291-459-5136…Email:Thispaperintroducesanovelapproachtomodelwarrantcomputationinaskepticalabstractargu-mentationframework.Weshowthatsuchsearchspacecanbede“nedasalattice,andillustratehowtheso-calleddialecticalconstraintscanplayaroleforguidingtheef“cientcomputationofwarrantedarguments.1IntroductionandMotivations Thispaperisbasedonapreliminaryworkshoppaperfromthesameauthors(AnAbstractModelForComputingWarrantinSkep-ticalArgumentationFrameworksŽ,Proc.ofthe11thNMR2006Workshop,Windermere,UK,June2006). Seeanin-depthdiscussionininPrakken,2005 Figure1:(a)ExhaustivedialecticaltreeforEx.6;(b)resultingtreeafterapplyingand-ormarking(Def.14);(c)…(d)twootherex-haustivedialecticaltreesbelongingtotheequivalenceclass Givenanargument,thereisaone-to-onecorrespon-dencebetweenabundlesetofargumentationlinesrootedandthecorrespondingequivalenceclassofdialecticaltreesthatsharethesamebundlesetastheirorigin(asspeci-“edinDef.8).Eachmemberofanequivalenceclassrepre-sentsadifferentwayinwhichatreecouldbebuilt.Eachpar-ticularcomputationalmethodusedtogeneratethedialecticaltreefromthebundlesetwillproduceaparticularmemberoftheequivalenceclass.De“nition10beanargumentativetheory,andletbeabundlesetofargumentationlinesrootedinanargument.Wede“nethemappingLines\{\n} Tree ,where TreeisthequotientsetofTree,and theequivalenceclassofProposition1Foranyargumentinanargumentativetheorythemappingisabijection.Asthemappingisabijection,wecanalsode“netheinversemapping.Inwhatfollows,wewilluseindistinctlyasetnotation(abundlesetofargumentationlinesrootedinanargument)oratreenotation(adialecticaltreerootedin),astheformermappingsallowustogofromanyofthesenotationstotheother.Proposition2beatheory,andadialecticaltreeinThenitholdsthatanysubtree,rootedin,isalsoadialecticaltreewrt2.3AcceptabledialecticaltreesDe“nition11beatheory.Adialecticaltreeiffeveryargumentationlineintheassociatedbundleset isacceptable.WewilldistinguishthesubsetATree(resp.ATree)ofallacceptabledialecticaltreesinTree(resp.TreeAsacceptabledialecticaltreesareasubclassofdialecticaltrees,allthepropertiespreviouslyshownapplyalsotothem.Inthesequel,wewilljustwritedialecticaltreesŽtorefertoacceptabledialecticaltrees,unlessstatedotherwise. Proofsnotincludedforspacereasons.De“nition12Adialecticaltreewillbecalledexhaustiveiffitisconstructedfromthesetofallpossibleexhaustiveargumen-tationlinesrootedin,otherwisewillbecalledTheexhaustivedialecticaltreeforanyargumentcanbeproventobeunique.Proposition3beatheory,andletbeanargumentinThenthereexistsauniqueexhaustivedialecticaltree(uptoanequivalencewrtasgiveninDef.9)Acceptabledialecticaltreesallowtodeterminewhethertherootnodeofthetreeistobeaccepted(ultimatelyorrejected(ultimately).Amarkingfunctionvidesade“nitionofsuchacceptancecriterion.Formally:De“nition13beatheory.AmarkingcriterionforisaMarkTree.WewillwriteMark(resp.Mark)todenotethattherootnodeofismarked-node(resp.Severalmarkingcriteriacanbede“nedforcapturingskep-ticalsemanticsforargumentation.Aparticularcriterion(whichwewilllateruseinouranalysisforstrategiesforcom-putingwarrant)istheand-ormarkingofadialecticaltree,whichcorrespondstoDungsgroundedsemanticssemanticsDung,1995].De“nition14beatheory,andletbeadialecticaltree.Theand-ormarkingofisde“nedasfollows:1)Ifhasnosubtrees,thenMark2)Ifhassubtrees,...,thena)MarkiffMark,forall.b)MarkiffsuchthatMark,forsomeProposition4beatheory,andletbeadialecticaltree.Theand-ormarkingde“nedinDef.14assignsthesamemarktoallthemembersof De“nition15beanargumentativetheoryandMarkmarkingcriterionfor.Anargumentisawarrantedargument(orjustawarrant)inifftheexhaustivedialecticaltreeissuchMarkExample7ConsidertheexhaustivedialecticaltreeinEx.6showninFig.1(a).Fig.1(b)showsthecorrespondingmarkingbyapplyingDef.14,showingthat…therootof…isanultimatelydefeatedargument,i.e.,Mark.Henceisnotawarrantedargument.Fig.1(c)…(d)showstwomarkeddialecticaltreesbelongingtothesameequivalenceclass 3WarrantComputationviaDialecticalTreesOurmainconcernistomodelwarrantcomputationinskepti-calargumentationframeworks,andinsuchacasetreestruc-tureslendthemselvesnaturallytoimplementation.Infact,someimplementationsofskepticalargumentationsystemse.g.,.,Garc´šaandSimari,2004)relyontreestructures(suchasdialecticaltrees)whichcanbecomputedbyperform-ingbackwardchainingattwolevels.Ontheonehand,ar-gumentsarecomputedbybackwardchainingfromaquery(goal)usingalogicprogrammingapproach(e.g.,SLDreso-lution).Ontheotherhand,dialecticaltreescanbecomputedbyrecursivelyanalyzingdefeatersforagivenargument,de-featersforthosedefeaters,andsoon.Inparticular,inmorecomplexandgeneralsettings(suchasadmissibilityseman-tics)dialecticalproofprocedureshavebeendeveloped De“nition20=(beanargumentationtheory,andbeacceptabledialecticaltrees.Wewillsaythatthereismovefrom,denotedasMove,iffItmustberemarkedthataproperconceptualizationofmoveinargumentationdemandsmoreparameters,suchasidentify-ingtheargumentationlineinwhichaargumentisintroduced,whoistheplayer(ProponentorOpponent)makingthemove,etc.SuchanapproachhasbeenformalizedbybyPrakken,2001;2005.Ourapproachinthiscaseisintentionallyover-simpli“ed,asitjustaimstorelatethenotionofrelevanceandthenotionofminimallysettleddialecticaltrees.Infact,notethatDef.20allowsustoformalizethecomputationofanacceptabledialecticaltreerootedinasasequenceofmovesMoveMove,...,Moveisadialecticaltreewithasinglenode.Infact,Prakkensnotionofrelevantmovecanbestatedinourset-tingasfollows:amoveMoverelevantiffMarkMarkThefollowingpropositionshowsthatminimallysettledtreesareonlythoseobtainedbyperformingasequenceofrelevantmovesendinginasettleddialecticaltree.Proposition6beanargumentationtheory,andletbeadialecticaltree.Thenisminimallysettlediffthereisasequenceofmoves,...,suchthateverymoveisrelevant,andresultsinasettleddialecticaltree.5RelatedWorkDialecticalconstraintshavemotivatedresearchinargumenta-tiontheoryindifferentdirections.Inourproposaldialecticalconstraintsareleftasaparticularparametertobeincludedintheargumentationtheory.Differentargument-basedproofprocedureshaveincludedparticulardialecticalconstraintsaspartoftheirspeci“cation.InInBesnardandHunter,2001theauthorspresentalogicofargumentationwhichdisal-lowsrepetitionofargumentsinargumenttreestreesBesnardandHunter,2001,p.215Otherapproachesforcomputingwell-foundedsemanticsviatrees(e.g.,.,KakasandToni,1999defensenodes(whichaccountforProponentsargumentinanargumentationline)cannotattackanyotherdefensenodeinthetree.Similarly,ininDungetal.,2006,forcomputingassumption-basedadmissiblesemanticsthereisafurtherre-quirementintheproofprocedurethattheproponentdoesnotattackitselfŽ.Suchkindofrestrictionscanbeseenaspartic-ulardialecticalconstraintsinthecontextofourproposal.Recentlytherehavebeenotherresearchorientedtowardsformalizingdialecticalproofproceduresforargumentation.Tothebestofourknowledge,noneofsuchworksformalizesthedialecticalsearchspacethroughalatticeaspresentedinthispaper.Ourworkcomplementspreviousresearchcon-cerningthedynamicsofargumentation,notablynotablyPrakken,en,Brewka,2001.AlthoughPrakkendevelopsaverycomprehensivegeneralframework,someimportantcomputationalissues(e.g.,searchspaceconsiderations)arenottakenintoaccount.6Conclusions.FutureWorkInthispaperwehavepresentedanovelapproachtomodelthesearchspaceassociatedwithwarrantcomputationinanabstractargumentationframework.Wehaveshownhowthenotionofdialecticaltreecanbeusedconstructivelytomodeldifferentstagesintheprocessofcomputingwarrantedargu-ments.Wehavealsoshownhowtheprocessofcomputingwarrantcanberecastintocomputingdialecticaltreeswithinalattice,illustratinghowdialecticalconstraintscanplayaroleforguidinganef“cientcomputationofwarrantedliter-als.Partofourfutureworkisrelatedtostudyingtheoreticalpropertiesoftheproposedframework,analyzingtheirinci-dencefordevelopingef“cientargument-basedinferenceen-gines.Researchinthisdirectioniscurrentlybeingpursued.AcknowledgementsWethankanonymousreviewersforcommentswhichhelpedtoim-provethe“nalversionofthispaper.Thisresearchwaspartiallysup-portedbytheDept.ofComputerScienceandEng.,SeCyT,U.N.Sur,CONICET,and(PICT13096,15043,PAV076),Argentina.PartialresearchsupportwasalsoobtainedfromProjectTIN2004-07933-C03-01/03andRamonyCajalProgram(MCyT,Spain).ReferencesencesBesnardandHunter,2001P.BesnardandA.Hunter.Alogic-basedtheoryofdeductivearguments.Arti“cialIntelligence1:2(128):203…235,2001.2001.Brewka,2001G.Brewka.Dynamicargumentsystems:Aformalmodelofargumentationprocessesbasedonsituationcalculus.ofLogicandComputation,11(2):257…282,2001.2001.Chesnevaretal.,2005C.Chesnevar,G.Simari,andL.Godo.Computingdialecticaltreesef“cientlyinpossibilisticdefeasiblelogicprogramming.LNAI/LNCSSeries(Proc.8thIntl.LPNMRConf.),pages158…171,2005.2005.Dungetal.,2006P.Dung,R.Kowalski,andF.Toni.Dialecticproofproceduresforassumption-based,admissibleargumenta-Artif.Intelligence,170(2):114…159,2006.2006.Dung,1995P.Dung.Ontheacceptabilityofargumentsanditsfundamentalroleinnonmonotonicreasoning,logicprogrammingandn-persongames.Artif.Intelligence,77(2):321…358,1995.1995.Garc´šaandSimari,2004A.GarcšaandG.Simari.DefeasibleLogicProgramming:AnArgumentativeApproach.TheoryandPracticeofLogicProg.,4(1):95…138,2004.2004.Hamblin,1970C.L.Hamblin.Fallacies.Methuen,London,London,Hunter,2004A.Hunter.TowardsHigherImpactArgumenta-tion.InProc.19thAmericanNationalConf.onArt.Intelligence,pages275…280.MITPress,2004.2004.KakasandToni,1999A.KakasandF.Toni.Computingargu-mentationinlogicprogramming.JournalofLogicProgramming9(4):515…562,1999.1999.PrakkenandVreeswijk,2002H.PrakkenandG.Vreeswijk.Log-icalSystemsforDefeasibleArgumentation.InD.GabbayandF.Guenther,editors,HandbookofPhilosophicalLogic,pages219…318.KluwerAcademicPublishers,2002.2002.Prakken,2001H.Prakken.Relatingprotocolsfordynamicdisputewithlog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