Methods for Solving Reasoning Problems in Abstract Argumentation – A Survey

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Artificial Intelligence

www.elsevier.com/locate/artint

Methods for solving reasoning problems in abstract argumentation – A survey

Günther Charwat

^a

, Wolfgang Dvoˇrák

^b

, Sarah A. Gaggl

^c

, Johannes P. Wallner

^a

, Stefan Woltran

^a,∗

aViennaUniversityofTechnology,InstituteofInformationSystems,Austria bUniversityofVienna,FacultyofComputerScience,Austria

cTechnischeUniversitätDresden,InstituteofArtificialIntelligence,Germany

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received29March2013

Receivedinrevisedform19November2014 Accepted28November2014

Availableonline15December2014 Keywords:

Abstractargumentation Algorithms

Argumentationsystems

Withinthelastdecade,abstractargumentationhasemergedasacentralfieldinArtificial Intelligence.Besidesprovidingacoreformalismformanyadvancedargumentationsystems, abstractargumentationhasalsoservedtocaptureseveralnon-monotoniclogicsandother AIrelatedprinciples.Althoughthe ideaofabstractargumentationis appealinglysimple, severalreasoningproblemsinthisformalismexhibithighcomputationalcomplexity.This callsforadvancedtechniqueswhenitcomestoimplementationissues,achallengewhich hasbeen recently faced from different angles. In this survey, wegive an overview on differentmethodsforsolvingreasoningproblemsinabstractargumentationandcompare their particular features. Moreover, we highlight available state-of-the-art systems for abstractargumentation,whichputthesemethodstopractice.

^{2015TheAuthors.PublishedbyElsevierB.V.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Argumentationisahighlyinterdisciplinaryfieldwithlinkstopsychology,linguistics,philosophy,legaltheory,andformal logic. Sincetheadventofthecomputer age,formalmodels ofargumenthavebeenmaterializedindifferentsystemsthat implement—oratleastsupport—creation,evaluation,andjudgment ofarguments.However,untilDung’sseminalpaper on abstractargumentation[1],the heterogeneityof theseapproacheswas severelyhampering a strongandjointdevelop- ment of afield like “computational argumentation”.In fact, Dung’sidea ofevaluating argumentson an abstract levelby takingonlytheirinter-relationshipsintoaccount,not onlyhasbeenshowntounderliemanyoftheearlierapproachesfor argumentation,butalsouniformlycapturesseveralnon-monotoniclogics.Yetthissecond contributionlocatedArgumenta- tion asasub-discipline ofArtificial Intelligence[2].Theincreasing significanceofargumentation asa research areaofits ownhasalsobeenwitnessedbythebiennialCOMMAConferenceonComputationalModelsofArgument,¹ whichfromthe secondmeetingonwardsprovidessessionsforsoftwaredemonstrationsofimplementedsystems,theIJCAIWorkshopSeries onTheoryandApplicationsofFormalArgumentation(TAFA),² the2010establishedJournalofArgumentandComputation,³ ortheTextbookonArgumentationinArtificialIntelligence[3].

*

Correspondingauthor.

E-mailaddresses:gcharwat@dbai.tuwien.ac.at(G. Charwat),wolfgang.dvorak@univie.ac.at(W. Dvoˇrák),sarah.gaggl@tu-dresden.de(S.A. Gaggl), wallner@dbai.tuwien.ac.at(J.P. Wallner),woltran@dbai.tuwien.ac.at(S. Woltran).

1 http://www.comma-conf.org/.

2 http://homepages.abdn.ac.uk/n.oren/pages/TAFA-13/.

3 http://www.tandfonline.com/toc/tarc20/current.

http://dx.doi.org/10.1016/j.artint.2014.11.008

0004-3702/^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

One particularfeature ofabstract argumentationframeworks istheir simple structure.In fact, abstractargumentation frameworksarejustdirectedgraphswhereverticesplaytheroleofargumentsandedgesindicateacertainconflictbetween thetwoconnectedarguments.Theseargumentationframeworksareusuallyderivedduringaninstantiationprocess(see,e.g., [4,5]), where structured argumentsare investigated withrespectto their ability to contradictother such arguments; the actualnotionof“contradicting”canbeinstantiatedinmanydifferentforms(see,e.g.,[6]).Havinggeneratedtheframework insuchaway,theprocessof“conflict-resolution”,i.e.,thesearchforjointlyacceptablesetsofarguments,isthendelegated tosemanticswhichoperateontheabstractlevel.Thus,semanticsforargumentationframeworkshavealsobeenreferredto ascalculiofopposition[7].

One directionofresearch inabstract argumentationwas devoted to develop the “right”forms ofsemantics (see, e.g., [8–10] where propertiesfor argumentation semantics are proposed andevaluated). Thishas lead to what G. Simari has called a “plethoraofargumentationsemantics”.⁴ Today there seems to be agreement within the communitythat different semanticssuitdifferentapplications,hencemanyofthemareinuseforavarietyofapplicationdomains.⁵ Itisclearthat thissituationimpliesthatsuccessfulsystemsforabstractargumentationareexpectedtooffernotonlyasinglesemantics.

Thecentral roleofabstractargumentationframeworksalsoboostedresearchonefficientprocedures forthisparticular formalism.However,itwassoonrecognizedthatalreadythesesimpleframeworksshowhighcomplexity(see,e.g.,[12–14]);

duetothelinktonon-monotoniclogicandtologicprogramminginparticular,thiscamewithoutahugesurprise.Together withthefactthatmanydifferentsemanticsexist,generalimplementationmethodsforabstractargumentationthusrequire

• ^{acertainlevelof}generality,suchthatnotonlyasinglesemanticscanbetreated;and

• ^{asufficientlevelofefficiencytofacethehighinherentcomplexityoftheproblemsathand.}

Scopeofthesurvey Inthisarticle,wepresentaselectionofevaluationmethodsforabstractargumentationwhichwebelieve tomeettheserequirements.Wegroupthesemethodsintotwocategories:thereductionapproachandthedirectapproach.

Theunderlyingideaofthereductionapproachistoexploitexistingefficientsoftwarewhichhasoriginallybeendeveloped for other purposes. To this end, one has to formalize the reasoning problems within other formalisms like constraint- satisfaction problems (CSP) [15], propositional logic [16] or answer-set programming (ASP) [17]. In this approach, the resulting argumentation systems directly benefit from the high level of sophistication today’s systems for SAT (satisfia- bilityinpropositional logic)orASPhavereached. Thereduction approachwillbe presentedindetail inSection3 ofthis article.Hereby,wewillfirstfocuson

• ^SAT-basedargumentationsystems.ThisdirectionhasbeenadvocatedbyBesnardandDoutre[18],andlaterextendedby meansofquantifiedpropositionallogic[19,20].Wewillfirstdiscussthetheoreticalunderpinningsofthisapproachand thencontinuewithanintroductiontotheCEGARTIXsystem[21]andtheArgSemSATsystem[22],whichbothrelyon iterativecallstoSATsolversforargumentationsemanticsofhighcomplexity(i.e.,beinglocatedonthesecond levelof thepolynomialhierarchy).

• ^{CSP-based approach. Reductions to CSP have been addressed by Amgoud and Devred [23] and Bistarelli and San-} tini[24–28];the latterwork led tothe development ofthe ConArg system. Weintroduce the formalizationof argu- mentation frameworksintermsofCSPs, wherethe argumentsofthegivenframework representthevariables ofthe CSPwithdomainsof0and1.Theconstraintsthendependonthespecificsemantics.

• ^{ASP-based approach.The useofthis}logic-programmingparadigm tosolve abstractargumentationproblemshasbeen initiatedbyseveralauthors(thesurveyarticlebyToniandSergot[29]providesagoodoverview).Wefocushereonthe ASPARTIXapproach[30]whichincontrasttotheaforementionedSATmethodsreliesonaquery-basedimplementation wheretheargumentationframeworktobeevaluatedisprovidedasaninputdatabase(fromthispointofview,theSAT orCSPmethodscanbeseenasacompiler-likeapproachtoabstractargumentation,whiletheASPmethodactslikean interpreter).A largecollectionofsuchASPqueriesisprovidedbytheASPARTIXsystem.Wewilldiscussstandardways ofASPencodings,butalsosomerecentmethodswhichexploitadvancedASPtechniques[31].

Intheremainder ofSection3weshallpresenttheconceptsbehindotherreduction-basedapproaches,forinstance,the equationalapproachasintroducedbyGabbayin[32]andthereductionstomonadicsecondorderlogicasproposedin[33].

In Section 4, we collect methods andalgorithms which havebeen developed from scratch (instead of using another formalismlikeSATorASP).Theobviousdisadvantageofthisdirectapproachisduetothefactthatexistingtechnologycannot bedirectlyemployed.Ontheotherhand,suchargumentation-tailoredalgorithms easetheincorporationofshort-cutsthat arespecifictotheargumentationdomain.Indetail,wewilldiscussthefollowingideas:

• ^{The labelingapproach[34–38] givesa more}fine-grainedhandleon thestatusof argumentswhen evaluatedw.r.t. se- manticsandalsoprovidesasolidbasisfordedicatedalgorithms.Wepresenttwodifferentproposalsforimplementing

4 Duringthepresentationof[11]atCOMMA2006.

5 However,inconnectionwithparticularinstantiationschemes,itisoftenclaimedthatonlysemanticsthatfollowtheprincipleofadmissibility(argu- mentsshallonlybejointlyacceptedifeachoftheselectedargumentsisdefendedbytheselectedset;wewillmaketheconceptsmoreclearinSection2) shouldbeconsidered(see,e.g.,[5]).

Fig. 1.Example argumentation framework.

the enumerationofpreferredextensions,one along thelines of[35] andanotherfollowing [34] usingimprovements from [36]. Furthermore, we discussan algorithm dedicatedto credulous reasoning with preferredsemantics follow- ing thework of[38].Labeling-basedalgorithms havebeenmaterializedinthe ArguLabsystemaswellasinVerheij’s CompArgsystem.

•CharacterizationsviaDialogueGames.Heretheacceptancestatusofanargumentisgivenintermsofwinningstrategies incertain gamesontheargumentationframework. Typicallysuchgames aretwo-player gameswhereoneplayer, the proponent,arguesinfavoroftheargumentinquestionandasecondplayer,theopponent,arguesagainstit.Suchgames canbeusedtodesignalgorithms[35,39],whichareemployedinsystemslikeDungineandDung-O-Matic.

•^{Finally,wewilltake alookondynamic}programmingapproaches[40] whichoperateondecompositionsofframeworks.

Notably, therunning timesinthisapproach arenot mainlydependent on thesize ofthegivenframework, butona structuralparameter.We focusontheparametertree-widthandtheconceptoftreedecomposition.Thismethodwas firstadvocatedbyDunne[41]andlaterrealizedinthedynPARTIXsystem[42].

As already hinted above, many of the methods we presenthave found their way into an available software system.

Therefore, we will not only explain these methods in this survey, but shall also give the interested readerpointers to concretesystemswhichcanbeusedtoexperiment.Section5containsacomparisonofthesystemsw.r.t.theirfeatures(e.g.

supported semanticsandreasoningproblems)andtheunderlyingconcepts.Someofthesystemshavebeenevaluatedand comparedw.r.t.theirperformance(seee.g.,[31,36,43–45]),butnoexhaustiveperformancecomparisonshavebeendoneso far.Infact,anorganizedcompetitioncomparabletotheonesfromtheareasofSAT[46]orASP[47]isplannedtotakeplace in2015forthefirsttime[48].⁶ Thus,weabstainherefromasystematiccomparisonofthesystems’performance.

To summarize, ourgoal isto introduce aselection of methodsfor evaluating abstractargumentation frameworks;we shallexplainthekeyconceptsindetailforselectedsemanticsandgivepointerstotheliteraturefortheremainingsemantics or when it comes to more subtle aspects like optimization. Concerningabstract argumentation itself, we give a concise introduction inSection 2. Forreadersnot familiar withabstractargumentation, wehighly recommendthe recentsurvey articlebyBaronietal.[49].

SincethefocusofthisarticleisontheevaluationofsemanticsforDung’sabstractargumentationframework,advanced systems including instantiation (e.g., ASPIC [50] and Carneades [51]), assumption-based argumentation [52], or systems based on defeasible logic [53] are out ofthe scope ofthis article.⁷ Likewise, we will not consider the vastcollection of extensionstoDung’sframeworks,⁸ likevalue-based[56],bipolar[57],extended[58],constrained[59],temporal[60],prac- tical [61], andfibringargumentation frameworks [62], aswell asargumentation frameworks withrecursive attacks [63], argumentationcontextsystems[64],andabstractdialecticalframeworks[65].Wealsoexcludeabstractargumentationwith uncertainty orweightshere; recentarticlesby Hunter[66] andrespectively Dunne etal.[67] introduce thesevariantsin detail.

2. Background

In thissectionwe introduce (abstract)argumentationframeworks[1]andrecallthe semanticswe studyinthispaper (seealso[10,49,68]).

Definition1. An argumentationframework(AF)is a pair F=(A,R) where A is a setof argumentsand R⊆^A×^{A is the} attackrelation.Thepair(a,b)∈^{Rmeansthataattacksb.Wesaythatanargumenta}∈^{A isdefended(in F)byaset S}⊆^A if,foreachb∈^{A suchthat}(b,a)∈^{R,thereexistsac}∈^{S suchthat}(c,b)∈^R.

Anargumentationframeworkcanberepresentedasadirectedgraph.

Example1.LetF=(A,R)beanAFwith A= {^a,b,c,d,e}^andR= {(a,b),(b,c),(c,b),(d,c),(d,e),(e,e)}^.Thecorrespond- inggraphrepresentationisdepictedinFig. 1.

Asemanticsforargumentationframeworksisdefinedasafunction

σ

whichassignstoeachAFF=(A,R)aset

σ

(F)⊆ 2^A ofextensions.

6 Seehttp://argumentationcompetition.orgforfurtherinformation.

7 Anoverviewontheseapproachesisgivenin[54].

8 Abriefoverviewonsuchapproachesisgivenin[55].

Fig. 2. Relationsbetween argumentationsemantics: An arrowfrom asemantics σ toanother semantics τ denotes that eachσ-extensionis also a τ-extension.

We considerfor

σ

the functionsnaive,stb,adm, com,grd,prf, semand stg which standfor naive,stable, admissible, complete,grounded,preferred,semi-stableandstageextensions,respectively.Towardsthedefinitionofthesesemanticswe introduceafewmoreformalconcepts.

Definition 2. Given an AF F =(A,R), the characteristicfunction F^F :^2A→^{2A of F is defined as} F^F(S)= {^x∈ ^A | xis defended byS}^{.Foraset S}⊆^{Aandanargumenta}∈^{A,wewrite S}!^Ra(resp.a!^{R S)incasethereisanargument} b∈^{S,such that} (b,a)∈^{R (resp.}(a,b)∈^R). Furthermore,we write S&!^{Ra (resp.a}&!^{R S) in casethereis no argument} b∈^S,suchthat(b,a)∈^{R (resp.}(a,b)∈^R).

Moreover,fora set S⊆^{A,we denotetheset ofargumentsattackedby (resp.attacking) S asS⊕}R = {^x|^S!^Rx} ^(resp.

S⁽_R = {^x|^x!^RS}^{),anddefinetherangeof S asS+}_R =^S∪^S⊕_R ^{andthenegativerangeof S asS−}_R =^S∪^S(_R^.

Thenextdefinitionformallydefinesthesemanticswewillfocusoninthissurvey.Allofthemarebasedonconflict-free sets,i.e.itisnotallowedtojointlyacceptargumentswhichareadjacentintheframework.Differentadditionalcriteriaare then usedforthe concretedefinition: naive extensionsarejustmaximal (with respectto set-inclusion) conflict-freesets, stableextensionshavetoattackallotherarguments,admissiblesetsdefendthemselvesfromattackers,completeextensions in addition have to contain all defended arguments. The grounded extension is given by the subset-minimal complete extension. Preferred extensions are subset-maximal admissible sets (equivalently: subset-maximal complete extensions);

finally,semi-stableandstageextensionsarecharacterizedbymaximizingtheconceptofrange.

Definition3.Let F=(A,R)beanAF.Aset S⊆^{A is}conflict-free(in F),iftherearenoa,b∈^{S,such that}(a,b)∈^R.cf(F) denotesthecollectionofconflict-freesetsofF.Foraconflict-freesetS∈^cf(F),itholdsthat

• ^S∈^naive(F),ifthereisnoT∈^cf(F)withT⊃^S;

• ^S∈^stb(F),ifS⁺_R=^A;

• ^S∈^adm(F),ifS⊆FF(S);

• ^S∈^com(F),ifS=FF(S);

• ^S∈^grd(F),ifS∈^com(F)andthereisnoT∈^com(F)withT⊂^S;

• ^S∈^prf(F),ifS∈^adm(F)andthereisnoT∈^adm(F)withS⊂^T;

• ^S∈^sem(F),if S∈^adm(F)andthereisnoT∈^adm(F)withS⁺_R⊂^T+_R;

• ^S∈^stg(F),ifthereisnoT∈^cf(F),withS⁺_R ⊂^TR⁺.

Werecallthat foreachAF F,thegroundedsemanticsyields auniqueextension, thegroundedextension,whichisthe least fixed-point of the characteristic function FF.Furthermore, Fig. 2 showsthe relations betweenthe aforementioned semantics. Thefigure is completein thesense that ifthere isno arrowfrom semantics

σ

to semantics

τ

,then thereis someAFF suchthat

σ

(F)!

τ

(F).Forallsemantics

σ

weintroducedhere,exceptstablesemantics,itholdsthatforanyAF F wehave

σ

(F)&=∅^.

Example2.ConsidertheAFfromExample 1.Then:cf(F)= {∅,{^a},{^b},{^c},{^d},{^a,c},{^a,d},{^b,d}}^;naive(F)= {{^a,c},{^a,d}, {^b,d}}^;adm(F)= {∅,{^a},{^d},{^a,d}}^;andstb(F)=^com(F)=^grd(F)=^prf(F)=^sem(F)=^stg(F)= {{^a,d}}^.

Labeling-basedsemantics So far we haveconsidered so-calledextension-based semantics. However, there are severalap- proachesdefiningargumentationsemanticsviacertainkindsoflabelingsinsteadofextensions.Asanexampleweconsider thepopularapproachbyCaminadaandGabbay[69]andinparticulartheircompletelabelings.Basically,such alabelingis athree-valuedfunctionthat assignsoneofthelabelsin,outandundectoeachargument,withtheintuitionbehind these labelsbeingthefollowing.Anargumentislabeledwith:inifitisaccepted,i.e.,itisdefendedbytheinlabeledarguments;

outiftherearestrongreasonstorejectit,i.e.,itisattackedbyanacceptedargument;undeciftheargumentisundecided, i.e.,neitheracceptednorattackedbyacceptedarguments.WedenotelabelingfunctionsL^{alsobytriples}(Lin,Lout,Lundec),

where L^{in is the set of arguments labeled by in,} L^{out is the set of arguments labeled by out and} Lundec is the set of argumentslabeledbyundec.

Asanexample,wegivethedefinitionofcompletelabelingsfrom[69].

Definition4.GivenanAF F=(A,R),a functionL:^A→{ⁱⁿ,out,undec} ^{isacompletelabeling iffthe followingconditions} hold:

• L(a)=^{iniffforeachb with}(b,a)∈^R,L(b)=^out.

• L(a)=^{outiffthereexistsbwith}(b,a)∈^R,L(b)=^in.

There is a one-to-one mapping betweencomplete extensionsand completelabelings, such that the set ofarguments labeled within corresponds tothecomplete extensionandthearguments labeledwithout correspondtothe arguments attacked by the complete extension. Having complete labelings athand we can also characterize preferred labelings as follows:

Definition5.Givenan AF F=(A,R).Thepreferredlabelingsare thosecompletelabelingswhereLin is⊆^{-maximalamong} allcompletelabelings.

Right by thedefinitions,we havethesameone-to-one mappingbetweenpreferredextensionsandpreferredlabelings asforcompletesemantics.Onecandefinelabeling-basedversionsforallofoursemantics(see[69]),butthisisoutofthe scopeofthissurvey.

Reasoninginargumentationframeworks WerecallthemostimportantreasoningproblemsforAFs:Givenanargumentation framework F anda semantics

σ

, Enumσ(F) resultsin an enumeration ofall extensions. Asimpler notionis Countσ(F), whichonlycountsthenumberofextensions.Query-basedproblemsareCredσ(a,F)andSkeptσ(a,F)fordecidingcredulous (respectivelyskeptical)acceptanceofanargumenta.Theformerreturnsyesifaiscontainedinatleastoneextensionunder

σ

,whileforthelattertoreturnyes,amustbecontainedinallextensionsunder

σ

.Finally,we alsoconsidertheproblem Verσ(S,F)ofverifyinga givenextension, i.e.,testingwhethera givenset S isa

σ

-extension of F.Thisproblemtypically occursasasubroutineofareasoningprocedure.

Definition6.GivenanAF F=(A,R),asemantics

σ

andanargumenta∈^A,then

• ^Enumσ(F)=

σ

(F)

• ^Countσ(F)= |

σ

(F)|

• ^Credσ(a,F)=!_{yes ifa}∈"

S∈σ(F)S no otherwise

• ^Skeptσ(a,F)=!_{yes ifa}∈#

S∈σ(F)S no otherwise

• ^Verσ(S,F)=!_{yes ifS}∈

σ

(F) no otherwise

Example3. Considerthe AF F givenin Example 1.Fornaive semantics,the reasoningproblemsresultinEnumnaive(F)= {{^a,c},{^a,d},{^b,d}}^andCountnaive(F)=^3.Furthermore,forargumentaweobtain Cred_naive(a,F)=^{yes andSkept}naive(a,F)

=^{no. For preferred semantics, F has a single extension Enum}prf(F)= {{^a,d}}^{, Count}prf(F)=^{1, andthus credulous and} skepticalacceptancecoincide(e.g.,Credprf(a,F)=^Skeptprf(a,F)=^yes).

Next, let us turn to the complexity of reasoning in abstract argumentation frameworks. We assume the reader has knowledgeaboutstandardcomplexityclasses,i.e.,P,NP andL (logarithmicspace).Furthermore,webrieflyrecapitulatethe conceptoforaclemachinesandrelatedcomplexityclasses.LetC ^{denotesomecomplexityclass.Bya}C^{-oraclemachinewe} meana(polynomialtime)Turingmachinewhichcanaccessanoraclethatdecides agiven(sub)-probleminC ^withinone step. Wedenotesuch machinesasNP^C iftheunderlyingTuring machineisnon-deterministic. Theclass Σ₂^P =NP^NP thus denotesthesetofproblemswhichcanbedecidedbyanon-deterministicpolynomialtimealgorithmthathas(unrestricted) accesstoanNP-oracle.TheclassΠ₂^P=^{coNPNPisdefinedasthe}complementaryclassofΣ₂^P,i.e.,Π₂^P=^coΣ₂^P.Therelation betweenthecomplexityclassesisasfollows:

L

⊆

P

⊆

_coNP^NP

⊆ Σ

₂^P

Π

₂^P

The computational complexityof credulousandskepticalreasoninghasbeenstudied extensivelyinthe literature(see [70] fora starting point). Table 1summarizes thecomputational complexity classifications ofthe defineddecisionprob- lems[68,12,1,13,71,14,72,73],whereC^{-cdenotesthatthe}correspondingproblemiscompleteforclassC^.

Table 1

ComputationalcomplexityofreasoninginAFs.

σ Cred_σ Skept_σ Ver_σ

naive in L in L in L

stb NP-c coNP-c in L

adm NP-c trivial in L

com NP-c P-c in L

grd P-c P-c P-c

prf NP-c Π₂^P-c coNP-c

sem Σ₂^P-c Π₂^P-c coNP-c

stg Σ₂^P-c Π₂^P-c coNP-c

3. Reduction-basedapproaches

In thissection we will discussreduction-based approachesin abstractargumentation. Asimplied by thename,these methodsreduceortranslateareasoningproblemtoanother,typicallytoanotherformalism.Fromacomputationalpointof view,weassurethatthisreductionisefficientlycomputable,i.e.,achievableinpolynomialtime,andthattheanswerforthe originalprobleminstancecanbeimmediatelyobtainedfromtheanswertothenewprobleminstance.Suchmethodsoffer thegreatbenefitofexploitingexistingandhighlysophisticatedsolversforwell-knownandwell-studiedproblemdomains.

Naturally,reduction-basedmethodscanbedistinguishedbythetargetsystem.Manysuchapproacheshavebeenstudied for abstract argumentation ranging from propositional logic [19,18,21,20], constraint satisfaction problems (CSP) [27,23, 26,28] and answer-set programming(ASP) [30,74–76] to equational systems [32,77]. We will give an overview of these approachesandinparticularfocusonthefirstthreevery prominenttarget systems,thereductionstopropositionallogic, CSPandASP.

3.1. Propositional-logicbasedapproach

Propositionallogicistheprototypicaltargetsystemformanyapproachesbasedonreductions,astheBooleanSATprob- lemiswellstudiedandmoreoveraccompaniedwithmanymatureandefficientsolverssuchasMiniSat[78]andGRASP[79].

First,werecallthenecessarybackgroundofBooleanlogicandquantifiedBooleanformulae(QBF)sincetheyserveasour targetsystems.

Thebasis ofpropositionallogic isasetofpropositional variablesP^{,to whichwealsorefer toasatoms.}Propositional formulaeare builtasusual fromtheconnectives∧,∨,→ ^and¬^{,denoting thelogical}conjunction, disjunction,(material) implicationandnegationrespectively.Weusethetruthconstants0^{todenotetrueand}⊥^{forfalse.Inaddition,weconsider} quantifiedBooleanformulaewiththeuniversalquantifier∀andtheexistentialquantifier∃(bothoveratoms),thatis,given aformulaφ,then Q pφisaQBF,withQ ∈{∀,∃}^andp∈P^.Furthermore, Q{^p1,. . . ,p_n}φisashorthandfor Q p₁· · ·^{Q p}nφ. Theorderofvariablesinconsecutivequantifiersofthesametypedoesnotmatter.

A propositional variable p in a QBF φ is free if it does not occur within the scope of a quantifier Q p and bound otherwise.Ifφ containsnofreevariable,thenφissaidtobeclosedandotherwiseopen.Wewillwrite φ[^p/ψ] ^todenote theresultofuniformlysubstitutingeachfreeoccurrenceofpwithψ informulaφ.

AninterpretationI⊆P^{definesforeach}propositionalvariableatruthassignmentwherep∈^{Iindicatesthatpevaluates} totruewhile p∈/^{I indicatesthat pevaluates tofalse.This}generalizestoarbitraryformulaeinthestandardway:Givena formulaφandaninterpretation I,thenφ evaluatesto trueunder I (i.e., Isatisfies φ)ifoneofthefollowingholds(with p∈P^).

• φ=^pandp∈^I

• φ= ¬^pandp∈/^I

• φ=ψ1∧ψ2 andbothψ1 andψ2 evaluatetotrueunderI

• φ=ψ1∨ψ2 andoneofψ1andψ2evaluatestotrueunderI

• φ=ψ₁→ψ₂ andψ₁ evaluatestofalseorψ₂evaluatestotrueunderI

• φ=∃^pψ andoneofψ[^p/0]^andψ[^p/⊥]^{evaluatestotrueunderI}

• φ=∀^pψ andbothψ[^p/0]^andψ[^p/⊥]^{evaluatetotrueunderI.}

IfaninterpretationI satisfiesaformulaφ,denotedby I|4φ,wesaythat Iisamodelofφ.

The approaches in Section 3.1.1 and Section 3.1.2 share the basic idea of translating a given AF, a semantics and a reasoning problem to a propositional formula, thereby reducing the problemto Boolean logic. In generalthis works by eitherinspectingthemodelsoftheresultingformula,whichareincorrespondencetotheextensionsoftheAF,ordeciding whetheraformulaissatisfiableorunsatisfiable,tosolvequery-basedreasoning.Notethatwerestrictourselvesheretothe semanticswhichweconsidertobe sufficient forillustratingthemainconcepts. Ingeneral,theapproachescanbeapplied tomanyothersemantics.

3.1.1. Reductionstopropositionallogic

The firstreduction-based approach[18,20]we considerhereuses propositionallogic formulae(withoutquantifiers) to encode the problemoffindingadmissible sets.Givenan AF F=(A,R), foreachargumenta∈^{A a}propositional variable v_a isused.Then, S⊆^A isan extensionundersemantics

σ

iff{^va|^a∈^S}|4φ,withφbeingapropositional formulathat evaluates AF F under semantics

σ

(below we will present in detail how to translate AFs into formulae). Formally, the correspondencebetweensetsofextensionsandmodelsofapropositionalformulacanbedefinedasfollows.

Definition7.LetS⊆^{2A beacollectionofsetsofargumentsandlet}I⊆^{2P beacollectionof}interpretations.Wesaythat S ^andI^{correspondtoeachother,insymbols}S∼=I^,if

1. foreach S∈S^{,thereexistsan I}∈I^,suchthat{^a|^va∈^I,a∈^A}=^S;

2. foreach I∈I^{,thereexistsan S}∈S^,suchthat{^a|^va∈^I,a∈^A}=^S;and 3. |S|= |I|^.

GivenanAF F=(A,R)thefollowingformulacanbeusedtosolvetheenumerationproblemofadmissiblesemantics.

adm_A,R

= $

a∈^A

%%

v_a

→ $

(b,a)∈^R

¬

^vb

&

∧

%

v_a

→ $

(b,a)∈^R

% '

(c,b)∈^R v_c

&&&

(1) The models of adm_A,R now correspondto the admissible sets of F, i.e., Enum_adm(F)∼= {^M|^M|4^admA,R}. Taken into considerationthatbydefinitionasatisfiableformulahasinfinitelymanymodels(thusviolatingitemthreeinDefinition 7), itisnowpossibletorestrictthesetofmodelstothosecontainingonlyatomsoccurringintheformula.Thefirstconjunction in (1) ensures that the resulting set of arguments is conflict-free, that is, whenever we accept an argument a (i.e., v_a evaluatestotrueunderamodel),allitsattackerscannotbeselectedanyfurther.Thesecondconjunctexpressesthedefense ofargumentsbystatingthat,ifweaccepta,thenforeachattackerb,somedefenderc mustbeacceptedaswell.Notethat anemptyconjunctionistreatedas0^{,whereastheempty}disjunctionistreatedas⊥^.

Example4.Thepropositionalformulaforadmissiblesetsoftheframework F=(A,R)inExample 1isgivenby

adm_A,R

≡ (

v_a

→ 0 ) ∧

⁽²⁾

(

v_b

→ ( ¬

^va

∧ ¬

^vc

) )

∧

⁽³⁾

(

v_c

→ ( ¬

^vb

∧ ¬

^vd

) )

∧

⁽⁴⁾

(

v_d

→ 0) ∧

(5)

(

v_e

→ ( ¬

^vd

∧ ¬

^ve

) )

∧

⁽⁶⁾

(

v_a

→ 0 ) ∧

⁽⁷⁾

(

v_b

→ (

⊥ ∧ (

v_b

∨

^vd

) ))

∧

⁽⁸⁾

(

v_c

→ (

(

v_a

∨

^vc

) ∧ ⊥ ))

∧

⁽⁹⁾

(

v_d

→ 0) ∧

(10)

(

v_e

→ ( ⊥ ∧

^vd

) )

(11) Lines (2)to (6)encode the conflict-freeproperty, while lines(7) to(11) ensurethat arguments inan admissible set are defended. Note that for convenience, the conjuncts are arranged in a different order than in the definition of adm_A,R. Consider forinstance argumentb.Line (3)specifiesthat ifwe accept b we cannot accept a andc anymore (conflict-free property).Likewise,line(8)statesthatbcanonlybeacceptedincaseitisdefendedagainstitsattackers.Fortheattackerc eitherbitselfordmustbeaccepted.However,sinceattackeraisnotattackedbyanyotherargument,thereisnomodelof adm_A,R wherev_b evaluatestotrue.

Anotherinteresting translation tocapturesemantics ofAFs within propositionallogic isdone by Gabbay[80].Here, a correspondencebetweenAFsandpropositionallogicisshownviathePeirce–Quinedagger(“nor”)connective.

Furthermore,severalpapersdealwiththeconversetranslation,i.e.translatingaBooleanformulainCNFtoan AF.Simi- larasbefore,foreachatominaformulaacorrespondingargumentisconstructed.Acceptingsuchanargument,e.g.under stablesemantics,istheninterpretedassettingtheatomtotrue.Theresultisacorrespondencebetweenextensionsunder a specificsemanticsandsatisfyingassignmentsoftheformula. Usually,thesetranslationsincorporateauxiliary arguments, which areusedtosimulatethelogicalconnectives.In[12,13] and[81] suchmethodsare studiedandused toshowcom- plexityboundsorfortranslationsbetweenformalisms.

3.1.2. ReductionstoquantifiedBooleanformulae

ProblemsbeyondNP requireamoreexpressiveformalismthanBooleanlogic.Preferredsemantics,forexample,isdefined as subset-maximaladmissible (or complete) sets. Intuitively, we can compute a preferred extension by searching for an admissiblesetandadditionallymaking surethatthereisnopropersupersetwhichisalsoadmissible.Inordertoexpress subsetmaximalitydirectlyinsidethelogic,auniversal(or,equivalently,anegatedexistential)quantifiercanbeused,making quantified Boolean formulae a well-suited formalism. It is possible to specify preferredsemantics in QBFs either via an extension-based,oralabeling-basedapproach.

First, we consider the extension-based approach from [20]. Here, we encode the maximality check with an auxiliary formula.Forconveniencewedenoteby A⁷= {^a7|^a∈^A}^{thesetofrenamedargumentsinA.Likewise,wedefinearenaming} fortheattack relationasR⁷= {(a⁷,b⁷)|(a,b)∈^R}^{.Thefollowingdefinesashorthand forcomparingtwo setsofatomsan} interpretationisdefineduponwithrespecttothesubset-relation.

A

<

A⁷

= $

a∈^A

(

v_a

→

^va⁷

) ∧ ¬ $

a⁷∈A⁷

(

v_a7

→

^va

)

Inother words,thisformulaensuresthatanymodelM|4(A<A⁷)satisfies{^a∈^A|^va∈^M}⊂{^a∈^A|^va⁷∈^M}.Nowwe canstatetheQBFforpreferredextensions.Letthequantifiedvariablesbe A⁷_v= {^va⁷|^a7∈^A7}^.

prf_A,R

=

^admA,R

∧ ¬∃

^A7v

((

_A

<

A⁷

)

∧

^admA⁷,R⁷

)

(12) Inshort,wecheckwhethertheacceptedargumentsforman admissiblesetandwhetherthereexistsapropersuperset of it which isalso admissible. Ifthe former check succeeds and inthe latter no such set exists, then we have found a preferredextension.ForanarbitraryAF F=(A,R),itspreferredextensionsareina1-to-1correspondencetothemodels ofprf_A,R,i.e.,Enum_prf(F)∼= {^M|^M|4^prfA,R}^.

The second approach is based on complete labelings (see Definition 4) instead of extensions [19].⁹ To this end, we employ four-valuedinterpretations to expressmore thantwo possible statesforeach argument. Inaddition tothe truth valuestrueandfalsewealsoaddvaluesundecidedandinconsistent.Thethreelabelingsin,out andundecidedcorrespond to thefirstthree truthvalues.The wholeapproach canbe encodedinclassical two-valuedQBFs. Hereby,thetruth value of p∈P ^{isencodedvia p⊕ and p(.Now every classical two-valued}interpretationassigns valuesto thesetwo atomsas usual.Fortwovariableswehavefourdifferentcases,whichcorrespondtothefourtruthvalues:{^p⊕,p⁽}⊆^Iisinterpreted asassigning inconsistentto p,true(resp. false)isassignedto p ifonly p^⊕ (resp.p⁽)isinI,andundecidedisassignedif neither p^⊕nor p⁽isinI.

Forpreferredsemanticstheencoding ismorecomplexthan(12),buttheideasaresimilar.Webeginwithformulaefor thefourtruthvalues.Notethatweslightlyadaptedtherepresentationandformulaefrom[19]tobettermatchtheprevious encodings,buttheimportantconceptsremainthesame.

val

(

p

,

v

) =

 



 

p^⊕

∧

^{p( ifv}

=

ⁱ p^⊕

∧ ¬

^{p( ifv}

=

^t

¬

^p⊕

∧

^p( ifv

=

^f

¬

^p⊕

∧ ¬

^{p( ifv}

=

^u

Here,val(p,v)encodesthefourpossibletruthvaluesforavirtualatom pthatcanbereferredtoonasortofmeta-level.

Actually,insteadofptheauxiliaryatomsp^⊕andp⁽arepresentintheconcreteformula.Usingthisconcept,wecanspecify thelabelingformulaforeachargumentinanAF F=(A,R).

lab^t_A,R

(

a

) =

^val

(

v_a

,

t

) → $

(b,a)∈^R

val

(

v_b

,

f

)

(13)

lab_A^f_,R

(

a

) =

^val

(

v_a

,

f

) → '

(b,a)∈^R

val

(

v_b

,

t

)

(14)

lab^u_A,R

(

a

) =

^val

(

v_a

,

u

) →

%%

¬ $

(b,a)∈^R

val

(

v_b

,

f

)

&

∧

%

¬ '

(b,a)∈^R

val

(

v_b

,

t

)

&&

(15) TheseformulaereflectDefinition 4:Theformulae(13),(14)and(15)encodethein,outandundecidedlabelings,respec- tively.Forexample,(13)canbeinterpretedinthefollowingway:Ifanargumenta issettotrue,thenallitsattackersmust befalse.(14)canbeinterpretedsimilarly,exceptthatifanatomdenotesthatanargumentisfalse,thenoneofitsattackers must betrue. Finally,(15)states thatfor anyargumentto whichwe assign undecided,itcannot be the casethat all its attackersarefalseorthatoneofthemistrue.

9 Asimilarapproachwasrecentlyrealizedforabstractdialecticalframeworks(andthusalsoforAFs)inthesystemQADF:http://www.dbai.tuwien.ac.at/ proj/adf/qadf/.

Threevaluesaresufficienttoreflectthethreelabelings.Toavoidproblemswiththefourthtruthvalue(inconsistent),we excludeitfromoccurringintheevaluationbythefollowingformula.

3valA

= $

a∈A

¬

^val

(

v_a

,

i

)

Now,completeextensionsarecharacterizedbythefollowingformula.WewilluseLassuperscriptincom^L_A,R todenote thatthisformulahandleslabelingsinsteadofextensions.

com^L_A,R

:=

^3valA

∧ $

a∈^A

(

lab^t_A,R

(

a

) ∧

^labA^f,R

(

a

) ∧

^labuA,R

(

a

) )

The formula com^L_A,R expresses that all the arguments are assignedeither true,false or undecidedvia the 3valA sub- formula. Foreach argument,the three conjuncts onthe right ofthe formulaencode implications which ensure that the labels are assigned asspecified forcomplete labelings. For example,ifa is true,then all its attackersmust be false. By applying this, one can encode complete labelings andhence completeextensions. Preferred extensions(or labelings)are expressedasbeforebysubsetmaximization.

A

<

^LA⁷

= $

a∈^A

(

val

(

v_a

,

t

) →

^val

(

v_a7

,

t

) )

∧ ¬ $

a⁷∈^A7

(

val

(

v_a7

,

t

) →

^val

(

v_a

,

t

) )

(16) Then, similar asin prf_A,R,the preferredextensionsortheir labelingscan beencodedwitha QBFasfollows,withthe quantifiedatoms A⁷_v= {^v⊕_a7,v⁽_a7|^a7∈^A7}^.

prf^L_A,R

=

^comLA,R

∧ ¬∃

^A7v

((

_A

<

^LA⁷

)

∧

^comL_A7,R⁷

)

(17)

For an AF F=(A,R)the followingnotion ofcorrespondenceholds: Letthe setofatoms evaluatedto trueunder the four-valuedinterpretationbeM^t= {^p|^p⊕∈^M,p⁽∈/^M}^,thenEnumprf(F)∼= {^Mt|^M|4^prfLA,R}^{.NotethatprfL}A,R differsfrom prf_A,R notonlybyusingalabeling-basedapproach,butalsobymaximizingcompletelabelingsratherthanadmissiblesets.

Utilizing the expressive power of quantifiers and the labeling approach, the authors of [19] also encode a range of other semantics,forinstancesemi-stablereasoning,whereonecan applythesameidea asoutlined above,butinsteadof maximizingtheargumentsthatarein,theargumentsthatarelabeledundecidedareminimized.

Thisresultsinageneralsystemforencodingmanysemantics,butonehastobecarefulwithchoosingtherighttarget system.Forexample,groundedsemanticscaneasilybespecifiedinthisformalismusingaQBF,butcomputingthegrounded extensioncanbedoneusinganalgorithmwithpolynomialrunningtime.Thus,anappropriateencodingwouldyieldaQBF from afragment which isknown tobe efficientlydecidable, forinstance,2-QBF(the generalization ofKrom formulae to QBFs).However,wearenotawareofanyworkwhichdealswithsuch“complexity-sensitive”encodingsintermsofQBFs.

3.1.3. IterativeapplicationofSATsolvers

The lastpropositional-logic basedapproachwe outlinehereisbasedontheideaofiteratively searchingformodelsof propositionalformulaeandhasbeeninstantiatedinthesystemsArgSemSAT[82,22]andCEGARTIX[21,83].Theideaistouse an algorithmwhichiterativelyconstructsformulaeandsearchesformodelsoftheseformulae.A newformulaisgenerated basedonthemodelofthepreviousone(orbasedonthefactthatthepreviousformulaisunsatisfiable).Atsomepointthe algorithm reachesa final decisionand terminates.This isin contrastto so-calledmonolithic encodings,whichformulate the whole problemin a single formula. The encodingsin previous sectionsare examples forsuch monolithic encodings.

Theiterativeapproachissuitablewhentheproblemtobesolvedcannotbedecidedingeneral(understandardcomplexity theoretic assumptions) by the satisfiability of a single propositional formula (constructible in polynomial time) without quantifiers. Thisis, for instance,the case withskeptical acceptance under preferredsemantics where the corresponding decisionproblemisΠ₂^P complete.InsteadofreducingtheproblemtoasingleQBFformula,wedelegatethesolvingtaskin theiterativeschemetoanalgorithmqueryingaSATsolvermultipletimes.

Thealgorithmsforpreferredsemanticsworkroughlyasfollows.Tocomputepreferredextensionswetraversethesearch space ofacomputationally simplersemantics. Forinstance,we caniteratively search foradmissible setsorcomplete ex- tensions anditeratively extendthem until we reacha maximal set,which isa preferredextension. Bygenerating anew candidateadmissibleset/completeextension,whichisnotcontainedinanalreadyvisitedpreferredextensionwecanenu- merateallpreferredextensionsinthismanner.Thisallowsansweringcredulousandskepticalreasoningaswell.

Fordecidinge.g.skepticalacceptanceofanargumentunderpreferredsemanticsonerequiresingeneralanexponential numberofcalls totheSATsolver(understandardcomplexitytheoreticassumptions).However,thenumberofSATcalls in theiterativeSATschemeisdependentonthenumberofpreferredextensionsofthegivenAF,see[21].

In thefollowing,we firstsketch theCEGARTIXapproach from[21] forskepticalacceptance underpreferredsemantics andsubsequentlyoutlinethePrefSatapproach[82],implementedintheArgSemSATsystem,forenumeratingallpreferred extensions.Again,weslightlyadaptedthealgorithmsforauniformsettingandpresentation.

Algorithm 1decidesskepticalacceptanceunderpreferredsemanticsofanargumentainanAFF.Theideaistoproceed fromonepreferredextensiontothenextandcheckingwhetheraisinoneoftheextensions.Thisisencodedintheouter

Algorithm1Skept_prf(a,F). Require:AFF=(A,R),argumenta∈^A,

Ensure:returnsyesiffaisskepticallyacceptedunderpreferredsemantics 1:φ←^admA,R

2:while ∃^I,I|4φdo

3: while∃^I7,I⁷|4ψ^I(A,R)∧¬^vado 4: I←^I7

5: end while

6: if∃^I7,I⁷|4ψ^I(A,R)then 7: φ←φ∧γ^I

8: else

9: no

10: end if 11:end while 12:yes

while loop, lines 2 to 11. The models offormula φ representthe remaining admissible sets in the currentstate of the algorithm.In thebeginning,φ encodesall admissiblesets of F.Westart withan admissiblesetanditerativelyextendit while making sure that a is not accepted in thisadmissible set. This is done inthe second loop (lines 3 to 5) and by adding¬^va tothequery.Theformulaψ^I incorporatesthemodelI andstatesthatamodelofitmuststillcorrespondtoan admissibleset,butalsohastobeasupersetofthecurrentone,specifiedby I.

Ifwecannotaddfurtherargumentstotheadmissibleset,wecheckwhetherwecanextenditwithhavinga inside,in line6.Ifthisis thecase,every preferredextensionthat isasupersetofthecurrentadmissiblesetcontains a.Hence,we canproceed toadifferentadmissiblesetnot containinga.Incasewecannot adda to theadmissibleset,wehavefound apreferredextension withouta,herebyrefutingits skepticalacceptancein F.In theformercase(I doesnotrepresenta preferredextension)we strengthenthemainqueryφ byadding

γ

^I inline7,statingthatatleastoneargumentcurrently notacceptedinI mustbeacceptedfromnowon.Thisensuresthatinfutureiterationswecomputeadmissiblesetsthatare notcontainedinpreviouslyfoundpreferredextensions.

Theformulaearedefinedasfollows.

ψ

^I

(

A

,

R

) =

^admA,R

∧ $

a∈^A,va∈^I

v_a

∧ % '

a∈^A,va∈/^I v_a

&

γ

^I

₌ ^'

a∈^A,va∈/^I v_a

.

Example5. Forthe AF F fromExample 1 we cancheck the skepticalacceptance ofb. Thecondition ofthe firstloop is satisfiedasthereexisttheadmissiblesets ∅^,{^a}^,{^d}^and{^a,d}^{in F.The algorithmnow}non-deterministicallyselectsone oftheadmissiblesets.Saywepick∅.Thesecondwhileloopthencreatesasubsetmaximaladmissibleset(excludingb)in two iterations,say firstaddinga andthend.As {^a,d} ^{isnowsubset maximal,the second loop}terminates. Sincethisset cannotbeextendedifweallowtoalsoacceptb,weknowthatwehavefoundapreferredextension.Thismeanswerefute theskepticalacceptanceofb.

ThePrefSatapproach[82]isdesignedtoenumerateallpreferredextensionsutilizingasimilaridea.Hereby,alsoasimpler semanticsfortraversingthesearchspaceisused,buttheencodingsrelyontheconceptoflabellings(seealsoSection4.1).

WeoutlinethePrefSatprocedureinAlgorithm 2.

PrefSatencodes labelings ofan AF F=(A,R) by generatingthree variables per argument,i.e., the setof variables in the constructed formula are {^Ia,Oa,Ua|^a∈^A}^{. In the final result these variables correspond naturally to a labeling.In} particular,a three-valuedlabeling K corresponds toamodel J if K=(I,O,U)with I= {^a|^Ia∈ ^J}^{, O}= {^a|^Oa∈ ^J}^and U= {^a|^Ua∈ ^J}^{.Thefollowingconstraintencodesthatforeveryargumentexactlyonelabelingisassigned.}

$

a∈A

( (

I_a

∨

^Oa

∨

^Ua

) ∧ ( ¬

^Ia

∨ ¬

^Oa

) ∧ ( ¬

^Ia

∨ ¬

^Ua

) ∧ ( ¬

^Oa

∨ ¬

^Ua

) )

Furthermore,one canencodetheconditionsforalabelingtobe completebyconjoining certainsubformulae.Forinstance, theformula

$

x∈^A,∃(y,x)∈^R

% $

(x⁷,x)∈^R

( ¬

^Ix

∨

^Ox⁷

)

&

encodesthatwecan acceptxifeachattackerisout.Theremainingconstraintsforalabelingtobecompleteare encoded similarly. Severalequivalent formulae for encoding complete labelings have been investigated by Cerutti et al. [82]. Let com⁷_A,R beoneofthesechoicesforencodingcompletelabelings.TheformulaeforAlgorithm 2arethendefinedasfollows.