Methods for solving reasoning problems in abstract argumentation – A survey

Contents lists available atScienceDirect

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

,

∗

a_{ViennaUniversityofTechnology,InstituteofInformationSystems,Austria} b_{UniversityofVienna,FacultyofComputerScience,Austria}

c_{TechnischeUniversitätDresden,InstituteofArtificialIntelligence,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

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 strongandjoint

develop-ment of afield like “computational argumentation”.In fact, Dung’sidea ofevaluating argumentson an abstract levelby

takingonlytheirinter-relationshipsintoaccount,not onlyhasbeenshowntounderliemanyoftheearlierapproachesfor

argumentation,butalsouniformlycapturesseveralnon-monotoniclogics.Yetthissecond contributionlocated

Argumenta-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:[email protected](G. Charwat),

[email protected]

(W. Dvoˇrák),

[email protected]

(S.A. Gaggl),

[email protected](J.P. Wallner),

[email protected]

(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

•

acertainlevelofgenerality,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 useofthislogic-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 morefine-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 alookondynamicprogrammingapproaches[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

)

}

.The

correspond-inggraphrepresentationisdepictedinFig. 1.

Asemanticsforargumentationframeworksisdefinedasafunction

σ

whichassignstoeachAFF

=

(

A

,

R

)

aset

σ

(

F

)

⊆

2A ofextensions.

6 _See

_http

_{://argumentationcompetition.orgforfurtherinformation.} 7 _{Anoverviewontheseapproachesisgivenin[54].}

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

R_a(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

R_x

_}

_(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 isconflict-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

⊆

F

F

(

S

)

;

•

S

∈

com

(

F

)

,ifS

=

F

F

(

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

⊂

T_R+.

Werecallthat foreachAF F,thegroundedsemanticsyields auniqueextension, thegroundedextension,whichisthe

least fixed-point of the characteristic function

F

F.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 several

ap-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,

where

L

in is the set of arguments labeled by in,

L

out is the set of arguments labeled by out and

L

undec is the set of

argumentslabeledbyundec.

Asanexample,wegivethedefinitionofcompletelabelingsfrom[69].

Definition4.GivenanAF F

=

(

A

,

R

)

,a function

L

:

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 thosecompletelabelingswhere

L

in 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

•

Skept_σ

(

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 Crednaive

(

a

,

F

) =

yes andSkeptnaive

(

a

,

F

)

=

no. For preferred semantics, F has a single extension Enumprf

(

F

)

= {{

a

,

d

}}

, Countprf

(

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.Let

C

denotesomecomplexityclass.Bya

C

-oraclemachinewe

meana(polynomialtime)Turingmachinewhichcanaccessanoraclethatdecides agiven(sub)-problemin

C

withinone

step. Wedenotesuch machinesasNPC iftheunderlyingTuring machineisnon-deterministic. Theclass

Σ

₂P

=

NPNP thus

denotesthesetofproblemswhichcanbedecidedbyanon-deterministicpolynomialtimealgorithmthathas(unrestricted)

accesstoanNP-oracle.Theclass

Π

P

2

=

coNP

NP_{isdefinedasthecomplementaryclassof}

_Σ

P

2,i.e.,

Π

2P

=

co

Σ

2P.Therelation

betweenthecomplexityclassesisasfollows:

L

⊆

P

⊆

NP coNP

⊆

Σ

₂P

Π

₂P

The computational complexityof credulousandskepticalreasoninghasbeenstudied extensivelyinthe literature(see

[70] fora starting point). Table 1summarizes thecomputational complexity classifications ofthe defineddecision

Table 1

ComputationalcomplexityofreasoninginAFs.

σ Credσ Skeptσ Verσ

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 ΠP 2-c coNP-c sem Σ2P-c Π P 2-c coNP-c stg Σ2P-c Π P 2-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,astheBooleanSAT

prob-lemiswellstudiedandmoreoveraccompaniedwithmanymatureandefficientsolverssuchasMiniSat[78]andGRASP[79].

First,werecallthenecessarybackgroundofBooleanlogicandquantifiedBooleanformulae(QBF)sincetheyserveasour

targetsystems.

Thebasis ofpropositionallogic isasetofpropositional variables

P

,to whichwealsorefer toasatoms.Propositional

formulaeare builtasusual fromtheconnectives

∧,

∨,

→

and

¬

,denoting thelogicalconjunction, disjunction,(material)

implicationandnegationrespectively.Weusethetruthconstants todenotetrueand

⊥

forfalse.Inaddition,weconsider

quantifiedBooleanformulaewiththeuniversalquantifier

∀

andtheexistentialquantifier

∃

(bothoveratoms),thatis,given

aformula

φ

,then Q p

φ

isaQBF,withQ

∈ {∀,

∃}

andp

∈

P

.Furthermore, Q

{

p1

,

. . . ,

pn

}φ

isashorthandfor Q p1

· · ·

Q pn

φ

.

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.If

φ

containsnofreevariable,then

φ

issaidtobeclosedandotherwiseopen.Wewillwrite

φ[

p

/ψ]

todenote

theresultofuniformlysubstitutingeachfreeoccurrenceofpwith

ψ

informula

φ

.

AninterpretationI

⊆

P

definesforeachpropositionalvariableatruthassignmentwherep

∈

Iindicatesthatpevaluates

totruewhile p

/

∈

I indicatesthat pevaluates tofalse.Thisgeneralizestoarbitraryformulaeinthestandardway:Givena

formula

φ

andaninterpretation I,then

φ

evaluatesto trueunder I (i.e., Isatisfies

φ

)ifoneofthefollowingholds(with

p

∈

P

).

•

φ

=

pandp

∈

I

•

φ

= ¬

pandp

/

∈

I

•

φ

=

ψ

1

∧

ψ

2 andboth

ψ

1 and

ψ

2 evaluatetotrueunderI

•

φ

=

ψ

1

∨

ψ

2 andoneof

ψ

1and

ψ

2evaluatestotrueunderI

•

φ

=

ψ

1

→

ψ

2 and

ψ

1 evaluatestofalseor

ψ

2evaluatestotrueunderI

•

φ

= ∃

p

ψ

andoneof

ψ[

p

/]

and

ψ

[

p

/⊥]

evaluatestotrueunderI

•

φ

= ∀

p

ψ

andboth

ψ

[

p

/

]

and

ψ

[

p

/

⊥]

evaluatetotrueunderI.

IfaninterpretationI satisfiesaformula

φ

,denotedby I

|

φ

,wesaythat Iisamodelof

φ

.

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 sufficientforillustratingthemainconcepts. Ingeneral,theapproachescanbeapplied

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 apropositional variable

va isused.Then, S

⊆

A isan extensionundersemantics

σ

iff

{

va

|

a

∈

S

} |

φ

,with

φ

beingapropositional formulathat

evaluates AF F under semantics

σ

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

correspondencebetweensetsofextensionsandmodelsofapropositionalformulacanbedefinedasfollows.

Definition7.Let

S

⊆

2A _{beacollectionofsetsofargumentsandlet}

_I

_⊆

₂P _{beacollectionofinterpretations.Wesaythat}

S

and

I

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.

admA,R

=

a∈A

va

→

(b,a)∈R

¬

vb

∧

va

→

(b,a)∈R (c,b)∈R vc

(1)

The models of admA,R now correspondto the admissible sets of F, i.e., Enumadm

(

F

)

∼

= {

M

|

M

|

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., va

evaluatestotrueunderamodel),allitsattackerscannotbeselectedanyfurther.Thesecondconjunctexpressesthedefense

ofargumentsbystatingthat,ifweaccepta,thenforeachattackerb,somedefenderc mustbeacceptedaswell.Notethat

anemptyconjunctionistreatedas ,whereastheemptydisjunctionistreatedas

⊥

.

Example4.Thepropositionalformulaforadmissiblesetsoftheframework F

=

(

A

,

R

)

inExample 1isgivenby

admA,R

≡

(

va

→

)

∧

(2) vb

→

(

¬

va

∧ ¬

vc)

∧

(3) vc

→

(

¬

vb

∧ ¬

vd)

∧

(4)

(

vd

→

)

∧

(5) ve

→

(

¬

vd

∧ ¬

ve)

∧

(6)

(

va

→

)

∧

(7) vb

→

⊥ ∧

(

vb

∨

vd)

∧

(8) vc

→

(

va

∨

vc)

∧ ⊥

∧

(9)

(

vd

→

)

∧

(10) ve

→

(

⊥ ∧

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 admA,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

admA,R wherevb 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 toshow

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 A

= {

a

|

a

∈

A

}

thesetofrenamedargumentsinA.Likewise,wedefinearenaming

fortheattack relationasR

= {(

a

,

b

) |

(

a

,

b

) ∈

R

}

.Thefollowingdefinesashorthand forcomparingtwo setsofatomsan

interpretationisdefineduponwithrespecttothesubset-relation.

A

<

A

=

a∈A

(

va

→

va

)

∧ ¬

a∈A

(

va

→

va)

Inother words,thisformulaensuresthatanymodelM

|

(

A

<

A

)

satisfies

{

a

∈

A

|

va

∈

M

}

⊂ {

a

∈

A

|

va

∈

M

}

.Nowwe canstatetheQBFforpreferredextensions.Letthequantifiedvariablesbe A_v

= {

va

|

a

∈

A

}

.

prf_A,R

=

admA,R

∧ ¬∃

Av

A

<

A

∧

admA,R

(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

ofprfA,R,i.e.,Enumprf

(

F

)

∼

= {

M

|

M

|

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-valuedinterpretationassigns 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 pisinI.

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⊕andparepresentintheconcreteformula.Usingthisconcept,wecanspecify

thelabelingformulaforeachargumentinanAF F

=

(

A

,

R

)

. labt_A,R

(

a

)

=

val

(

va,t

)

→

(b,a)∈R val

(

vb,f

)

(13) lab_Af_,R

(

a

)

=

val

(

va,f

)

→

(b,a)∈R val

(

vb,t

)

(14) labu_A,R

(

a

)

=

val

(

va,u

)

→

¬

(b,a)∈R val

(

vb,f

)

∧

¬

(b,a)∈R val

(

vb,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

(

va,i

)

Now,completeextensionsarecharacterizedbythefollowingformula.WewilluseLassuperscriptincomL

A,R todenote

thatthisformulahandleslabelingsinsteadofextensions.

comL_A,R

:=

3valA

∧

a∈A

labt_A,R

(

a

)

∧

lab_Af_,R

(

a

)

∧

labu_A,R

(

a

)

The formula comL_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

<

LA

=

a∈A val

(

va,t

)

→

val

(

va

,

t

)

∧ ¬

a∈A val

(

va

,

t

)

→

val

(

va,t

)

(16)

Then, similar asin prfA,R,the preferredextensionsortheir labelingscan beencodedwitha QBFasfollows,withthe

quantifiedatoms A_v

= {

v⊕_a

,

va

|

a

∈

A

}

.

prfL_A,R

=

comL_A,R

∧ ¬∃

A_v

A

<

LA

∧

comL_A,R

(17)

For an AF F

=

(

A

,

R

)

the followingnotion ofcorrespondenceholds: Letthe setofatoms evaluatedto trueunder the

four-valuedinterpretationbeMt

_{= {}

_p

_|

_p⊕

_∈

_M

_,

_p

_∈

_/

_M

_}

_,thenEnum

prf

(

F

)

∼

= {

Mt

|

M

|

prfLA,R

}

.NotethatprfLA,R differsfrom

prfA,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

decisionproblemis

Π

P

2 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,whichisnotcontainedinanalreadyvisitedpreferredextensionwecan

enu-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

Algorithm1Skeptprf

(

a

,

F

)

.

Require:AFF=(A, R),argumenta∈A,

Ensure:returnsyesiffaisskepticallyacceptedunderpreferredsemantics 1:φ←admA,R 2:while ∃I, I|φdo 3: while∃I, I|ψI_{(A, R) ∧ ¬v} ado 4: I←I 5: end while 6: if∃I, I|ψI_{(A, R)then} 7: φ←φ∧γI 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.Theformula

ψ

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 strengthenthemainquery

φ

byadding

γ

I _{inline7,statingthatatleastoneargumentcurrently}

notacceptedinI mustbeacceptedfromnowon.Thisensuresthatinfutureiterationswecomputeadmissiblesetsthatare

notcontainedinpreviouslyfoundpreferredextensions.

Theformulaearedefinedasfollows.

ψ

I

(

A

,

R

)

=

admA,R

∧

a∈A,va∈I va

∧

a∈A,va∈I/ va

γ

I

₌

a∈A,va∈/I va

.

Example5. Forthe AF F fromExample 1 we cancheck the skepticalacceptance ofb. Thecondition ofthe firstloop is satisfiedasthereexisttheadmissiblesets

∅

,

{

a

}

,

{

d

}

and

{

a

,

d

}

in F.The algorithmnownon-deterministicallyselectsone

oftheadmissiblesets.Saywepick

∅

.Thesecondwhileloopthencreatesasubsetmaximaladmissibleset(excludingb)in

two iterations,say firstaddinga andthend.As

{

a

,

d

}

isnowsubset maximal,the second loopterminates. 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

(

Ia

∨

Oa

∨

Ua)

∧

(

¬

Ia

∨ ¬

Oa)

∧

(

¬

Ia

∨ ¬

Ua

)

∧

(

¬

Oa

∨ ¬

Ua)

Furthermore,one canencodetheconditionsforalabelingtobe completebyconjoining certainsubformulae.Forinstance,

theformula

x∈A,∃(y,x)∈R (x,x)∈R

(

¬

Ix

∨

Ox

)

encodesthatwecan acceptxifeachattackerisout.Theremainingconstraintsforalabelingtobecompleteare encoded

similarly. Severalequivalent formulae for encoding complete labelings have been investigated by Cerutti et al. [82]. Let

Algorithm2Enumprf

(

F

)

.

Require:AFF=(A, R),argumenta∈A, Ensure:returnsEnumprf(F)

1:S← ∅ 2:φ←comA,R 3:while ∃J, J|φdo 4: while∃J, J|ψJ(A, R)do 5: J←J 6: end while 7: S←S∪ {{a|Ia∈J}} 8: φ←φ∧γJ 9:end while 10: returnS

ψ

J

(

A

,

R

)

=

com_A,R

∧

a∈A,Ia∈J Ia

∧

a∈A,Ia∈/J Ia

γ

J

=

a∈A,Ia∈/J Ia

.

Algorithm 2 traverses the spaceof complete labelings in two while loops. The outer while loop computes candidate

completelabelings,whichare notsmallerthanpreviouslyfoundpreferredlabelings(when comparingonlythearguments

assignedin).The innerwhileloop searchesforamaximalpreferredlabeling.Iftheinnerwhileloop terminates,then the

corresponding preferredlabeling/extensionisaddedtothesolutionset

S

.Inline8weexcludesmallercompletelabelings

from subsequent iterations. In [82] the algorithm for enumeratingall preferredextensions contains further refinements,

suchasrestrictingthesearchspacetonon-emptycompleteextensions.

3.1.4. Reasoningproblems

Thefirsttwo reductionspresentedinSection 3.1.1andSection 3.1.2immediatelysolveproblemsofenumerating

exten-sions.Decidingcredulousandskepticalreasoningistypicallyeasytoachieve.InordertodecideCredσ

(

a

,

F

)

onecanconjoin

thebaseformulawithanatomcorrespondingtotheacceptanceofargumenta.Ifthereexistsamodel,aiscredulously

ac-cepted.Addingtheatomforainnegatedformtotheformuladecidesifa isnotskepticallyaccepted,i.e.,ifthereexists a

model,anextensiondoesnotcontaina.

Similarly, the iterativeSAT scheme ofAlgorithm 1 (see Section 3.1.3) can be adapted tosolve credulous reasoningby

addingtheatom tobequeriedpositively insteadofnegatively inline3.Regarding theenumerationproblem,Algorithm 2

enumeratesallpreferredextensionsusingiterativeSAT.

Counting thenumberof extensionscannot beeasily encodedinthe formulae,buttheSAT solveritself mayofferthis

featurebycountingthenumberofmodels.

3.2. CSP-basedapproach

In this subsection we consider reductions to Constraint Satisfaction Problems (CSPs) [84], which allow solving

com-binatorial search problems. The approach of CSP is inherently relatedto propositional logic reductions asintroduced in

Subsection3.1.1,seealso[85]foraformalanalysisoftherelationbetweenthetwoapproaches.

ACSP cangenerallybedescribed byatriple

(

X

,

D

,

C

)

,where X

= {

x1

,

. . . ,

xn

}

istheset ofvariables, D

= {

D1

,

. . . ,

Dn

}

is a set offinite domains for thevariables and C

= {

c1

,

. . . ,

cm

}

a set ofconstraints. Eachconstraint ci isa pair

(

hi

,

Hi

)

where hi

=

(

xi1

,

. . . ,

xik

)

is a k-tuple of variables and Hi is a k-ary relation over D. In particular, Hi is a subset of all

possible variablevalues representingtheallowed combinations ofsimultaneous valuesforthevariables inhi.An

assign-ment v isa mappingthat assigns to every variable xi

∈

X an element v

(

xi

)

∈

Di. Anassignment v satisfiesa constraint

((

xi1

,

. . . ,

xik

),

Hi

) ∈

C iff

(

v

(

xi1

),

. . . ,

v

(

xik

)) ∈

Hi. Finally,asolution isan assignment v toall variables such thatall con-straintsaresatisfied,denotedby

(

v

(

x1

),

. . . ,

v

(

xn

))

.

Finding a valid assignment of a CSP is in general NP-complete. Nevertheless, several programming libraries support

constraintprogramming, likeECLiPSe,SWIProlog,Gecode, JaCoP,Choco,Turtle(justtomentionsome ofthem) andallow

forefficientimplementationsofCSPs.Theseconstraintprogrammingsolversmakeuseoftechniqueslikebacktrackingand

localsearch.

ComputingargumentationsemanticsviaconstraintprogramminghasbeenaddressedmainlybyAmgoudandDevred[23]

andBistarelli andSantini[26–28],wherethe latterprovidethesystemConArgwhichisable tocomputea widerangeof

semanticsforabstractargumentationframeworks.

3.2.1. MappingsofAFstoCSPs

GivenanAFF

=

(

A

,

R

)

,theassociatedCSP

(

X

,

D

,

C

)

isspecifiedasX

=

Aandforeachai

∈

X,Di

= {

0

,

1

}

.Theconstraints

are formulateddepending onthespecificsemantics

σ

.Forexample,solutions thatcorrespondtoconflict-free setscanbe

setto1 atthesametime.Here,theconstraintisoftheform

((

a

,

b

),

((

0

,

0

),

(

0

,

1

),

(

1

,

0

)))

whichisequivalent tothecases

whenthepropositionalformula

(

a

→ ¬

b

)

evaluatestotrue.

Inthefollowing,wewillusethenotation from[23],becauseitreflectsthesimilarities betweentheCSP approachand

thereductionstopropositionallogicasoutlinedabove.

Foradmissiblesemanticswegetthefollowingconstraints.

Cadm

=

a

→

b:(b,a)∈R

¬

b

∧

a

→

b:(b,a)∈R c:(c,b)∈R c

a

∈

A

(18)

Thefirstpartensuresconflict-freesetsandthesecondpartencodesthedefenseofarguments.Then,foranAFF

=

(

A

,

R

)

anditsassociatedadmissibleCSP

(

X

,

D

,

Cadm

)

,

(

v

(

x1

),

. . . ,

v

(

xn

))

isasolutionoftheCSPifftheset

{

xj

,

. . . ,

xk

}

s.t. v

(

xi

)

=

1 isanadmissiblesetin F.

Example6.FortheAFofExample 1onPage30weobtainthefollowingadmissibleCSP

(

X

,

D

,

Cadm

)

. X

=

A,foreachai

∈

X wehaveDi

= {

0

,

1

}

and Cadm

=

(

a

→

)

∧

(

a

→

), (

b

→ ¬

a

∧ ¬

c

)

∧

(

b

→ ⊥ ∧

d

),

(

c

→ ¬

b

∧ ¬

d

)

∧

c

→

(

a

∨

c

)

∧ ⊥

, (

d

→

)

∧

(

d

→

),

(

e

→ ¬

d

∧ ¬

e

)

∧

(

e

→ ⊥ ∨

d

)

.

ThisCSPhasthefollowingsolutions:

(

0

,

0

,

0

,

0

,

0

)

,

(

1

,

0

,

0

,

0

,

0

)

,

(

0

,

0

,

0

,

1

,

0

)

,

(

1

,

0

,

0

,

1

,

0

)

whichcorrespondtothe admis-siblesetsofF,namely

{}

,

{

a

}

,

{

d

}

and

{

a

,

d

}

.

MostCSP solvers do not support subset maximization. Thus, for preferredsemantics, the authors in[28] propose an

approachthat iteratively computesadmissible/completeextensionsandaddsconstraints toexcludecertain sets,such that

onefinallyobtainsthepreferredextensions.

3.2.2. Reasoningproblems

Similarlyto thereductionstopropositional logic,one candecide thefollowing reasoningproblemswithCSPs, namely

verification,skepticalandcredulousreasoning.Furthermore,enumerationisusuallysupportedbymodernCSPsolvers.

3.3. ASP-basedapproach

Answer-set programming (ASP, for short) [86,87], also known as A-Prolog [88,89], is a declarative problem solving

paradigm,rootedinlogicprogrammingandnon-monotonicreasoning.Duetocontinuousrefinementsoverthelastdecade,

answer-setsolvers(e.g.,[90,91])nowadaysnotonlysupportarichlanguagebutalsoare capableofsolvinghardproblems

efficiently.Furthermore,thedeclarativeapproachofASPresultsinreadableandmaintainablecode(comparedtoCcode,for

instance),thusallowingtodefinetheproblemsathandinanaturalway.

SolvingproblemsinabstractargumentationviaASPhasbeenstudiedbyseveralauthors(see[29]forasurvey),including

theapproachproposedbyNievesetal.[74]wheretheprogramisre-computedforeveryinputinstance,WakakiandNitta

[76]whouselabeling-basedsemanticsandtheapproachbyEglyetal.[30]whichfollowsextension-basedsemantics.Here,

we focuson thelatter, sincethisapproach isput intopracticeby the ASPARTIXsystemwhich supports awide rangeof

differentsemanticsandadditionallyoffersawebfront-end.

In the following, we first give a brief introduction to ASP.We then presenthow the computation of admissible and

preferredextensions canbe encoded inASP.In order toobtain preferredextensions,it is necessaryto check for

subset-maximality of admissible sets. We sketch two approaches for this in ASP, one based directly on a certain saturation

technique [92] (which is unfortunately hardly accessible for non-experts in ASP) and a second one which makes useof

metasp

encodings [93] (allowingtospecifysubset minimization via asingle simple statement).Additionally, we discuss

howreasoningproblemscanbespecified.

3.3.1. Answer-setprogramming

Wegiveabriefoverviewofthesyntaxandsemanticsofdisjunctivelogicprogramsundertheanswer-setsemantics[94];

forfurtherbackground,see[95,91].

We fix a countable set

U

of (domain)elements, also called constants, andsuppose a total order

<

over the domain

elements.Anatomisanexpression p

(

t1

,

. . . ,

tn

)

,where pisapredicateofarityn

≥

0 andeachtiiseitheravariableoran

elementfrom

U

.Anatomisgroundifitisfreeofvariables. B_U denotesthesetofallgroundatomsover

U

.

A(disjunctive)rule rwithn

≥

0,m

≥

k

≥

0,n

+

m

>

0 isoftheform a1

∨ · · · ∨

an

←

b1

, . . . ,

bk,not bk+1

, . . . ,

not bm