On the Input/Output behavior of argumentation frameworks

www.elsevier.com/locate/artint

On the Input/Output behavior of argumentation frameworks

Pietro Baroni

^a

, Guido Boella

^b

, Federico Cerutti

^c

, Massimiliano Giacomin

^a

^,∗ , Leendert van der Torre

^d

, Serena Villata

^e

aDipartimentodiIngegneriadell’Informazione,UniversityofBrescia,viaBranze,38,25123,Brescia,Italy bDipartimentodiInformatica,UniversityofTorino,Italy

cDepartmentofComputingScience,UniversityofAberdeen,UK dUniversityofLuxembourg,Luxembourg

eINRIASophiaAntipolis- Mediterranee,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received21March2014

Receivedinrevisedform22August2014 Accepted25August2014

Availableonline28August2014

Keywords:

Argumentationframeworks Argumentationsemantics Modularity

Decomposability Equivalence

This paper tackles the fundamental questions arising when looking at argumentation frameworksasinteractingcomponents,characterizedbyanInput/Outputbehavior,rather than as isolated monolithical entities. This modeling stance arises naturally in some application contexts, like multi-agent systems, but, more importantly, has a crucial impact onseveral generalapplication-independentissues, like argumentation dynamics, argumentsummarizationand explanation,incrementalcomputation,and inter-formalism translation. Pursuing this research direction, the paper introduces a general modeling approach and provides a comprehensive set of theoretical results putting the intuitive notionofInput/Outputbehaviorofargumentationframeworksonasolidformalground.

Thisis achieved by combining three main ingredients. First, several novel notions are introduced at the representation level, notably those of argumentationframeworkwith input,of argumentationmultipole, and ofreplacement of multipoles within a traditional argumentationframework.Second, severalrelevant features ofargumentationsemantics areidentifiedandformallycharacterized.Inparticular,thecanonicallocalfunction provides an input-aware semantics characterization and a suite of decomposability properties are introduced, concerning the correspondences betweensemantics outcomes at global and local level. The third ingredient glues the former ones, as it consists of the investigationof somesemantics-dependent properties ofthe newly introduced entities, namely S-equivalence of multipoles, S-legitimacy and S-safeness of replacements, and transparency of asemantics with respect to replacements. Altogether they provide the basis and draw the limits of sound interchangeability of multipoles within traditional frameworks.The paper developsan extensive analysis of all the conceptslisted above, covering seven well-known literature semantics and taking into account various, more orlessconstrained,waysofpartitioninganargumentationframework.Diverseexamples, takenfromtheliterature,areusedtoillustratetheapplicationoftheresultsobtainedand, finally,anextensivediscussionoftherelatedliteratureisprovided.

©^{2014ElsevierB.V.}All rights reserved.

*

Correspondingauthor.

E-mailaddresses:pietro.baroni@ing.unibs.it(P. Baroni),guido@di.unito.it(G. Boella),f.cerutti@abdn.ac.uk(F. Cerutti),massimiliano.giacomin@ing.unibs.it (M. Giacomin),leon.vandertorre@uni.lu(L. van der Torre),serena.villata@inria.fr(S. Villata).

http://dx.doi.org/10.1016/j.artint.2014.08.004 0004-3702/©^{2014ElsevierB.V.}All rights reserved.

Roughlyspeaking,modularityinvolvestwomainproperties,namelyseparability andinterchangeability ofmodules.Asto the former, ithas to be possibleto describe andanalyze the globalbehavior of an artifact in termsof the combination ofthelocalbehaviorsofthemodulescomposing it.Eachlocalbehaviorcanbe characterizedindividually inawaywhich is independentofthe internal details ofthe other modules(and, ina sense, ofthe moduleitself)andcaptures onlythe connections andmutualinteractions betweenthemoduleandtheother ones. Toputitinother words,each modulecan bedescribed asablack-box whoseInput/Outputbehaviorfullydeterminesitsrole intheglobalbehaviorofanyartifactit ispluggedin.Astothelatter,theinterestinreplacingamodulewithanotheroneisverycommonandarisesfromalarge varietyofmotivations,eitherattheoperationalordesignlevel.Interchangeabilityoftwomodulesrequiresfirstofallthat theyarecompatibleasfarastheconnectionswiththerestoftheartifactareconcerned,i.e.thattheinterfacestheyexpose aresuchthatwhereveroneofthemodulescanbe“pluggedin”,theothercantoo.Besidesthisplug-level interchangeability, itisofgreatinteresttocharacterizethebehavior-level interchangeabilityofmodules,namelytoidentifythesituationswhere internallydifferentmodulescanbefreelyinterchangedwithoutaffectingtheglobalbehavioroftheartifacttheybelongto, sincetheirInput/Outputbehaviorisequivalentinthisrespect.

While the formalism of abstract argumentation frameworks [25] and the relevant argumentation semantics (see [3]

fora survey)donot appearto havebeendesignedwithmodularity inmind,investigatingtheir relevantproperties isan importantresearch topicwhich,afterhavingbeensomehowoverlooked,isattracting increasingattentioninrecentyears.

Anargumentationframeworkisbasicallyadirectedgraphrepresentingtheconflictsbetweenasetofarguments(thenodes ofthegraph)andanargumentationsemanticscanberegardedasamethodtoanswer(typicallyinanon-univocalway,i.e.

producingasetofalternativeanswers)the“justificationquestion”:“Whichisthejustificationstatusofargumentsgiventhe conflict?”

Referring to a representative set of semantics proposed in the literature, (namely admissible, complete, grounded, preferred,stable,semi-stableandidealsemantics)thispaperprovidesasystematicandcomprehensiveassessmentofmod- ularityinabstractargumentation,byidentifyingandanalyzinginthiscontexttheformalcounterpartsofthegeneralnotions ofseparabilityandinterchangeabilitydescribedabove.

Givenapartitionofanargumentationframeworkintopartial (orlocal)interactingsubframeworks,analyzingseparability consistsinaddressingthefollowingissues:

•

^{Isitpossibletodefinealocal}counterpartofthenotionofsemantics?i.e.Isthereamethodtoproducelocalanswers tothejustificationquestion,takingintoaccounttheinteractionswithothersubframeworks?

•

^{Can the set of} justification answers prescribed by the (global) semantics be obtained by properly combining (in a bottom-upfashion)thesetsoflocalanswersproducedinthesubframeworksbyitslocalcounterpart?

•

symmetrically,Canthesetsoflocalanswersbeobtained(inatop-downfashion)asprojectionsontothesubframeworks oftheglobalanswers?

As to the first issue, we introduce the notion of localfunction for a subframework¹ and show that under very mild requirements,satisfiedbyallsemanticsconsideredinthispaper,itispossible(andeasy)toidentifythecanonicallocalfunc- tion foraglobalsemantics.Astothesecondandthirdissues,weintroducetheformalnotionsoftop-downandbottom-up decomposability,which,jointly,correspondtothenotionof(full)decomposabilityofanargumentationsemantics.

Strongasitmayseem,fulldecomposabilitywithrespecttoeveryarbitrarypartitionofeveryargumentationframework is not unattainable. Indeed,we show that itis satisfied by some of thesemantics considered in thispaper, while some othersareabletoachieveatleasttop-downdecomposabilityandtheremainingoneslackalldecomposabilityproperties.

Asarbitrarypartitionscorrespondtoacompletelyfree(ifnotanarchical)notionofmodularity,wealsoconsidera“tidier”

styleofpartitioning,involvingthegraph-theoreticalnotionofstronglyconnectedcomponents.Itturnsoutthat,restrictingthe setofpartitionsthisway,helpssome,butnotall,semanticstorecoverfulldecomposability.

Turningtointerchangeability,wedealwithbothitsplug-levelandbehavior-levelaspects.Astotheplug-level,borrowing some terminologyfrom circuittheory,we introduce the notionof argumentationmultipole as a generic replaceable argu- mentationcomponent,namelyapartialframeworkinteractingthroughaninputandoutputrelationwithanexternalsetof invariantarguments.

Plug-levelcompatibilityoftwomultipolesisaveryrelaxednotion,sinceitisonlyrequiredthattwomultipolesreferto thesamesetofexternal arguments.Thisismotivatedbythe factthat imposingatighter correspondencebetweenInput/

1 Technically,asubframeworkiscapturedbytheformalnotionofargumentationframeworkwithinput providedinDefinition 11.

multipolesareequivalentwithrespecttosomesemanticsandnotequivalentwithrespecttoanothersemantics.

Input/Outputequivalenceisthebasis forthe analysisoftheoperation ofreplacementwithin anargumentation frame- work.Basically,areplacementconsistsinsubstitutingapartoftheframeworkwithaplug-levelcompatiblemultipole.While thisnotionperse allowsforarbitrarysubstitutions,one isinterested inanalyzing thosereplacementswhichhaveasound basis. Inthisperspective,buildingon multipoleequivalence,itis possibletoidentifythe semantics-dependentnotions of legitimate andcontextuallylegitimate replacement. Briefly,areplacementislegitimateifitinvolves“fully”equivalentmulti- poles,whileitiscontextuallylegitimateifitinvolvesmultipoleswhichareequivalentinthecontextwherethereplacement takesplace,whilenotnecessarilybeingequivalentinothercontexts.Cleary,legitimatereplacementsarea(typicallystrict) subsetofcontextuallylegitimatereplacements.

One might expect that, given a semantics, legitimate (with respect to that semantics) replacements ensure that the invariant part of the framework is unaffected (in a sense, that it does not notice the change). This property is called semanticstransparency.A strongerexpectation (sincethe requirementon thereplacements is weaker)would be that the invariant part ofthe framework is unaffected forany contextually legitimate replacement: thisproperty is called strong transparency.

Natural asit may seem,transparency isnot achieved by all semantics andrequires a detailedanalysis, showingthat differentlevels oftransparencyare achievedbythe semanticsconsidered inthispaper,alsotakingintoaccount different restrictionsonthesetofallowedreplacements.

Theseresultsprovideareferencecontextandfundamentalanswerstomodularity-relatedissuesinabstractargumenta- tion,which,uptonow,havebeenconsideredintheliteraturefocusingonspecificaspectsandhenceobtainingpartialand problem-specificresults.Moreover,whilebeingtheoreticalbynature,theachievementsofthispaperhaveseveralsignificant application-orientedimplications.

Ontheonehand,semanticsdecomposabilitypropertiesprovideasoundbasisforexploitingvariousformsofincremen- tal computation which maydeliver important efficiencygains intwo mainrespects. First, they enable (and characterize the limitsof) theapplicationof divide-and-conquerstrategies inthe designofalgorithms for computationalproblems in abstractargumentationframeworks.Asmostoftheseproblemsareintractableintheworst case,facingreduced-sizesub- problemsseparatelyandthencombiningthepartialresultsinanefficientmannermaysignificantlyimproveperformances on theaverage.Second, thereis asignificant applicationinterestin argumentationdynamics,whichcapturesall contexts wherea givenframeworkisupdated incrementally,asaconsequenceoftheacquisition ofnewinformationand/or ofthe actionsoftheparticipantstoamulti-agentsystem.Clearly,ifthemodificationtotheinitialframeworkislimited,oneisin- terestedtopartiallyreusetheresultsofpreviouscomputationsinthenewframeworkratherthanredoingallcomputations fromscratch.Again,decomposabilitypropertiesenable(andcharacterizethelimitsof)theuseofincrementalcomputation techniquesbasedontheseparationbetweenmodifiedandunmodifiedpartsintheupdatedframework.

Ontheotherhand,thenotionsandpropertiesconcerningmultipoleequivalenceandsemanticstransparencyareappli- cable inallcontexts wherethere isan interestinreplacing apartofa framework withanother one.As anexample,the activities ofsummarizationandexplanationinvolvedinreasoningandcommunicatingatdifferentlevelsofgranularityare, basically, alternative forms ofreplacement. In the former,a complex part ofan argumentationprocess (e.g. the analysis anddiscussion offactual evidencesina legalcase) issummarized(i.e.replaced)by amore syntheticrepresentation(e.g.

focusing onthefacts whichturnout tohavean actualimpactonthecasedecision)which,whileleavingoutunnecessary details,mustensurethattheglobaloutcomeispreserved.Dually,explanationcanberegardedasthereplacementofasyn- theticrepresentationwitha moredetailed/articulated one,againensuring thatthisdoesnotinduce undesiredside-effects outsidethereplacedpart.Further,andmorespecific oftheabstractargumentationfield,the basicformalism ofargumen- tation frameworksis oftenused asa “groundlevel” representation forother richerand/or morespecific formalisms.For instance,formalismsinvolvingtheexplicitrepresentationofpreferences,values,andattackstoattackscanbetranslated(or flattened)tothebasicformalismthroughsuitableprocedures.Astheseprocedurestypicallyconsistofasetoflocalreplace- mentrules,multipoleequivalenceandsemanticstransparencyareveryeffectivetoolstoanalyzetheir behavior,soundness andapplicabilityundervarioussemantics.

Thepaperisorganizedasfollows.AfterrecallingthenecessarybackgroundinSection2,thegeneralnotionsconcerning semanticsdecomposabilityareintroducedanddiscussedinSection 3,whileSection4providesdecomposabilityresultsfor thesevensemanticsconsideredinthispaper.Section5dealswiththekeytechnicalnotionofeffect-dictatedsemanticsand Section 6thenintroducesthefundamentalconcepts concerninginterchangeability,namelyargumentationmultipoles,their Input/Output equivalence,thereplacement operator andthepropertiesof semanticstransparency. Section 7analyzes the relationships betweendecomposability andtransparencyata generallevel,whileSection 8provides transparencyresults forthesevensemanticsconsideredinthispaper.Application examplesaregiveninSection9,Section 10discussesrelated worksand,finally,Section11concludesthepaper.AllproofsaregiveninAppendix A.

(

Args

,

att

∩ (

^Args

×

^Args

))

.

Inthispaperweusethelabelling-based approachtothedefinitionofargumentationsemantics.Asshownin[21,3],for the semanticsconsidered inthis paperthereis a directcorrespondencewiththe “traditional”extension-based approach, hencetheresultspresentedinthispaperarevalidinbothapproaches. Thelabelling-baseddefinitionshavebeenadopted onlybecausetheyallowsimplerproofs.

Alabellingassignstoeachargumentofanargumentationframeworkalabeltakenfromapredefinedset

Λ

.Fortechnical reasons,wedefinelabellingsbothforargumentationframeworksandforarbitrarysetsofarguments.

Definition2.Let

Λ

beasetoflabels.GivenasetofargumentsArgs,alabelling ofArgs isatotalfunctionLab

:

^Args

−→ Λ

^. Thesetofalllabellings of Args isdenotedas

L

Args.Givenanargumentationframework AF

= (

^Ar

,

att

)

,alabelling ofAF isa labellingofAr.Thesetofalllabellings ofAF isdenotedas

L(

AF

)

.Foralabelling Lab ofArgs,therestrictionofLab toaset ofargumentsArgs^

⊆

^{Args,denotedasLab}

↓

Args^,isdefinedasLab

∩ (

^Args

× Λ)

^.

We adopt the most common choice for

Λ

, i.e.

{in, out, undec }

^{, where the label}

in

means that the argument is accepted,thelabel

out

meansthattheargumentisrejected, andthelabel

undec

meansthatthestatusoftheargument isundecided.AsexplainedafterDefinition 8,anexception ismadeforstablesemantics,which canbemoreconveniently definedassuming

Λ = {in, out }

^{.GivenalabellingLab,wewrite}

in(

Lab

)

for

{

^A

|

^Lab

(

A

) = in}

^,

out(

Lab

)

for

{

^A

|

^Lab

(

A

) = out }

^and

undec(

Lab

)

for

{

^A

|

^Lab

(

A

) = undec}

^.

Alabelling-basedsemanticsprescribesasetoflabellingsforeachargumentationframework.

Definition3.GivenanargumentationframeworkAF

= (

^Ar

,

att

)

,alabelling-basedsemanticsS associateswithAF asubsetof

L(

AF

)

,denotedasLS

(

AF

)

.

In general, a semantics encompasses a set of alternative labellings for a single argumentation framework. However, a semantics may be definedso that a unique labelling isalways prescribed,i.e. forevery argumentation framework AF,

|

^LS

(

AF

)| =

^{1.Inthiscasethesemanticsissaidtobe}single-status,whileinthegeneralcaseitissaidtobemultiple-status.

Inthelabelling-basedapproach,asemanticsdefinitionreliesonsomelegality constraintsrelatingthelabelofan argu- menttothoseofitsattackers.

Definition4.LetLab bealabellingoftheargumentationframework

(

Ar

,

att

)

.An

in

-labelledargumentissaidtobelegally

in

iffall itsattackersare labelled

out

.An

out

-labelledargumentissaidtobelegally

out

iffithasatleastoneattacker thatislabelled

in

.An

undec

-labelledargumentissaidtobelegally

undec

iffnotallitsattackersarelabelled

out

andit doesnothaveanattackerthatislabelled

in

.

Wenowintroducethedefinitionsoflabellingscorrespondingtotraditionaladmissible² andcompletesemantics.

Definition5. Let AF

= (

^Ar

,

att

)

be an argumentation framework. An admissiblelabelling is a labelling Lab where every

in

-labelledargumentislegally

in

andevery

out

-labelledargumentislegally

out

.

Definition6.Acompletelabelling isalabellingwhereevery

in

-labelledargumentislegally

in

,every

out

-labelledargument islegally

out

andevery

undec

-labelledargumentislegally

undec

.

On this basis, the labelling-based definitions of several argumentation semantics can be introduced. To simplify the technicaltreatmentinthefollowing,groundedandpreferredsemanticsaredefinedbyreferringtothecommitmentrelation betweenlabellings[3].

2 Itcanberemarkedthat(unliketheothersemantics)admissiblelabellingsarenotinaone-to-onecorrespondencetoadmissiblesetssinceseveral admissiblelabellingsmightcorrespondtothesameadmissibleset.

A preferredlabelling ofAF is amaximal (w.r.t.

^{) labelling among all complete}labellings. Theideallabelling of AF isthe maximal (under

) complete³ labelling Lab that islessorequally committedthan each preferredlabelling of AF (i.e.for each preferred labelling Lab_P it holds that Lab

^LabP). A semi-stable labelling of AF is a complete labelling Lab where

undec (

Lab

)

isminimal(w.r.t.setinclusion)amongallcompletelabellings.

Whilestablesemanticsisdefinedbyassuming

Λ = {in, out, undec }

^{,thedefinitionofstablelabellingentailsthatstable} semantics canbe equivalentlydefinedwithreferenceto thesetoflabels

Λ = {in, out }

^{.Inthiscase,astablelabelling is} simply acomplete labelling,since thecodomain

Λ

doesnotinclude

undec

.In thesequel weimplicitlyassume that,for stablesemanticsonly,

Λ = {in, out }

^{:thisallowsasimplertreatmentofsuchsemanticswithoutanylossof}generality.

Theuniquenessofthegroundedandtheideallabellinghasbeenprovedin[22].Accordingly,groundedandidealseman- tics aresingle-status,the othersemantics aremultiple-status.Admissible, complete,stable,grounded,preferred, idealand semi-stablesemanticsaredenotedinthefollowingasAD,CO,ST,GR,PR,ID andSST,respectively.

Wealsorecallthetraditionalnotionsofskepticalandcredulousjustificationofanargumentwithrespecttoasemantics.

Definition9.Givenalabelling-basedsemanticsS andanargumentationframeworkAF,anargument A isskepticallyjustified underS if

∀

^Lab

∈

^LS

(

AF

)

Lab

(

A

) = in

^{;anargument A is}credulouslyjustified underS if

∃

^Lab

∈

^LS

(

AF

) :

^Lab

(

A

) = in

^.

Finally,acommentisinorderonaspecialcaseofargumentationframeworkthatisexplicitlyconsideredinthepaper, i.e. the empty argumentation framework AF_∅

 (∅, ∅)

^{. By definition the only possible labelling of AF}_∅ ^{is the empty set,} thus asemanticscan eitherprescribe

∅

^forAF∅ oritcanprescribenolabelling atall.Inthisrespect,foranysemanticsS introduced aboveitholdsLS

(

AF_∅

) = {∅}

^{,i.e.theemptysetisactuallyprescribedby S.Noteinparticularthat}

∅

^isastable labelling,sinceitiscompleteanddoesnotinclude

undec

-labelledarguments.

3. Decomposabilityofargumentationsemantics 3.1. Thenotionoflocalfunction

The first stepto define thenotion ofsemantics decomposabilityis tointroduce a formal settingtoexpressthe inter- actions between thepartial frameworksinduced by an arbitrary partitioningofan argumentation framework. Intuitively, givenanargumentationframeworkAF

= (

^Ar

,

att

)

andasubsetArgs ofitsarguments,theelementsaffectingAF

↓

Argsinclude theargumentsattackingArgs fromtheoutside,calledinput arguments,andtheattackrelationfromtheinputargumentsto Args,calledconditioningrelation.

Definition10.GivenAF

= (

^Ar

,

att

)

andasetArgs

⊆

^{Ar,theinput ofArgs,denotedasArgsinp,istheset}

{

^B

∈

^Ar

\

^Args

| ∃

^A

∈

Args

, (

B

,

A

) ∈

^att

}

^,theconditioningrelation ofArgs,denotedasArgs^R,isdefinedasatt

∩ (

^Argsinp

×

^Args

)

.

Example1. ConsiderAF

= ({

^A

,

B

,

C

,

D

}, {(

^A

,

B

), (

B

,

C

), (

C

,

A

), (

A

,

D

), (

D

,

A

)})

^{withreferenceto thepartialframeworksin-} ducedbythesets

{

^A

,

B

,

C

}

^and

{

^D

}

^{(seeFig. 1).Itholdsthat}

{

^A

,

B

,

C

}

^inp

= {

^D

}

^and

{

^A

,

B

,

C

}

^R

= {(

^D

,

A

)}

^,while

{

^D

}

^inp

= {

^A

}

and

{

^D

}

^R

= {(

^A

,

D

) }

^.

Given a partialargumentation framework AF

↓

Args (possibly AF itself)affected by a (possibly empty) setof arguments Args^inp attacking Args according to Args^R, onemay wonder whetherfixing thelabelling assigned tothe input arguments allowsonetodeterminethesetoflabellingsofAF

↓

Args.Asshowninthefollowing,thisquestioncannotbeansweredonce andforall,sincedifferentsemanticsexhibitdifferentbehaviors inthisrespect,and,forsomesemantics,adependencyholds underspecific constraintson theconsideredpartitionoftheargumentationframework.Inordertoexpresssucha depen- dency(wheneveritholds),weintroducethenotionsofargumentationframeworkwithinput,consistingofanargumentation frameworkAF

= (

^Ar

,

att

)

(playingtheroleofapartialargumentationframework),asetofexternalinputarguments

I

^{,a la-} bellingL_I assignedtothemandanattackrelationR_I from

I

^{toAr,andofalocal}function which,givenanargumentation frameworkwithinput,returnsacorrespondingsetoflabellingsofAF.

3 Literally,theoriginaldefinitionreferstoanadmissiblelabellingratherthanacompletelabelling.However,thedefinitionadoptedhereisequivalentto theoriginalone,sinceitcanbeshownthattheideallabellingisacompletelabelling[22].

Fig. 1. Running example: a partition of a simple framework (Examples 1–5).

Definition11.An argumentationframeworkwithinput is atuple

(

AF

, I ,

L_I

,

R_I

)

,includingan argumentationframework⁴ AF

= (

^Ar

,

att

)

,asetofarguments

I

^suchthat

I ∩

^Ar

= ∅

^,alabellingL_I

∈ L

_I ^{andarelationR}_I

⊆ I ×

^{Ar.Alocalfunction} assigns to anyargumentation framework withinput a (possibly empty) set oflabellings of AF, i.e. F

(

AF

, I ,

^L_I

,

R_I

) ∈

2^L(AF).

Foranysemantics,a “sensible”localfunction, calledcanonicallocalfunction,isthe onethat describesthelabellingsof theso-calledstandardargumentationframeworks.

Definition12.Givenanargumentationframeworkwithinput

(

AF

, I ,

^L_I

,

R_I

)

,thestandardargumentationframeworkw.r.t.

(

AF

, I ,

^L_I

,

R_I

)

isdefinedasAF^

= (

^Ar

∪ I

^

,

att

∪

^R_I

)

,where

I

^

= I ∪ {

^A

|

^A

∈ out(

^L_I

) }

^andR_I

=

^R_I

∪ {(

^A

,

A

) |

A

∈ out(

^L_I

) } ∪ {(

^A

,

A

) |

^A

∈ undec(

^L_I

) }

^.

Roughly,thestandardargumentationframeworkputsAF undertheinfluenceof

(I ,

L_I

,

R_I

)

,byadding

I

toAr andR_I toatt,andbyenforcing⁵ thelabelL_I fortheargumentsof

I

^inthisway:

•

^{foreachargumentA}

∈ I

^{suchthat L}_I

(

A

) = out

^{,anunattackedargumentAisincludedwhichattacksA,inorderto} get A labelled

out

byalllabellingsofAF^;

•

^{foreach argument A}

∈ I

^{such that L}_I

(

A

) = undec

^,aself-attackisaddedto A inordertogetitlabelled

undec

by alllabellingsofAF^;

•

^{eachargumentA}

∈ I

^{suchthat L}_I

(

A

) = in

^isleftunattacked,sothatitislabelled

in

byalllabellingsofAF^. Definition 13. Given a semantics S, the canonical localfunction of S (also called local function of S) is defined as F_S

(

AF

, I ,

^L_I

,

R_I

) = {

^Lab

↓

Ar

|

^Lab

∈

^LS

(

AF^

)}

^,whereAF

= (

^Ar

,

att

)

andAF^ isthestandardargumentationframeworkw.r.t.

(

AF

, I ,

^L_I

,

R_I

)

.

Notethatinthecaseofstablesemanticsundec

∈ Λ /

^,thusR_I ^{doesnotinclude}self-attacks.

Incase

I = ∅

^{(entailing L}_I

= ∅

^{and R}_I

= ∅

^{) the canonical localfunction returnsthe labellingsofAF, asshownby} Proposition 1.

Proposition1.GivenasemanticsS andanargumentationframeworkAF,F_S

(

AF

, ∅, ∅, ∅) =

^LS

(

AF

)

.

While the canonical local function is defined for any semantics, its definition is best suited for complete-compatible semantics,i.e.semanticssatisfyinganumberofintuitiveconstraints.

Definition14.AsemanticsS iscomplete-compatible iffthefollowingconditionshold:

1. ForanyargumentationframeworkAF

= (

^Ar

,

att

)

,everylabellingL

∈

^LS

(

AF

)

satisfiesthefollowingconditions:

•

^ifA

∈

^{Ar isinitial,thenL}

(

A

) = in

•

^ifB

∈

^{Ar andthereisaninitialargumentA whichattacks B,thenL}

(

B

) = out

•

ifC

∈

Ar isself-attacking,andtherearenoattackersofC besidesC itself,thenL

(

C

) = undec

2. for anysetofarguments

I

^{andanylabelling L}_I

∈ L

_I^,theargumentation frameworkAF^

= (I

^

,

att^

)

, where

I

^

= I ∪ {

^A

|

^A

∈ out(

^L_I

) }

^andatt

= {(

^A

,

A

) |

^A

∈ out(

^L_I

) } ∪ {(

^A

,

A

) |

^A

∈ undec(

^L_I

) }

^{,admitsa(unique)labelling,i.e.}

|

^LS

(

AF^

) | =

^1.

4 Inthefollowing,unlessotherwisespecified,wewillimplicitlyassumeAF= (^Ar,att).

5 Actually,theenforcementisabitdifferentforadmissiblesemantics.Thisexceptionhasnoconsequencesonthetechnicaldevelopmentofthepaper.

Fig. 2. The standard argumentation framework w.r.t.(AF↓{A,B,C},{D}, {(D, out)}, {(D,A)})(Example 2).

It should benotedthat, incase

undec ∈ Λ /

^{,the thirdbulletof condition 1entailsthat thereisno labelling ifaself-} attackingargumentC isattackedbyC only,andincondition2itnecessarilyholdsthat

undec(

L_I

) = ∅

^.

AsshownbyProposition 2,therequirementsofthepreviousdefinitionguaranteethattheconstructionofthestandard argumentationframeworkmakessense,i.e.givenastandardargumentationframeworkw.r.t.

(

AF

, I ,

^L_I

,

R_I

)

,acomplete- compatiblesemanticsenforcesthelabellingL_I fortheargumentsof

I

^{asdescribedabove.}

Proposition2.LetS beacomplete-compatiblesemanticsandletAF^

= (

^Ar

∪ I

^

,

att

∪

^R_I

)

bethestandardargumentationframe- workw.r.t.anargumentationframeworkwithinput

(

AF

, I ,

^L_I

,

R_I

)

.ThenforanyLab

∈

^LS

(

AF^

)

itholdsthatLab

↓

_I^

= {(

^A

, in) |

A

∈ out(

^L_I

) } ∪

^L_I ^andLab

↓

_I

=

^L_I^.

Moreover,whenappliedtotheemptyargumentationframework(whichbydefinitiondoesnotreceiveattacksfrom

I

⁾ thecanonicallocalfunctionofacomplete-compatiblesemanticsalwaysreturnstheemptysetasauniquelabelling.

Proposition 3. Given a complete-compatible semantics S, a set of arguments

I

^{and a labelling L}_I

∈ L

_I^{, it holds that} FS

(

AF_∅

, I ,

^L_I

, ∅) = {∅}

^.

Taking into account Proposition 1 thisresult entails that LS

(

AF_∅

) = {∅}

^, corresponding to the second requirement of Definition 14with

I = ∅

^.

Allthesemanticsconsideredinthepaperarecomplete-compatible,withtheexceptionofadmissiblesemantics.

Proposition4.GR

,

CO

,

ST

,

PR

,

SST

,

ID areallcomplete-compatiblesemantics.

Admissible semantics is not complete-compatible, asit canbe seen by considering e.g. the argumentationframework AF

= ({

^A

}, ∅)

^,whereLAD

(

AF

) = {(

^A

, undec), (

A

, in) }

^.

Thefollowingexampleclarifiesthenotionofcanonicallocalfunction,consideringinparticularcompletesemantics.

Example2. Letusreferagain tothe argumentationframework AF of Fig. 1.Forthe canonical localfunction ofcomplete semanticsitholdsthat FCO

(

AF

↓

{^A,B,C}

, {

^D

}, {(

^D

, out) }, {(

^D

,

A

) }) = {{(

^A

, undec), (

B

, undec), (

C

, undec) }}

^{,duetothefact} that the standard argumentation framework w.r.t.

(

AF

↓

{A,B,C}

, {

^D

}, {(

^D

, out )}, {(

^D

,

A

)})

^{, shown in Fig. 2, admits as the} uniquecompletelabelling

{(

^D

, in), (

D

, out), (

A

, undec), (

B

, undec), (

C

, undec) }

^{.Inasimilarway,itiseasytoshowthat} FCO

(

AF

↓

{^A,B,C}

, {

^D

}, {(

^D

, in) }, {(

^D

,

A

) }) = {{(

^A

, out), (

B

, in), (

C

, out) }}

^andFCO

(

AF

↓

{^A,B,C}

, {

^D

}, {(

^D

, undec) }, {(

^D

,

A

) }) = {{(

^A

, undec ), (

B

, undec ), (

C

, undec )}}

^. Considering the application of F_CO to AF

↓

{D}, F_CO

(

AF

↓

{D}

, {

^A

}, {(

^A

, out )}, {(

^A

,

D

) }) = {{(

^D

, in) }}

^{, F}CO

(

AF

↓

{^D}

, {

^A

}, {(

^A

, in) }, {(

^A

,

D

) }) = {{(

^D

, out) }}

^{and F}CO

(

AF

↓

{^D}

, {

^A

}, {(

^A

, undec) }, {(

^A

,

D

) }) = {{(

^D

, undec) }}

^.

As showninSection 4,foranysemantics consideredinthispaperthelocalfunctionadmitsacompactrepresentation, withouttheneedtorefertostandardargumentationframeworks.

3.2. Decomposabilitypropertiesofargumentationsemantics

We now aimat introducing a formal notion ofsemantics decomposability. To thispurpose, consider a generic argu- mentation frameworkAF

= (

^Ar

,

att

)

andanarbitrarypartition ofAr, i.e.aset

{

^P1

, . . . ,

Pn

}

^suchthat

∀

ⁱ

∈ {

¹

, . . . ,

n

}

^Pi

⊆

^Ar and Pi

= ∅

^,



i=1...nPi

=

^{Ar and P}i

∩

^Pj

= ∅

^fori

=

^{j.Suchapartition identifiestherestricted}argumentationframeworks AF

↓

P1

, . . . ,

AF

↓

Pn,thataffecteachother withtherelevantinputargumentsandconditioningrelationsasstatedinDefini- tion 10.Intuitively asemanticsS isdecomposableifS canbeputincorrespondencewithalocalfunction F suchthat:

•

^{every labelling prescribed by S on AF, namely every element ofL}S

(

AF

)

, corresponds to the union ofn “compatible”

labellingsLP₁

, . . . ,

LPn oftherestrictedargumentationframeworks,allofthemobtainedapplyingF ;

Definition15. A semantics S is fullydecomposable (or simply decomposable) iffthere is a local function F such that for every argumentation framework AF

= (

^Ar

,

att

)

andevery partition

P = {

P1

, . . . ,

Pn

}

^{of Ar, L}S

(

AF

) = U (P,

AF

,

F

)

where

U (P,

^AF

,

F

)  {

^LP1

∪ . . . ∪

^LPn

|

^LPi

∈

^F

(

AF

↓

Pi

,

P_i^inp

, ( 

j=¹...n,j=ⁱL_Pj

)↓

_Pi^inp

,

P_i^R

)}

^.

Example3.Consideringagaintheargumentationframework AF ofFig. 1andthepartition

{{

^A

,

B

,

C

}, {

^D

}}

^{,fulldecompos-} ability of complete semantics requires a local function such that the labellings of AF are exactly those obtained by the unionofthecompatiblelabellingsofAF

↓

{^A,B,C} andAF

↓

{^D} givenbythelocalfunctionitself.Letusconsiderthecanonical localfunction⁶ of CO (referto Example 2). Thelabelling

{(

^A

, out), (

B

, in), (

C

, out) }

^{iscompatible with}

{(

^D

, in) }

^,since thefirstisobtainedby F_COwithD labelled

in

,andthelatterisobtainedby F_CO withA labelled

out

.Ontheotherhand, thelabelling

{(

^A

, out), (

B

, in), (

C

, out) }

^{isnotcompatiblee.g.with}

{(

^D

, out) }

^{.Overall,exactlytwogloballabellingsarise} fromthecombinationsofthecompatibleoutcomesof FCO,namely

{(

^A

, undec), (

B

, undec), (

C

, undec), (

D

, undec) }

^and

{(

^A

, out), (

B

, in), (

C

, out), (

D

, in)}

^,correspondingtothecompletelabellingsofAF.

Thebehaviorofcompletesemanticsinthisexampleisnotincidental:wewillproveinSection4thatcompletesemantics isfullydecomposable.

Proposition 5showsthat,ifacomplete-compatiblesemanticsS isfullydecomposable,thenthelocalfunctionappearing inDefinition 15coincideswiththecanonicallocalfunctionF_S.

Proposition5.Givenacomplete-compatiblesemanticsS,ifS isfullydecomposablethenthereisauniquelocalfunctionsatisfyingthe conditionsofDefinition 15,coincidingwiththecanonicallocalfunctionFS.

Full decomposability can be viewed as the conjunction of two partial decomposability properties, namely top-down decomposability andbottom-up decomposability.

Inwords, a semanticsis top-down decomposableifthe procedureto compute theglobal labellingsidentified byDef- inition 15 is complete, i.e.all of the globallabellings can be obtainedby combiningthe labellings prescribed by F_S for the restrictedsubframeworks, even if puttingtogether labellings of the restrictedsubframeworks may give rise to some

“spurious”labellingsbesidesthecorrectones.Thefollowingdefinitionformalizesthisintuition.

Definition16. A complete-compatible semantics S is top-downdecomposable iff for anyargumentation framework AF

= (

Ar

,

att

)

andanypartition

P = {

^P1

, . . . ,

P_n

}

^{ofAr,itholdsthatL}S

(

AF

) ⊆ U (P,

^AF

,

F_S

)

.

Whiletop-down decomposabilitycorrespondsto completenessoftheprocedureidentifiedby Definition 15,bottom-up decomposability requires its soundness, i.e.that anycombination oflocal labellings is a global labelling, while it is not guaranteedthatallgloballabellingscanbeobtainedinthisway.

Definition17. A complete-compatiblesemantics S isbottom-updecomposable iff forany argumentation framework AF

= (

Ar

,

att

)

andanypartition

P = {

^P1

, . . . ,

Pn

}

^{ofAr,itholdsthatL}S

(

AF

) ⊇ U (P,

^AF

,

FS

)

.

Acommentonthetwodefinitionsaboveisinorder.Whilethedefinitionoffulldecomposabilityappliestoanykindof semanticsandrequirestheexistenceofalocalfunctionsatisfyingthedecomposabilityproperty,Definitions 16 and17are restrictedtocomplete-compatiblesemanticsandrefertothecanonicallocalfunctionFStoavoidtriviality:thelocalfunction returningallthepossiblelabellingsofAF triviallysatisfiestheinclusionconditionofDefinition 16foranysemantics,while thelocalfunction always returningtheempty settrivially satisfiesthe conditionofDefinition 17.Thisis thereasonwhy bothdefinitionsrefertothespecificcanonicallocalfunction,whichmakessenseforcomplete-compatiblesemanticsinthe lightofProposition 5.Ifasemanticsisnotcomplete-compatible⁷thenthenotionofcanonicallocalfunctionismeaningless, sincethelabelling L_I wouldnotbeingeneralenforcedfortheargumentsof

I

^{inthestandard}argumentationframework w.r.t.

(

AF

, I ,

^L_I

,

R_I

)

(seeProposition 2).

6 ItisshowninProposition 5thatconsideringthecanonicallocalfunctioniswithoutlossofgenerality.

7 Besidesadmissiblesemantics,intheliteraturethereareafewexamplesofnon-complete-compatiblesemantics,likestagesemantics[42]andvarious formsofprudentsemantics[24].

F = (

ingasetofpartitionsofAr.

Apartition selectorisdefinedasafunctionofargumentationframeworks,since differentargumentationframeworkswith thesamesetofargumentsmayallowdifferentsetsofpartitions,dependingontheattackrelation.

The decomposabilitynotions introduced sofarcan then beextended totake intoaccount aspecific restrictionon the consideredpartitions.

Definition 19. Let

F

^{be a partition selector. A} complete-compatible semantics S is top-down decomposablew.r.t.

F

^iff for anyargumentation framework AF and anypartition

P = {

^P1

, . . . ,

Pn

} ∈ F (

^AF

)

, it holdsthat LS

(

AF

) ⊆ U (P,

^AF

,

FS

)

. A complete-compatible semantics S is bottom-updecomposablew.r.t.

F

^{iff forany} argumentationframework AF and any partition

{

^P1

, . . . ,

Pn

} ∈ F (

AF

)

, LS

(

AF

) ⊇ U (P,

AF

,

FS

)

.Acomplete-compatiblesemantics isfullydecomposable (orsimply decomposable)w.r.t.apartitionselector

F

^{iffitisbothtop-downandbottom-up}decomposablew.r.t.

F

^.

Of course, full decomposability, top-down decomposability and bottom-up decomposability as introduced in Defini- tions 15,16 and17,respectively,areequivalenttothecorrespondingdecomposabilitypropertiesw.r.t.

F

ALL,i.e.theselector returningallpossiblepartitions.

Definition20.ForanyargumentationframeworkAF

= (

^Ar

,

att

)

,

F

ALL

(

AF

)  {{

^P1

, . . . ,

Pn

} | {

^P1

, . . . ,

Pn

}

is a partition of Ar

}

^.

Apartfromthislimitcase,aparticularpartitionselectorthathasreceivedattentionintheliteratureandwillbeconsid- eredinthispaperistheonebasedonthenotionofstronglyconnectedcomponent(SCC)ofanargumentationframework.

Its importanceisduetothefactthatmostargumentationsemanticsintheliteratureare SCC-recursive[10],which,briefly, means that the semantics can be defined interms ofa basefunction operatingat the levelof single strongly connected components. Roughly, this also implies that an incremental computation procedure based on the decomposition of the frameworkintoitsstronglyconnectedcomponentscanbedefined,apropertyexploitedinseveralsubsequentworks[33,40, 23].Hereweintroducethenecessarybasicdefinitions,leavingfurtherdiscussiononthissubjecttoSection10.

Definition21.GivenanargumentationframeworkAF

= (

^Ar

,

att

)

,thesetofstronglyconnectedcomponentsofAF,denotedas SCCS_AF,consistsoftheequivalenceclassesofargumentsinducedbythebinaryrelationofpath-equivalence,i.e.therelation

ρ (

A

,

B

)

definedoverAr

×

^{Ar suchthat}

ρ (

A

,

B

)

holdsifandonlyifA

=

^{B ortherearedirectedpathsfromA to B and from} B to A inAF.

For instance,the argumentationframework of Fig. 1has aunique strongly connectedcomponent includingall of the arguments,whilefortheargumentationframeworkAF ofFig. 2itholdsthatSCCSAF

= {{

^D

}, {

^D

}, {

^A

,

B

,

C

}}

^.

At leasttwopartition selectors basedonstronglyconnectedcomponentscan beconsidered. Thesimplestselector,de- noted as

F

SCC,includes foreach argumentationframework AF the unique partition consistingofthestrongly connected componentsSCCS_AF.Asecond selector,denotedas

F

∪SCC,includesallthepartitionssuch thateveryelementistheunion ofsome(possiblyunconnected)stronglyconnectedcomponents.

Definition 22. For any argumentation framework AF

= (

^Ar

,

att

)

,

F

SCC

(

AF

)  {

^SCCSAF

} \ {∅}

^,

F

_∪SCC

(

AF

)  {{

^P1

, . . . ,

Pn

} | {

^P1

, . . . ,

P_n

}

is a partition of Ar and

∀

ⁱ

((

S

∈

^SCCSAF

∧

^Pi

∩

^S

= ∅) →

^S

⊆

^Pi

})

^.

It isimmediatetoseethat,foranyAF,

F

SCC

(

AF

) ⊆ F

∪SCC

(

AF

)

.As tothefirstpartofthe definition,note thatthe set SCCSAF includes

∅

^onlyincaseAF

=

^AF∅,whichdoesnotadmitanypartition(sincealltheelementsofapartitionmustbe nonempty),thus

F

SCC

(

AF_∅

) = ∅

^.

4. Analyzingsemanticsdecomposability

In thissection wediscussthe decomposabilitypropertiesofthe semanticsreviewedin Section 2.Asyntheticview of theresultsisgiveninTable 1(notethatforallsemanticsfull,top-downandbottom-updecomposabilityw.r.t.

F

∪SCC turn outtobesatisfiedifandonlyiffull,top-downandbottom-updecomposabilityw.r.t.

F

SCC aresatisfied,respectively).Since admissiblesemanticsisnotcomplete-compatible,onlythenotionoffulldecomposabilityisapplicabletoit.

4.1. Admissibleandcompletesemantics

Wefirstanalyzeadmissibleandcompletesemantics,sincetheyarethebasisfortheotheronesconsideredinthispaper:

accordingtoDefinition 8,stable,grounded,preferred,ideal,andsemi-stablesemanticsselectlabellingsamongthecomplete ones,whichareadmissiblebydefinition.Giventhis,itwouldbeveryunpleasantifcomplete(andthusadmissible)semantics wouldnotbedecomposable.AsshownbyTheorems 1 and 3,luckily bothadmissibleandcompletesemanticsturnoutto befullydecomposable.

Thefollowingdefinitionintroduces thecanonicallocalfunctionofadmissiblesemantics,byextendingthedefinitionof admissiblelabellinginordertoaccountfor“external”inputargumentsintheobviousway.Theproofthatthedefinitionis correctisprovidedbyTheorem 2.

Definition 23. Given an argumentation framework with input

(

AF

, I ,

L_I

,

R_I

)

, FAD

(

AF

, I ,

L_I

,

R_I

)  {

^Lab

∈ L(

^AF

) |

Lab

(

A

) = in → ((∀

^B

∈

^Ar

: (

^B

,

A

) ∈

^att

,

Lab

(

B

) = out) ∧ (∀

^B

∈ I : (

^B

,

A

) ∈

^R_I

,

L_I

(

B

) = out)),

^Lab

(

A

) = out → ((∃

^B

∈

Ar

: (

^B

,

A

) ∈

^att

∧

^Lab

(

B

) = in) ∨ (∃

^B

∈ I : (

^B

,

A

) ∈

^R_I

∧

^L_I

(

B

) = in))}

^.

Theorem 1 provesthat admissiblesemantics isfullydecomposable,showingthat thelocalfunction FAD introduced in Definition 23satisfiestheconditionsofDefinition 15.

Theorem1.AdmissiblesemanticsAD isfullydecomposable,withFADsatisfyingtheconditionsofDefinition 15.

The following theorem confirms that Definition 23 actually corresponds to the canonical local function ofadmissible semantics.

Theorem2.ThecanonicallocalfunctionofadmissiblesemanticsisFAD,asdefinedinDefinition 23.

Also the canonical local function of complete semantics can be guessed on the basis of the definition of complete labelling.

Definition 24. Given an argumentation framework with input

(

AF

, I ,

^L_I

,

R_I

)

, F_CO

(

AF

, I ,

^L_I

,

R_I

)  {

^Lab

∈ L(

^AF

) |

Lab

(

A

) = in → ((∀

^B

∈

^Ar

: (

^B

,

A

) ∈

^att

,

Lab

(

B

) = out) ∧ (∀

^B

∈ I : (

^B

,

A

) ∈

^R_I

,

L_I

(

B

) = out)),

^Lab

(

A

) = out → ((∃

^B

∈

^Ar

: (

B

,

A

) ∈

^att

∧

^Lab

(

B

) = in) ∨ (∃

^B

∈ I : (

^B

,

A

) ∈

^R_I

∧

^L_I

(

B

) = in)),

^Lab

(

A

) = undec → (((∀

^B

∈

^Ar

: (

^B

,

A

) ∈

^att

,

Lab

(

B

) =

in) ∧ (∀

^B

∈ I : (

B

,

A

) ∈

^R_I

,

L_I

(

B

) = in)) ∧ ((∃

^B

∈

^Ar

: (

^B

,

A

) ∈

^att

∧

^Lab

(

B

) = undec) ∨ (∃

^B

∈ I : (

B

,

A

) ∈

^R_I

∧

^L_I

(

B

) = undec)))}

^.

Itis easytosee that FCO

(

AF

, I ,

^L_I

,

R_I

) ⊆

^FAD

(

AF

, I ,

^L_I

,

R_I

)

, i.e.every “locallycomplete”labelling isalso“locally admissible”.

Theorem 3showsthatalsocompletesemanticsisfullydecomposable.⁸ Sincetheproofadopts F_CO asthelocalfunction and CO is complete-compatible, by Proposition 5 it holds that FCO is actually the canonical local function of complete semantics.

Theorem3.CompletesemanticsCO isfullydecomposableandF_COisitscanonicallocalfunction.

4.2. Stablesemantics

Stable semantics inheritsfulldecomposability fromcompletesemantics: the reasonisthat the definitionofstablela- bellingcorrespondstothatofcompletelabellingwiththeadditionalrequirementthatnoargumentislabelled

undec

,and

8 Proposition3of[40]provesaweakerpropertyofcompletesemantics,correspondingtobottom-updecomposabilityintheextension-basedapproach.