an introduction to abstract argumentation - epcl · an introduction to abstract argumentation ......

An introduction toAbstract Argumentation

Pietro BaroniDII - Dip. di Ingegneria dell’Informazione

University of Brescia (Italy)

EPCL Basic Training Camp 2013

EPCL BTC 2013 – Abstract Argumentation – P. Baroni

RoadmapPART I: Basics

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)

PART II: Advanced

Taking topology seriouslyDifferences vs commonalities: semantics agreementComparing argumentationframeworksSkepticism and related propertiesA richer notion of justificationstatusSkepticism-related criteria for semanticsSatisfying all criteriaComputational issuesBeyond Dung’s framework


Roadmap

What’s in the words ?


Abstract (for good)

General, independent of specific details,able tocapture a variety of situations(MW: disassociated from any specific instance, theoretical)In computer science, abstraction is the process by which data and programs are defined with a representation similar in form to its meaning (semantics), while hiding away the implementation details. Abstraction tries to reduce and factor out details so that the programmer can focus on a few concepts at a time. A system can have several abstraction layers whereby different meanings and amounts of detail are exposed to the programmer (Wikipedia).


Abstract (for bad)

Poorly related (or totally unrelated) with realproblems(MW: insufficiently factual, difficult to understand, abstruse)


Argumentation (for what?)

Argumentation is a multi-faceted word, with a variety of informal/intuitive and also formalmeaningsThe abstraction process detaches the word from some/most/all of its meanings and properties, keeping only those required by the desiredabstraction level (and possibly adding other ones)

Abstract arguments are not arguments


Roadmap

What’s in the words?From the Big Bang to now


The Big BangLess than 20 years ago

Dung’s 1995 paper on Artificial Intelligence JournalTheory of Abstract Argumentation (AA) FrameworksBasic notions related to Abstract Argumentation Semantics(conflict freeness, admissibility, 4 “traditional” semantics: complete, grounded, stable, preferred)Ability to capture a variety of other (less abstract) formalcontexts:» default logic» logic programming with negation as failure» defeasible reasoning» N-person games» stable-marriage problem


The explosion

Not an easy to read paper, but a very general (and basically simple) formalism with ideas and resultsregarded as extremely powerfulEnormous impact on the literature (675 citations ISI, 1098 Scopus, 2041 Google Scholar)Probably the most cited paper in the computationalargumentation literatureOriginated entire new research lines: manyfollowers (and some criticisms)


The AA universe

Abstractargumentationframework

“Traditional” semanticsnotions

Extensions/variationsof the definition of the basic framework (and relevant semantics)

“New” semanticsFormalisms and properties for old and new semantics


A note on notation

There is no universally adopted notation even forthe most basic conceptsThe slides include excerpts taken directly from the original papers and reflect all these differencesShould not be too mysterious …… some notes along the wayfor any doubt, please ask


Roadmap

What’s in the words?From the Big Bang to nowDung’s framework


Dung’s framework is(almost) nothing

A directed graph (called defeat graph) where:» arcs are interpreted as attacks » nodes are called arguments “by chance” (let say historical

reasons)


Dung’s framework is(almost) nothing

α β

β

α γ


Dung’s framework is(almost) everything

Arguments are simply “conflictables”

Conflicts are everywhereConflict management is a fundamental need with potential spectacular/miserable failures both in real life and in formal contexts (e.g. in classical logic)A general abstract framework centered on conflictshas a wide range of potential applications


Dung’s framework is(almost) everything

The pervasiveness of Dung’s framework and semantics is witnessed by the correspondencesdrawn in the original paper with a variety of otherformal contextsMany extensions and variations of Dung’s framework allow a translation procedure back to the original framework to exploit its basic features


A conflict calculus: abstractargumentation semantics

A way to identify sets of arguments “surviving the conflict together” given the conflict relation only

In general, several choices of sets of “survivingarguments” are possible

Two main styles for semantics definition: extension-based and labelling-basedThese points will be discussed extensively after some “user instructions”


AA user manual1. Identify an interesting application domain where conflict

management plays a key role2. Define a suitable formalization of problem instances in the

selected domain3. Define the notions of argument and attack in your

formalization, i.e. a translation/abstraction method from problem instances to argumentation frameworks

4. Play at your will with “conflict calculus” at abstract level5. Map back the results of “conflict calculus” (extensions or

labellings) into entities at the problem level6. Are they meaningful? Do they provide useful/original

perspectives? Did you avoid to reinvent the wheel?


An example:the stable marriage problem

Steps 1 and 2



Step 3



Steps 4 and 5



Step 6



Step 6

The notion of preferred extension provides a naturalsolution to a non traditional version of the marriageproblem


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics


Abstract argumentationsemantics

A way to identify sets of arguments “surviving the conflict together” given the conflict relation only

Two main styles for semantics definition: extension-based and labelling-based

In general, several choices of sets of “surviving arguments” are possible (multiple-status semantics) but some semantics prescribe exactly one extension/labelling (single status semantics)


Extension-based semantics

A set of extensions is identified

Each extension is a set of arguments which can “survive together” or are “collectively acceptable” i.e. represent a reasonable viewpoint

The justification status of each argument can bedefined on the basis of its extension membership» skeptical justification = membership in all extensions» credulous justification = membership in one extension


Sets of extensions

α β E1 = {{α},{β}}

β

α γ

E1 = {{α},{β},{γ}}E2 = {∅}

E2 = {∅}


Labelling-based semantics

A set of labels is defined (e.g. IN, OUT, UNDECIDED) and criteria for assigning labels toarguments are given

Several alternative labellings are possible

The justification status of each argument can bedefined on the basis of its labels



αΙΝ

βOUT

αOUT

βIN

L1

αUND

βUND

L2

α β



βOUT

αIN

γUND

βIN

αUND

γOUT

βUND

αOUT

γIN

L1



βUND

αUND

γUND

L2


Labellings vs. extensions

Labellings based on {IN, OUT, UNDEC} and extensions can be put in direct correspondence

Given a labelling L, LabToExt(L) = in(L)

Given an extension E, a labelling L=ExtToLab(E) can be defined as follows:in(L)=Eout(L)=attacked(E)undec(L)=all other arguments



α β



αΙΝ β



αΙΝ

βOUT



αΙΝ

βOUT

α β



αΙΝ

βOUT

αOUT

βIN

L1



α β



αUND

βUND

L2



αΙΝ

βOUT

αOUT

βIN

αUND

βUND

L1

L2



βOUT

αIN

γUND

βIN

αUND

γOUT

βUND

αOUT

γIN

L1



βUND

αUND

γUND

L2


Defining argumentation semanticsMany different proposals in the literature, corresponding to different intuitionsThere are cases where each literature semanticsgives a different outcomeHow (and in how many ways) would you colour thisgraph?

η

ζ ι

θα βγ δ ε κ


Analyzing argumentationsemantics

A catalogue of literature semantics is notimmediately useful since different semanticsdefinitions don’t follow a common pattern and are not directly comparableTo start, examining a set of general propertiesdriving the definition (or marking the differences) of argumentation semantics is more usefulFocus on extension-based definitionWe will look at the semantics catalogue after that


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics properties


Conflict freeness

The conflict free principle states that an attackingand an attacked argument can not stay together

Notational remarkusually denotes the set of extensions

prescribed by semantics for the argumentationframework


Conflict freeness

The conflict free principle states that an attackingand an attacked argument can not stay together

Very basic idea, followed by all semanticsMinimal use of the attack relation (in fact the attackdirection does not count)The empty set is conflict free by definition


Conflict freeness

α β

α β

α β

α β


Conflict freeness

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α γ

β

α γ

β

α γ

β

α γ

β

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I-maximality

A given criterion can select sets which are in a relation of proper inclusion (this holds for conflictfreeness and for more articulated notions)One may consider as a constraint that no extensionis a proper subset of another one

Important for some properties (e.g. to avoid trivialityin the notion of skeptical justification) but notfundamental and poorly related with other principles


Defense related principles:admissibility

To survive the conflict you should be able to defendyourself, namely to reply to every attack with a counterattackA conflict free set is admissible if it defends itself

The attack direction countsThe empty set is admissible by definition


Admissibility

α β

α β

α β

α β


Admissibility

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α γ

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α γ

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Admissibility from a labellingperspective

You should have good reasons to assign IN or OUT labels, but you are free to be undecidedAll arguments UNDECIDED is an “admissible” labelling, as the empty set is always an admissibleset


Defense related principles:strong admissibility/defense

Admissibility includes self-defenseA stronger notion of defense requires that defensecomes from other arguments which are in turn strongly defended by other arguments

Rather strong requirement: defense chains rootedin unattacked argumentsThe empty set is strongly defended by definition


Strong defense

α β

α β

α β

α β


Strong defense

β

α γ

β

α γ

β

α γ

β

α γ

β

α γ

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α γ


Strong defense

α β γ

δε


Strong defense

αΙΝ

βOUT

γIN

δUND

εUND


Defense related principles:reinstatement

Reinstatement concerns effectiveness (or altruism) of defense: if you defend some argument youshould take it on board (include it in the extension)

Completeness requirement: can not leave out yourprotected onesThe empty set satisfies reinstatement only if there are no unattacked arguments


Reinstatement

α β γ

δε


Defense related principles:weak reinstatement

A weaker notion of reinstatement is obtainedconsidering strong defense instead of defense


Defense related principles:CF reinstatement

Reinstatement does not require explicitly conflictfreeness with the defended argument. Adding thisgives rise to CF reinstatement


Directionality

Basic idea: arguments affect each other following the direction of attacks. An argument (or set of arguments) is affected onlyby its ancestors in the attack relation


Directionality

In words the restrictions of the extensions to anunattacked part of the graph are the same whateveris the remaining part of the graph (if any)Local extensions (or labellings) are included in global extensions*

*Directionality implies that you may compute the restrictions ofextensions to unattacked sets without considering the rest of the graph, itdoes not however imply that there is a way to proceed incrementally tocompute the extensions for the rest of the graph


Directionality

α β γ

δε


Directionality

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ε

α

ε


Directionality

α β γ

δε

α β γ

δε


Directionality

α β

γε


Directionality

α β

γε

α β

γε

α β

γε


A hidden principle: language independence

Extensions only depend on the attack relation, i.e. on the graph topology not on argument “names” or on other underlying propertiesIsomorphic frameworks have the same (modulo the isomorphism) extensions


Implicit equal treatmentThe language independence principle implies“equal opportunity” for arguments, i.e. that“topologically equivalent” arguments are treated in the same wayNo semantics can prescribe an “asymmetric” set of extensions for a “symmetric” framework

α β


Implicit equal treatment

If there are reasons to prefer α to β, they should bereflected in the topology

α β


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semantics


Conflict-freeness + maximality=naive semantics

The so-called naive semantics prescribes asextensions the maximal conflict free sets of a frameworkAs already remarked, it ignores the attack direction an so does not reflect the whole amount of availableinformationIt blatantly violates several basic principles likeadmissibility and reinstatement (even for unattackedarguments)In a sense, it is not really a semantics and is notconsidered in Dung’s paper


Naive semantics

α β γ

α β γ

α β γ


Can naive semantics work?Naive semantics makes sense when the direction of the attacks does not count: symmetric frameworksIn any symmetric framework maximal conflict free-sets are reasonable extensions for any multiple-status semanticsIf the underlying “instantiated” formalism gives rise only to symmetric frameworks more sophisticated semantics notions do not make any differenceLimited utility of semantics notions or limitedexpressiveness of the underlying formalism?


A bit less naive: stage semantics

Stage semantics prescribes as extensions the conflict free sets with maximal range, i.e. thosemaximizing the union of the extension with the arguments it attacks (attack direction “partiallycounts”)

Some “extremely naive” behavior of naivesemantics are avoided, but there are still caseswhere unattacked arguments are excluded fromsome extensions


α β γ

α β γ not a stageextension

α β γ

α β γ

A bit less naive: stage semantics


Defense based semantics

Dung’s paper introduces several notions and semantics all based on the notion of defense(admissibility principle) + conflict-freenessThe direction of attacks counts!


Complete semantics

Adds to admissibility the reinstatement principle: ifyou defend an argument you have to include it

The empty set is complete only if there are no unattacked argumentsCompleteness does not imply I-maximality: a complete extension can be a proper subset of another one (obtained adding some self-defendingset of arguments)


Admissibility and completeness

α β γ

δε

Complete extensions: { {}, {ε,β}, {α,γ,δ} }

Admissible sets: { {}, {α}, {ε}, {α,γ}, {α,δ}, {ε,β}, {α,γ,δ} }



α β γ

δε

{α,γ} is admissible but not complete


α β γ

δε


Complete extensions: {{α,γ}, {α,γ,δ}, {α,γ,ε} }

Admissible sets:{ {}, {α}, {δ}, {ε}, {α,γ}, {α,δ}, {α,ε}, {γ,δ}, {α,γ,δ}, {α,γ,ε} }


α β γ

δε


{α,γ} is admissible and complete


α β γ

δε


{α,γ,δ} is also admissible and complete


Grounded semantics:accepting only the unquestionable

Given that complete extensions satisfy some basic properties, if one wants to be cautious s/he has tochoose the smallest (wrt inclusion) complete extensionActually, it can be proved that there is a uniquesmallest complete extension, called groundedextensionSo grounded semantics belongs to the unique-status approach


Grounded semantics:unquestionable=strong defense

Dung’s definition of grounded semantics takesanother routeFirst, the characteristic function of a frameworkassociates with a set S the arguments it defends


The characteristic function is well-behaved

This entails that it has a least fixed point, which, bydefinition, is called grounded extension

It can be proved that the least fixed point of the characteristic function coincides with the leastcomplete extension



This has a rather intuitive counterpart: start computing the arguments defended by the emptyset, repeat the computation (i.e. find the argumentsdefended by them), and so on, until you reach a fixed pointFor finitary frameworks the grounded extension can be obtained as:FAF(FAF(… FAF(∅)…)



Computing the grounded extension

α β γ

δεζ

θη

ι


Horror vacui: stable semanticsStable semantics prescribes that any argument iseither in the extension or attacked by the extensionIn the labelling version: no argument is undecided

In general there are several stable extensionsAny stable extension is admissible and complete (hence it includes the grounded extension)


A stable extension

α β γ

δεζ

θη

ι


Another stable extension

α β γ

δεζ

θη

ι


Being bold is not always possible

The very strong requirement posed by stablesemantics in some cases is not satisfiableThe empty set is not the “most basic” extension(unless the framework is empty too)There are frameworks where no stable extensions existThe absence of odd-lengthcycles is a sufficient conditionfor existence of stable extensions

β

α γ


Being bold is not always possible

A limited (possibly isolated) part of the frameworkmay prevent the existence of extensions for the whole frameworkStable semantics is not directional

α β γ

β

α

γ δ


Accept as much as you can defend:preferred semantics

Admissibility + maximality = preferred semantics

From a labelling perspective this corresponds to maximizing IN without banning UNDECPreferred extensions always exist (in general many)Any preferred extension is complete (hence itincludes the grounded extension) Stable extensions are preferred, not viceversa


Preferred vs. stable semantics

α β

δ

γ ε


Preferred vs. stable semantics

α β

δ

γ ε

α β

δ

γ ε


Minimizing indecision:semi-stable semantics

Maximizing acceptance does not mean minimizingindecision: to this purpose you should maximize the union of IN and OUTStable semantics achieves this implicitly by banning UNDEC, semi-stable semantics does this explicitly without banning UNDEC


Semi-stable extensions always exist (in generalmany)When stable extensions exist, semi-stable extensions coincide with stable extensionsSemi-stable extensions are preferred, not viceversaSemi-stable semantics is not as “fragile” as stable semantics, but it is non directional too

Minimizing indecision:semi-stable semantics


Single status from multiple status

A natural way to obtain a single status semanticsfrom a multiple status semantics consists in usingthe intersection Int of the extensionsFor all the semantics seen so far Int is a (possiblystrict) superset of the grounded extensionInt is conflict-free but not admissible in generalAdding the requirement of admissibility to the one of inclusion in all extensions gives rise to a generic“scheme” for semantics definition which has beenexplicitly considered in two cases


Ideal and eager semantics

The ideal extension is the maximal (wrt inclusion) admissible set included in all preferred extensionsThe eager extension is the maximal (wrt inclusion) admissible set included in all semi-stable extensionsBoth are supersets of the grounded extensionThe eager extension is a superset of the ideal extension


A less cautious single statusβ

α

γ δ

β

α

γ δ

The ideal/eagerextension is {δ}, while the groundedextension is empty


A less cautious single statusThe ideal/eagerextension is {α}, while the groundedextension is empty

α β

The eagerextension is {β,δ}, the ideal extensionis empty

α β

δ

γ ε


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)


Universal and almost universal properties

Conflict freeness and language independence are satisfied by all semanticsI-maximality is satisfied by single-status semanticsand is directly implied by several multiple-statusdefinitions, only complete semantics is not I-maximal


“Conflict-free only” semantics

Naive and stage extensions can exclude unattackedarguments and include undefended argumentshence they can not satisfy admissibility, strong defense, reinstatement, weak reinstatement, and directionalityThey only satisfy CF-reinstatement


“Complete-based” semantics

Grounded, stable, preferred, semi-stable, ideal and eager extensions are complete extensionsThis implies that they satisfy admissibility and reinstatement (including weaker forms)Strong defense is only satisfied by groundedsemantics


Directionality

Disproving directionality is immediate for stable, semi-stable and eager semanticsProving directionality is less immediate (butpossible) for the other complete-based semantics


A synopsis

YYYYYYYYYLanguage indip.

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STA

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CO

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GR

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STB

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PR

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SST

NYNDirectionality

YYNWeak reinstatement

Y

Y

N

Y

Y

Y

EAG

YYCF-principle

YYCF-reinstatement

YNReinstatement

NNStrong defense

YNAdmissibility

YYI-maximality

IDNA


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)Taking topology seriously


Following the graph topology

Directionality suggests that the graph topology“encodes” a form of dependency among argumentsThe intuitive algorithm for computing the groundedextension starts from unattacked arguments and proceeds orderly following attacks as far as possible… but it immediately terminates if no unattackedarguments exists (does not manage cyclicdependences)Is there a topological order to follow in presence of cycles? (and useful for other semantics too?)


Strongly Connected Components

In a graph the relation of mutual reachability is anequivalence relation (also called path equivalence)The relevant equivalence classes are called Strongly Connected Components (SCCs): intuitively they are islands (or clouds) of mutual dependenceThe graph obtained considering SCCs as single nodes is acyclic, i.e. there is an acyclic dependencerelation among different SCCsThe SCC decomposition of a graph can becomputed in polynomial time


α β

δ

γ ε

Strongly Connected Components


A partial order for evaluation


A partial order for evaluation

S1

S2

S3

S5

S6

S7

S4


A partial order for evaluation: level 1

S1

S2

S3

S5

S6

S7

S4


S1

S2

S3

S5

S6

S7

S4



SCC-recursiveness

Basic idea: a semantics is SCC-recursive if itsextensions can be built following the partial order of SCCsSCC-recursiveness requires two ingredients:» a semantics-specific base function to be applied to the

single SCCs starting from the initial ones and then alongthe way

» a generic SCC-recursive scheme for the application of the base function along the way

Since the two ingredients are intertwined with some intricacies, let us proceed informally


1. Extensions “are built” following the SCC order

2. A semantics-specific base function BF is given, whichdetermines the set of extensions prescribed for asingle-SCC argumentation framework

3. For any initial SCC S, the base function BF is appliedto determine the possible values of E ∩ S for anyextension E to be built by adding elements from other SCCs

SCC-recursiveness informally


Suppose that applying the base function BF on initial SCCs itemerges that

There is exactly one choice for E ∩ S1

S1

There is one choice for E ∩ S2(actually the empty set)

(in general multiple choices are possible)

S2

SCC-recursiveness at work: initial SCCs


4. For any possible choice of E ∩ S, every subsequent SCC S’ ispartitioned into three subsets






Given the choice of E ( ) in the preceding SCCs, the nodesof a SCC S are partitioned into– DAF(S, E) : directly attacked by E – UAF(S, E) : defended from outside attacks by E

– PAF (S, E) : not attacked by E nor defended from outside attacks by E

S1

S3

S5

S6

S7

S4

S2

Attack, extensions and SCCs


4. For any possible choice of E ∩ S, every subsequent SCC S’ ispartitioned in three subsets

5. Nodes in DAF(S’, E) are suppressed and the construction of Eproceeds by applying recursively the same steps on the remaining part of S’.The distinction between PAF (S, E) and UAF(S, E) may betaken into account in this step.






S1

S3

S5

S6

S7

S4

S2

SCC-recursiveness at work: propagating choices


S3

S5

The restriction of S3consists of a single SCC: the base function is appliedand there is exactly one choice for E ∩ S3restricted S3 restricted S3

restricted S5

The restriction of S5 consists of three SCCs: the SCC recursivescheme is applied starting from the first one. The difference between“green” and “yellow” nodes may also be taken into account. Assume that it turns out that there is exactly one choice for E ∩ S5

restricted S5

SCC-recursiveness at work: recursion on restricted SCCs


S1

S3

S5

S6

S7

S4

or ...

S7

S7

S2

SCC-recursiveness at work: reaching final SCCs


SCC-recursiveness formally

The second parameter of BF and GF identifies the “green” nodeswith respect to “yellow” nodes

The set of extensions is given by a generic function GFparametric with respect to the base function BFGF is recursive with respect to SCC decomposition

Single-SCC argumentation frameworks are the base of recursion


Defining a SCC-recursive semantics simply amounts to definea base function BF

Simple requirements on the base function BF are sufficientto ensure reasonable results:

• If BF is conflict free the resulting extensions are conflictfree

• If BF treats “correctly” an argumentation frameworkconsisting of a single non self-defeating node, the resulting extensions are all supersets of the groundedextension

SCC-recursiveness is well-founded


The four traditional Dung’s semantics (complete, grounded, stable, preferred) are SCC-recursiveSCC-recursiveness + universal existence of extensions implies directionalityStable semantics is SCC-recursive since stableextensions may not existOther non directional semantics are not SCC-recursiveThe reverse does not hold: ideal semantics isdirectional but not SCC-recursive

SCC-recursiveness in the wild


Ideal semantics is notSCC-recursive

β

α

γ δ ε

Ideal extension = ∅

β

α

γ δ ε

Ideal extension = {ε}

The base functionshould give differentresults with the same input for the same SCC


SCC-recursiveness for “easy” semantics definition

Defining a semantics simply amounts to define a base function for single SCCsFour semantics were defined in the SCC-recursiveness paperAmong them CF2 semantics has the mostinteresting properties


The base function is very simple: it selects the maximal conflict free sets within the SCC

It departs from the notion of admissibility

“Symmetric” treatment of odd- and even-length cycles

2 extensions

3 extensions

CF2 semantics


A motivation for symmetric treatment of cycles

Unreliable witnesses example (taken from Pollock)Witness A says that witness B is unreliable, witnessB says that witness C is unreliable, witness C saysthat witness A is unreliable. All witnesses say that the statement S is false.Witness A says that witness B is unreliable, witness B says that witness C is unreliable, witness C says that witness D is unreliable, witness D says that witness A is unreliable. All witnesses say that the statement S is false.



Should the number of witnesses make anydifference on the justification status of an argumentinvolving the statement S?In fact an odd or even number makes a difference according to stable, semi-stable, and preferred semantics (and to any admissibility-based multiple-status semantics)It might not be what you want: different semanticsare appropriate for different contexts and needs


Unreliabilityclaims amongwitnesses

Consonantwitnesses vs.claims

A dissonantclaim



Unreliabilityclaims amongwitnesses

Consonantwitnesses vs.claims

A dissonantclaim


The dissonant claim is skeptically acceptedwith three witnesses



The dissonant claim is NOT skeptically acceptedwith four witnesses


When AF consists of a single SCC, CF2 extensions coincidewith maximal conflict free sets of AF

In general, it turns out that a CF2 extension is a maximal conflict free set of AF but not vice versa.

S1

S2

CF2 semantics


CF2 semantics

First choice in the first SCC


Three extensions based on the first choice in the first SCC

CF2 semantics


A maximal CF set whichis not a CF2 extension

CF2 semanticsSecond choice in the first SCC Only one extension based

on the second choice


Evaluating CF2 semantics

Like naive or stage semantics CF2 is mainly basedon conflict-freeness and trades off the “symmetric” treatment of odd- and even-length cycles withadmissibility (and reinstatement)However thanks to the SCC-recursive scheme ithas other properties (directionality, weakreinstatement) that naive or stage semantics fail toachieve


Evaluating CF2 semantics

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YYYYYYYYYLanguage indip.

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CF2

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STB

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PR

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SST

NYNDirectionality

YYNWeak reinstatement

Y

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N

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Y

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EAG

YYCF-principle

YYCF-reinstatement

YNReinstatement

NNStrong defense

YNAdmissibility

YYI-maximality

IDNA


A “natural” development

Recently stage2 semantics has been proposedIt embeds the notion of stage extension in the SCC-recursive schemeSame gains, as in CF2 semantics, and “more maximization”It fixes some counterintuitive behavior of CF2 semantics for instance with “long cycles”


Stage2 vs. CF2Stage2 and CF2 extensions Other CF2 extensions


Roadmap

What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreement


Why are they so many?

Different intuitionsDifferent needsHow to “patch” a counterintuitive behavior in a previous semantics? Just define a new semantics!How to get some papers published? Just define a new semantics!Parametric definition schemes offer new opportunities of exploration of the semantics “design space”


Why are they all different?If it is not different it does not get publishedMany questions in semantics design/behavior admitreasonable alternative answers e.g.Is self-defense sufficient? (Grounded NO, Others YES)Is undecision admitted?(Stable NO, Others YES)Are multiple alternatives possible?(Grounded, Ideal, Eager NO, Others YES)Should odd- and even-length cycles be treated “equally”?(Grounded, CF2, Stage YES, Others NO)


Why are they all similar?

Different semantics are not behaving differently in all cases: they can’t, because of» very basic constraints» common (possibly implicit) underlying principles


Obvious commonalities

The empty argumentation framework: AF = <∅,∅>

Of course, for any semantics S, ES(AF) = ∅

For any pair of semantics S1 and S2 there is at leastone argumentation framework where S1 and S2agree


Obvious commonalities …

The “monadic” argumentation framework: AF = <α,∅>

“Of course”, for any semantics S, ES(AF) = {α}A “valid” argument not interacting with any otherargument can not be rejected (basic reinstatement)

α



The “multi-monadic” argumentation framework: AF = <Α,∅>

For any semantics S, ES(AF) = A = {α,β,γ,...}A “valid” argument not interacting with any other argument can not be rejected (and conflicts consistonly in explicit attack relations): let say the “unattacked-in principle” (basic reinstatement)

α β γ …



“Attack chain” argumentation framework

For any semantics S, ES(AF) = {α, γ}Several principles come into play with this verysimple example: “unattacked-in”, “admissibility”, “reinstatement” (and conflicts consist only in explicitattack relations)

α β γ



Acyclic argumentation framework

δ γ β α

ε


Obvious commonalities?Acyclic argumentation framework

For any semantics S, ES(AF) = {α, δ, ε}NO! The grounded prudent extension (we did not introduce) is {δ, ε}. α is not included due to indirect conflict with δ“conflicts consist only in explicit attack relations” does nothold in prudent semantics

δ γ β α

ε


Why caring about agreement?

Theory:» Identification of “universally accepted” behaviors» Confirmation of already identified principles and “reverse

engineering” of further principles

Practice» Actual relevance of choice among semantics in specific

contexts» Classification of application contexts according to their

agreement properties


Formalizing agreement

Two semantics S1 and S2 are in agreement about an argumentation framework AF if ES1(AF) = ES2(AF) We require that both S1 and S2 admit extensions for AF, namely ES1(AF) ≠ ∅ and ES2(AF) ≠ ∅

In other words, for a semantics S, agreement is evaluated only about argumentation frameworks where S admits at least one (possibly empty) extensionOnly stable semantics may not admit extensions on a given AF


Two main research lines

Identifying topological families of argumentationframeworks ensuring agreement among some semantics

Systematic identification of all possible agreement classes (given a reference set of semantics) using general set-theoretical properties of semantics extensions rather than topological properties of argumentation frameworks


Simple topological families: acyclic frameworks

If AF is well-founded (i.e. acyclic when AF is finite) grounded, stable and preferred semantics agreeThe absence of cycles enforces a “single-status” behavior of multiple-status semantics: grounded extension is the “natural and univocal” extension in these cases Cycles as generators of alternative extensionsAlmost out of discussion (except prudent semantics)Further results extend this agreement to other semantics


Simple topological families: symmetric frameworks

Naive, stable, stage, semi-stable, preferred, and CF2 semantics are in agreement about AF if AF is symmetric (all attacks are mutual) Symmetry of attacks (in absence of self-defeating arguments) prevents any difference among multiple-status semantics: all maximal conflict-free sets satisfy “any” requirementSymmetry leaves open only the choice between single-status and multiple-status


Simple (to state) topological families: frameworks without odd-length cycles

Stable and preferred semantics are in agreement about AF if AF is limited controversial (i.e. free of odd-length cycles when AF is finite) [Dung 95]Odd-length cycles justify the distinction between stable and preferred semanticsOdd-length cycles are the reason for non existence of stable extensions in some cases or for “missing” stable extensions in other onesThe treatment of odd-length and even-length cycles is unequal in Dung’s multiple status semantics


Simple (to compute) families: determined frameworks

A framework is determined when its groundedextension is also stable (this can be checked in polynomial time)There are no undecided (“provisionally defeated”) arguments according to grounded semantics

Determined Not determined


An argumentation framework AF is determined if and only if itis initial acyclic, namely AF is empty or

the initial SCCs of AF are monadicthe restriction of AF to the nodes not included in andnot attacked by initial SCCs is in turn initial acyclic

Dung’s sufficient condition of acyclicity is included

S1 S1

S2S2

S3

A topological characterization of determined arg. frameworks


An argumentation framework AF is determined if and only if itis initial acyclic, namely AF is empty or


Initial acyclic

S1 S2

S3S’1

S’2



An argumentation framework AF is determined if and only if itis initial acyclic, namely is is empty or


Initial acyclic Not initial acyclic

S’1S’1

S’2



Single-status agreement

Determined frameworks are a superset of acyclicframeworksExploiting SCC-recursiveness, agreement withgrounded semantics is extended to a larger family of frameworks and a larger set of semantics


If AF is determined any SCC-recursive semantics agreeswith grounded semantics provided that:

– is conflict free– treats “reasonably” any monadic AF

A monadic AF

α

The requirement on

Uniform single status behavior of a large family of reasonablesemantics (including classical ones) ondetermined argumentation frameworks



Combining this result with some “easy” observationson “well-known” properties of inclusion relation and coincidence among extensions of various semanticsit turns out that the following semantics agree on determined argumentation frameworks:Grounded, Complete, Preferred, Stable, Semi-stable, Stage, Ideal, CF2, Resolution-basedgrounded*


*to be introduced later


Simple (to compute) families: almost determined frameworks

α

AF is not determined because

Unability to prescribe extensions is anapparent cause of disagreement

Any semantics able to prescribe extensions should agree withgrounded semantics also in this kind of case


Simple (to compute) families: almost determined frameworksAn argumentation framework is almost determinedif any argument not included in and not attacked by thegrounded extension is self-defeating

Any SCC-recursive semantics which:is universally definedis conflict free“reasonably” treats monadic argumentation frameworks

is in agreement with grounded semantics on any almost determined argumentation framework

Larger class of argumentation frameworks providing a necessary and sufficient condition for agreement with grounded semantics of universally definedSCC-recursive semantics


Combining this result with some “easy” observationson “well-known” properties of inclusion relation and coincidence among extensions of various semanticsit turns out that the following semantics agree on almost determined argumentation frameworks:Grounded, Complete, Preferred, Semi-stable, Ideal, CF2, Resolution-based groundedStable semantics is missing for obvious reasonsStage semantics is missing for less obviousreasons: in presence of self-defeating arguments itmay exclude unattacked arguments

Single-status agreement (reprise)


Exploiting symmetry with SCCs

A SCC-recursive semantics is *-symmetric if for any

AF which is symmetric and free of self-defeating arguments

coincides with the set of maximal (wrt inclusion)

conflict free sets of AF

Several reasonable multiple-status semantics (includingpreferred, stable and CF2) are *-symmetric

Not extremely interesting per se but ...


Simple (to compute) families: SCC-symmetric frameworks

An argumentation framework is SCC-symmetricif its restriction to any of its SCCs is symmetric

In other words, non mutual attacks are possible, but any cycleof AF involves mutual attacks only

S1

Not SCC-symmetric

S2

S1

S2

SCC-symmetric


Multiple-status agreementAll *-symmetric semantics are in agreement on anyargumentation framework SCC-symmetric and free ofself-defeating arguments

In particular, they are in agreement with the stable semantics

In particular, preferred and CF2 semantics are in agreement

The class of SCC-symmetric argumentation frameworksis equivalent to other classes considered in the literature:

symmetric conflict + transitive preferencerebutting (à la Pollock) defeat only

This suggests that this class may be relevant in practicalapplications


SCC-symmetric vs. limited controversial

While the result on determined argumentationframeworks subsumes the original result on acyclicframeworks the result on SCC-symmetricframeworks complements the original result on limited controversial frameworks


SCC-symmetric vs. limited controversial

SCC-symmetricand limited controversial

SCC-symmetricnot limited controversial

Limited controversialnot SCC-symmetric

Not SCC-symmetricnot limited controversial


Summing up on topologies ensuring agreement

Determined (and almost determined) argumentation frameworks:

– are probably not sufficient for most practical applications– provide a strong (single-status) reference behavior

for any new semantics proposal

SCC-symmetric argumentation frameworks:

– may be relevant in practice– in application domains where only SCC-symmetric

argumentation frameworks are present one should notbother about (multiple-status) semantics differences


Agreement classes

Given a set of argumentation semantics, the set of argumentation frameworks where all semantics inagree will be denoted as

E.g. denotes the set of argumentation frameworks (agreement class) where preferred, stable and semi-stable semantics agree

Identifying agreement classes provides the reference points for investigations on agreementAgreement classes may admit, but do not necessarily have, a topological characterization


Basic properties

E.g.

It may be (and it is) the case that for some different sets of semantics and it holds that


Agreement classes: how many?

Considering the set of 7 argumentation semantics

any subset of such that gives rise, in principle,

to an agreement class (120 classes in total)

It is proved that most of these 120 classes are not actually

different: only 14 distinct classes exist

The literature results I am aware of concern 7 semantics (naive,stage, eager are not included)


An example of agreement class coincidence

A* = AGR({GR,ST})When stable extensions exist they coincide with semi-stableextensions ⇒ A* = AGR({GR, ST, SST})Every stable extension is a preferred extension (maximal complete extension) and the grounded extension is the leastcomplete extension ⇒A* = AGR({GR, ST, SST, CO, PR})The ideal extension is a superset of the grounded extension and is included in any preferred extension⇒A* = AGR({GR, ST, SST, CO, PR, ID})Every CF2 extension is a superset of the grounded extension and is a maximal conflict-free set + every stable extension is a maximal conflict-free set ⇒A* = AGR({GR, ST, SST, CO, PR, ID,CF2})


Agreement classes:which kind of analysis?

Agreement classes are denoted as Σ1 … Σ14

We proceed by partial order of inclusion: if Σi ⊂ Σj then j > i

For each class Σi three main steps have been carried out:

1. identifying which classes Σk, with k<i, are included in Σi

2. for each of these classes Σk, showing that Σi \ Σk ≠ ∅3. for any Σh with h<i and Σh ⊄ Σi , examining Σi∩Σh

For any set of semantics not directly corresponding to any

of Σ1 … Σ14 it is shown that coincides with one of them


Several kinds of inclusion relations between (sets of) extensions:for any argumentation framework AF

•

•

•

•

•

•

Agreement classes: which (known) properties can be exploited?



•

•

•

•

•

•


Inclusion of the wholeset of extensions



•

•

•

•

•

•


The groundedextension is includedin many kinds of extensions



•

•

•

•

•

•


Any extension of one kindis included in an extensionof another kind


Several properties based on the inclusion relationships can be derived

Agreement classes: further properties we can prove


Several lemmata based on the inclusion relationships can be derived

Agreement classes: which kind of properties we prove?

Implications of cardinality




Implications of inclusionin the set of extensions




Some agreementsimply others




Implications of extensionproperties


Coincidence of agreement classes follows (almost directly) from the lemmata. Examples are:

Agreement classes: coincidence


GR=ST=PR=CF2=SST=ID=CO

GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

AF1

AF2

AF3

AF4

AF5

AF6

AF7

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8AF8 GR=ID

Agreement classes: unique-status behavior



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

AF1

AF2

AF3

AF4

AF5

AF6

AF7

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8AF8 GR=IDΣ1 is the class where all the considered semantics agree (in particular with GR)We already know it : determined argumentation frameworks

Agreement classes: GR unique-status behavior



GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

AF1

AF2

AF3

AF4

AF5

AF6

AF7

Σ1

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8AF8 GR=ID

Σ2 is the class where all the considered semantics agree (but ST may be undefined)We already know it : almost determined argumentation frameworks

GR=PR=CF2=SST=ID=COΣ2



GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

AF4

AF5

AF6

AF7

Σ4

Σ5

Σ6

Σ7

Σ8AF8 GR=IDΣ3 is the class where all but CF2 semantics agree (while ST may be undefined)


AF1

AF2

AF3

Σ1

Σ3




GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

AF4

AF5

AF6

AF7

Σ4

Σ5

Σ6

Σ7

Σ8AF8 GR=ID

Σ3 \ Σ2 ≠ ∅ as it includes AF3=


AF1

AF2

AF3

Σ1

Σ3


β

αγ



The grounded extension belongs to the set of complete extensions (is the least complete extension)

For all the considered semantics, except CO, it holds that no extension can be a proper subset of another extension

For all the considered semantics, any extension is a superset of the grounded extension

It follows that agreement with CO is possible for a multiple-status semantics only if also agreement with GR holdsAs a consequence CO only appears in agreement classes

Σ1,Σ2,Σ3

A note on complete semantics


AF4

AF5

AF6

AF7

Σ4

Σ5

Σ6

Σ7

Σ8 is the only other class of agreement involving GR

Examples of argumentation frameworks in Σ8 \ Σ7 will be given later

GR=PR=SST =ID=CO

Σ8AF8 GR=ID


AF1

AF2

AF3

Σ1

Σ3


PR=CF2=SST=ID

PR=CF2=ST=SST=IDPR=ST=SST=ID

PR=SST=ID



Grounded semantics specifies a single-status behaviorbased on strong defense (self-defense is not enough)Ideal semantics specifies a single-status behavior based on defense (admissibility) and inclusion in all preferred extensions (no attacks from other admissiblesets)A simple “topological fragment” realizes this difference

Complete semantics is outside these agreements asalready seen

Agreement classes: ID unique-status behavior

α β


AF5

AF6

AF7

Σ5

Σ6

Σ7

Σ4 is the class where all but CO and GR semantics agree

Σ4 \ Σ1 ≠ ∅ as it includes AF4=

Σ8AF8 GR=ID

AF2

AF3Σ3


PR=CF2=SST=ID

PR=CF2=ST=SST=IDPR=ST=SST=ID

PR=SST=ID

AF4

Σ4


AF1Σ1

α β

GR=PR=SST =ID=CO



AF6

AF7

Σ6

Σ7

Σ5 is the class where all but CO and GR semantics agree,while ST may be undefined

Σ5 \ (Σ4∪Σ2)≠∅ as it includes AF5=

Σ8AF8 GR=ID

AF3Σ3

PR=CF2=SST=ID

PR=SST=ID

α β

AF5Σ5

PR=CF2=ST=SST=ID

AF4

Σ4

GR=PR=SST =ID=CO

AF2



AF1Σ1

PR=ST=SST=ID

γ



AF7

Σ7

Σ6 is the class where all but CO, GR, and CF2 semantics agree(excluding CF2 requires a 3-length cycle)

Σ6 \ (Σ4∪Σ5)≠∅ as it includes AF6=

Σ8AF8 GR=ID

AF3Σ3PR=SST=ID

AF5Σ5 AF2


AF6

Σ6

PR=CF2=ST=SST=ID

AF4

Σ4


AF1Σ1

GR=PR=SST =ID=CO

PR=CF2=SST=ID

PR=ST=SST=ID

β

αγ



PR=SST=ID GR=PR=SST =ID=COΣ7

Σ7 is the class where only PR, SST, and ID semantics agree

Σ7 \ (Σ6∪Σ5 ∪Σ3)≠∅ as it includes AF7=

AF8 GR=ID

AF3Σ3

AF6

Σ6

PR=ST=SST=ID

α βδ

γε

PR=CF2=SST=ID

Σ8

PR=CF2=ST=SST=ID

AF4

Σ4

AF2



AF1Σ1

AF7

AF5Σ5




GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF9

AF9'

AF10 AF10'

AF11 AF11'

AF12 AF12'

AF13

AF13'

AF13''

AF13'''

AF14

AF14'

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ9

Σ10

Σ11

Σ12

Σ13

Σ14

GR=ID

Agreement classes: multiple status behavior



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF10 AF10'

AF11 AF11'

AF12 AF12'

AF13'

AF13'''

AF14

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ10

Σ11

Σ12

Σ13

Σ14

GR=ID

AF13 AF14'AF9

AF9'AF13''Σ9

PR=CF2=ST=SST

AF9



Σ9 is the class where all multiple-status semantics (except CO) agree. It includes SCC-symmetric frameworks

(Σ9 \ Σ4)∩Σ8 ≠∅ as it includes AF9

GE(AF9 ) = ID(AF9 ) = ∅

Σ9 \ (Σ4∪Σ8) ≠∅ as it includes AF9’

GE(AF9 ) = ∅ ; ID(AF9 ) = {δ}

β

αγ δ

β

αγ δ

The Σ9 class



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=ST=SST

∃ STST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF9

AF9'

AF10'

AF11 AF11'

AF12 AF12'

AF13

AF13'

AF13''

AF13'''

AF14

AF14'

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8Σ11

Σ12

Σ13

Σ14

GR=ID

Σ9

Σ10PR=CF2=SST

AF10



Σ10 is the class where all multiple-status semantics (except CO) agree on a multiple-status behavior but ST may be undefined

(Σ10 ∩Σ8) \ (Σ9∪Σ5) ≠∅ as it includes AF10

Σ9 \ (Σ9∪Σ5 ∪Σ8) ≠∅ as it includes AF10’

α β γ

δ ε

α β γ

The Σ10 class





α β γ

δ ε

α β γ

The Σ10 class

prevents the existence of stable extensions





α β γ

δ ε

α β γ

The Σ10 class

prevents the existence of stable extensions

makes a difference between GR and ID



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF9

AF9'

AF10 AF10'

AF11

AF12 AF12'

AF13

AF13'

AF13''

AF13'''

AF14

AF14'

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ10

Σ11

Σ12

Σ13

Σ14

GR=ID

Σ9

AF11'



Σ11 is the class where only traditional PR and ST (and hence SST) agree while CF2 may differ

Σ11\ (Σ6∪Σ9 ∪Σ8) ≠∅ as it includes AF11

(Σ11 ∩Σ8) \ (Σ6∪Σ9) ≠∅ as it includes AF11’

δ ε

β

αγ

β

α γ

δ

The Σ11 class

GE(AF11) = ∅ ; ID(AF11) = {α}

GE(AF11) = ID(AF11) = ∅EPR(AF11) = EST(AF11) = {{α},{δ}}ECF2(AF11) = {{α},{β},{γ},{δ}}


Σ11 is the class where only traditional PR and ST (and hence SST) agree while CF2 may differ


(Σ11 ∩Σ8) \ (Σ6∪Σ9) ≠∅ as it includes AF11’

δ ε

β

αγ

β

α γ

δ

The Σ11 class

GE(AF11) = ∅ ; ID(AF11) = {α}makes a

difference with CF2

GE(AF11) = ID(AF11) = ∅EPR(AF11) = EST(AF11) = {{α},{δ}}ECF2(AF11) = {{α},{β},{γ},{δ}}



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=CF2

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF9

AF9'

AF10 AF10'

AF11

AF13

AF13'

AF13''

AF13'''

AF14

AF14'

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ10

Σ11

Σ13

Σ14

GR=ID

Σ9

AF11'

AF12

Σ12

PR=SST

AF12'



Σ12 is the class where PR and SST agree and may differ from CF2 while ST is undefined

(Σ12∩Σ8) \ (Σ7∪Σ10 ∪Σ11) ≠∅ as it includes AF12

Σ12\ (Σ7∪Σ10 ∪Σ11∪Σ8) ≠∅ as it includes AF12’

α β γ

εδ

α β γ

εδ ζ η

The Σ12 class


Σ12 is the class where PR and SST agree and may differ from CF2 while ST is undefined

(Σ12∩Σ8) \ (Σ7∪Σ10 ∪Σ11) ≠∅ as it includes AF12

Σ12\ (Σ7∪Σ10 ∪Σ11∪Σ8) ≠∅ as it includes AF12’

α β γ

εδ

α β γ

εδ ζ η

The Σ12 class

The play continues combining elementary “difference fragments”



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF9

AF9'

AF10 AF10'

AF11 AF11'

AF12 AF12'

AF14

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ9

Σ10

Σ11

Σ12

Σ14

∃ STST=SST

AF13'''Σ13

AF8’

AF13

AF13'

AF14'

Σ8 GR=ID

AF13''



Σ13 is the class where ST is defined (and hence agrees with SST) and may differ from any other

Σ13 has articulated intersections with Σ14 to be examined later

The Σ13 class



GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF8’

AF9

AF9'

AF10 AF10'

AF11 AF11'

AF12 AF12'

AF13

AF13'

AF13''

AF13'''

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ9

Σ10

Σ11

Σ12

Σ13

GR=ID

AF14'

PR=CF2

AF14Σ14



Σ14 is the class corresponding to agreement between the last pair to be considered: PR and CF2

Let us now examine the distinct regions related to Σ13 and

Σ14 in the Venn diagram

(Σ13\ Σ12)∩Σ8∩Σ14 ≠∅ as it includes AF13

(Σ13\ Σ12)∩(Σ8 \ Σ14) ≠∅ as it includes AF13’

α βγ

α βγε

δ η

ζ

The Σ14 class – Regions of interest


(Σ13\ Σ12)∩(Σ14 \ Σ8) ≠∅ as it includes AF13’’

Σ13\ (Σ8∪Σ12 ∪Σ14) ≠∅ as it includes AF13’’’

α βγ

η

ζ ι

θ

δ ε

α βγ δ ε




(Σ14 ∩ Σ8) \ (Σ12 ∪Σ13) ≠∅ as it includes AF14’’

α βγ δ ε ζ

α βγ δ




GR=PR=CF2=SST=ID=CO

GR=PR=SST =ID=CO

PR=CF2=ST=SST=ID

PR=CF2=SST=ID

PR=ST=SST=ID

PR=SST=ID

PR=CF2=ST=SST

PR=CF2=SST

PR=ST=SST

∃ STST=SST

PR=CF2

PR=SST

AF1

AF2

AF3

AF4

AF5

AF6

AF7

AF9

AF9'

AF10 AF10'

AF11 AF11'

AF12 AF12'

AF13

AF13'

AF13''

AF13'''

AF14

AF14'

AF15

Σ1

Σ2

Σ3

Σ4

Σ5

Σ6

Σ7

Σ8

Σ9

Σ10

Σ11

Σ12

Σ13

Σ14

AF8’

GR=ID



There are argumentation frameworks where GR and IDagree while all other semantics disagree (and ST is undefined)


α βγε

δ η

ζθ

Σ8 and nothing else


There are argumentation frameworks where no two semantics agree (while ST is undefined) like AF15

η

ζ ι

θα βγ δ ε κ

Universal disagreement is possible

EPCL BTC 2013 – Abstract Argumentation – P. BaroniΣ1

Σ2

Σ3

Σ8

Σ4

Σ5 Σ6

Σ7

Σ9

Σ10 Σ11

Σ13

Σ14

Σ12

ST=PR=CF2=SST=IDGR=PR=CF2=SST=ID=CO

GR=PR=SST=ID=CO

GR=ID


PR=CF2=SST=ID

PR=SST=ID

ST=PR=SST=ID

PR=CF2=SST

ST=PR=CF2=SST

PR=SST

ST=PR=SST

ST=SST

PR=CF2

A synthetic view


Σ2

Σ3

Σ8

Σ4

Σ5 Σ6

Σ7

Σ9

Σ10 Σ11

Σ13

Σ14

Σ12


GR=PR=SST=ID=CO

GR=ID


PR=CF2=SST=ID

PR=SST=ID

ST=PR=SST=ID

PR=CF2=SST

ST=PR=CF2=SST

PR=SST

ST=PR=SST

ST=SST

PR=CF2

A synthetic viewUnique-status agreement including GR


Σ2

Σ3

Σ8

Σ4

Σ5 Σ6

Σ7

Σ9

Σ10 Σ11

Σ13

Σ14

Σ12


GR=PR=SST=ID=CO

GR=ID


PR=CF2=SST=ID

PR=SST=ID

ST=PR=SST=ID

PR=CF2=SST

ST=PR=CF2=SST

PR=SST

ST=PR=SST

ST=SST

PR=CF2


Unique-statusagreement including ID not GR


Σ2

Σ3

Σ8

Σ4

Σ5 Σ6

Σ7

Σ9

Σ10 Σ11

Σ13

Σ14

Σ12


GR=PR=SST=ID=CO

GR=ID


PR=CF2=SST=ID

PR=SST=ID

ST=PR=SST=ID

PR=CF2=SST

ST=PR=CF2=SST

PR=SST

ST=PR=SST

ST=SST

PR=CF2


Unique-statusagreement including ID not GR

Multiple-status agreement


Do you agree that …The “limited” number of agreement classessuggests that “agreement implies agreement” and commonalities among semantics are significantUniversal disagreement shows anyway that alsodifferences among semantics are significantSome disagreements (and the distinction between some classes) seem to require relatively peculiarargumentation frameworks (self-defeatingarguments and or byzarre SCCs with “attack knots”)Considering further semantics may increase (butnot explosively) the number of distinct classes


Roadmap…From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworks


Isomorphism and syntactic equivalence of AFs

Equivalence between argumentation frameworkscan be defined from different perspectivesThe language independence principle is based on a relation of isomorphism between frameworks whichis essentially equality (modulo a renaming of arguments)Equality between frameworks is also calledsyntactical equivalenceA very basic, restrictive and semantics-independent correspondence between frameworks


Extension equivalence

A semantics-dependent notion of equivalenceinvolves the equality of the sets of extensionsTwo argumentation frameworks AF1 and AF2 are equivalent wrt a semantics S iff ES(AF1) = ES(AF2)Simple and intuitive relation, but:» does not account for the difference between rejected and

undecided arguments (OUT and UNDEC labels)» infinite equivalent frameworks with very little topological

similarity» poorly related to “replaceability” of frameworks


Strong equivalence

A relation involving equality of extensions after union with any other framework

Oriented to replaceability (in the context of the union operation) but quite strongIn absence of self-defeating arguments it reduces to syntactical equivalence for most semantics(admissible, complete, grounded, preferred, stable, semi-stable, ideal, eager, cf2, stage2)


Strong equivalence

In presence of self-defeating arguments:» the strong equivalence relation coincides for admissible,

preferred, ideal, semi-stable and eager semantics » stable, grounded, and complete semantics give rise to

different strong equivalence relations» for cf2 and stage2 it still coincides with syntactical

equivalence (succinctness property)


Variants of strong equivalence

Instead of comparing extensions one may compare the set of skeptically (or credulously) justifiedarguments: it is proved that this does not make anydifference for complete, grounded, preferred, stable, semi-stable, ideal, eager semanticsOne may restrict to local equivalence (i.e. impose that the union operation does not add any new argument): it is proved that this does not make anydifference for admissible, preferred, semi-stable, ideal, eager semantics


Input/Output equivalence

Instead of assuming an arbitrary union operationone may refer to a replacement of a partialframework with another partial framework and define equivalence in terms of the effects on the remaining part of the frameworkSeveral very recent ideas on this kind of approach, also related to decomposability of argumentationframeworksSeveral works in progress


Skepticism relations for AFs

When two argumentation frameworks are notequivalent (in any sense) one may wonder whetherthey are comparable wrt some criterion so that a partial order of AFs can be definedOne such criterion is skepticism: intuitively an AF ismore skeptical than another if it represents a “lesscommitted” or “more undecided” situationWe introduce a basic skepticism relation for AFs tobe used later in skepticism comparison forsemantics



An argumentation framework AF1 is “more committed” than AF2 if AF1 can be obtained fromAF2 by transforming some mutual attacks intounidirectional attacksIn a sense a mutual attack represents an “indecisionpoint” about which of two conflicting argumentsprevailsTransforming a mutual attack into a unidirectionalone can be seen as “resolving an indecision” i.e. moving to a more committed situation



α β α β

More skeptical(less committed)No clear winner

Less skeptical(more committed)A clear winner


Roadmap. . .Dung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related properties


A matter of attitude

Intuitively, in presence of a conflict, one may adopta more or less skeptical attitude towards the possible conflict solutionsA more skeptical attitude tends to accept (include in the extensions) less argumentsIn other words, a more skeptical attitude tends to express less committed judgments: smallerextensions, or, correspondingly, less IN/OUT and more UNDECIDED arguments


A matter of attitude

Different semantics may be more or less skepticalThis can be useful to model or compare different attitudes in a problem-solving or decision-makingprocessWe need a formal notion of skepticism and of skepticism comparison

Notational remark: the symbol < will mean “lesscommitted” (i.e. “more skeptical”)


Comparing single extensions/labellings

Considering a single extension the skepticismrelation coincides with the inclusion relationCorrespondingly, considering a single labelling the skepticism relation coincides with the inclusionrelation of IN labelled arguments and the inclusionrelation of OUT labelled arguments


Comparing single extensions/labellings

αΙΝ

βOUT

αOUT

βIN

αUND

βUND

not comparable

less committedwrt both


Skepticism relations for sets of extensions: ∩ and ∪

The simplest way to compare two sets of extensionsconsists in comparing their intersections

or (in a “more credulous” way) their unions


Comparing sets of extensions/labellings

α β α β

α β

emptyintersectionfor bothsets



α β α β

α β

not the sameoutcomewith union

x


emptyintersectionagain


β

α

γ

β

α

γ

β

α

γ


xnot emptywith one more argument


β

α

γ δ

β

α

γ δ

β

α

γ δ


Skepticism relations for sets of extensions: ∩ and ∪ with OUT nodes

When comparing single extensions the inclusion of the set of IN arguments implies the inclusion of the set of OUT argumentsThis does not hold when comparing sets of extensionsFor skepticism comparisons closer to the notion of labellingone has to consider the attacked arguments too


Considering extensionsindividually

Skepticism relations based on union or intersectionar rather rough, e.g. an empty intersection of manydifferent extensions “is the same” as a single emptyextensionA “finer” comparison may consider single extensions in the sets and require that inclusionholds considering pairs of individual extensionsOne may require either that:- each extension in the more committed set has a “smaller” counterpart in the other set - each extension in the less committed set has a “larger” counterpart in the other set


Skepticism relations consideringextensions individually

The first choice corresponds to a strengthening of the intersection-based relations the latter of the union-based relations

Of course one can impose both conditions


The relation

E1’’

E2’’’

can include unrelated extensions like E1*

E1’

E1*

E2’

E2’’


The relation

E1’’

E2*

can include unrelated extensions like E2*

E1’ E2

’

E2’’


A stronger relation

Imposing both conditions is a sort of bidirectionalconstraint

E1’

E1’’

E2’

E2’’

E2’’’

E1’’’

E2’’’’


Implication ordering

The relations we have presented are progressivelystronger according to the intersection or union perspectiveThe bidirectional relation implies all the other ones


Using skepticism relations between sets of extensions

Skepticism relations between sets of extensions are usefulto define skepticism relations between semanticsA semantics S1 is more skeptical than a semantics S2 if forany argumentation framework the set of extensionsprescribed by S1 is more skeptical than the set prescribedby S2 (the definition is parametric wrt the skepticism relation for sets of extensions)

The requirement is rather strong: two semantics may beincomparable due to “just one” framework


Comparing semantics wrtskepticism: ∩ based relations

The skepticism comparison gives the same results for the three “intersection-based” relationsThe case of AFs where stable extensions exist has to beconsidered separately

All AFsAFs withstable ext.


Comparing semantics wrtskepticism: ∪ based relations

The skepticism comparison gives the same results also forthe three “union-based” relations

All AFsAFs withstable ext.


Comparing semantics wrtskepticism: the strong relation

All AFs AFs withstable ext.


An exercise

Naive and eager semantics are not included in the skepticism comparisons available in the literatureTry to add them


Roadmap

. . . “Basic” semantics propertiesA catalogue of semanticsProperties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related propertiesA richer notion of justification status


Is three the perfect number?

Most works on labellings in the literature adopt the so called “Caminada-labelling” with three possiblelabels: IN, OUT, UNDECAs we have already seen, one can freely move from3-labellings to extensions and viceversaAccordingly, 3-labellings and extensions are alternative ways to express the same thingHowever labellings have an “unlimited” potential ifone goes beyond the three “standard” labels


Justification states

A semantics prescribes a set of labellings(extensions): an argument gets one or more different labels from a set of labellings

αΙΝ

βOUT

αOUT

βIN


Precising justification states

The notion of skeptical justification is “precise”: a skeptically justified argument is IN in all labellingsThe notions of credulous justification and of rejection can instead be refinedA credulously (not skeptically) justified argument may be indifferently OUT or UNDEC in the labellings where it is not INA rejected argument may be indifferently OUT or UNDEC in all labellings


Precising justification states

To summarize the justification state of an argumentmore precisely it seems “natural” to consider the set of labels the argument gets in the alternativesprescribed by a semanticsSeven states{IN} : accepted in all alternatives{OUT} : rejected in all alternatives{UND} : undecided in all alternatives{IN,OUT} : “controversial” accepted or rejected{IN,UND} : not always accepted, never rejected{OUT,UND} : not always rejected, never accepted{IN,OUT,UND} : anything possible – who knows


Is seven the perfect number?

One could adopt the seven justification statesdirectly as labels rather than as a derived conceptand define non-Dung semanticsFull redefinition of labelling principles neededFrom:if an argument has an attacker IN then it should be OUTTo:if an argument has an attacker CONTROVERSIAL then …it can not be IN

if an argument has all attackers CONTROVERSIAL then …it should be CONTROVERSIAL


Using directly the seven labels…

“Non standard” outcomes are possible

αCONT

βCONT

γCONT

δCONT


Hardly fits semantics notions like “maximal admissible set” implicitly based on the IN or OUT alternative Can be encompassed in directionality/topology centered approaches like the acceptance function of abstract dialectical frameworks or the SCC-recursive scheme

Using directly the seven labels…


Why just seven?

Human reasoning is rich of nuances and gradualevaluationsWhat makes a set of labels suitable forargumentation labellings?Identifying at least the cases of definite acceptance, definite rejection and an intermediate caseOrdering labels


Ordering labelsacceptance

rejection

IN

OUT

UNDEC

{OUT}

{IN}

{UNDEC} {IN, OUT, UNDEC}{IN, OUT}

{OUT, UNDEC}

{IN, UNDEC}


Commitment ordering

Ordering according to “acceptance level” is not the only meaningful/useful one in a set of labelsDifferent commitment levels can be identified:a label is more committed if it corresponds to a more clearcut choiceDefinite acceptance and definite rejection are equivalent according to commitmentThe commitment ordering may play a key role in defining principles for labellings and for skepticismcomparison


Commitment ordering

commitment

abstention

IN OUT

UNDEC

{OUT}{IN}

{UNDEC}

{IN, OUT, UNDEC}

{IN, OUT}

{OUT, UNDEC}{IN, UNDEC}


Another commitment ordering

commitment

abstention

IN OUT

UNDEC

{OUT}{IN}

{UNDEC} {IN, OUT, UNDEC}

{IN, OUT}



commitment

abstention

IN OUT

UNDEC

{OUT}{IN}

{UNDEC} {IN, OUT, UNDEC}

{IN, OUT}


A simpler commitment ordering


Roadmap

. . . A catalogue of semanticsProperties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related propertiesA richer notion of justification statusSkepticism-related criteria for semantics


Skepticism relations between sets of extensions can also be used to define further general properties toassess/compare semanticsTwo properties of this kind considered in the literature:» skepticism adequacy» resolution adequacy

Both are based on the skepticism comparisonbetween frameworks

Using skepticism relations between sets of extensions


Skepticism adequacy

Basic idea: the skepticism relation of argumentationframeworks is reflected by the skepticism relation between sets of extensions prescribed by a givensemantics


Resolutions of anargumentation framework

Given an argumentation framework, transforming allits mutual attacks into unidirectional ones leads to a framework with the highest possible level of commitment, called a resolution of the originalframework

Of corse there are 2N ways to reach the highestlevel of commitment, i.e. 2N resolutions, where N isthe number of mutual attacks


Resolutions of anargumentation framework

AF1

Resolutions of AF1


Resolution adequacy

Basic idea: if an argument is skeptically justified in all the resolutions of AF (it survives in all resolutionsscenarios) then it should be in all extension of AF (its surviving ability should be evident also in the original AF)More formally, the intersection of the union of the sets of extensions prescribed by a given semanticsS for all the resolutions of an AF should be includedin the intersection of the extensions prescribed by S for the original AF


Resolution adequacy

Comparing intersections is just one of the possibleforms of skepticism comparison: we can generalize

Given a skepticism relation between sets of extensions a semantics S is - resolution-adequateif for any argumentation framework AF, the union of the sets of extensions prescribed by S for allresolutions of AF is more skeptical (according to ) than the set of extensions prescribed by S for AF

E

E

E


Two critical criteria

NNNNNYNNNNStrong defense

YYYYYYYNNNReinstatement

YYYYYYYYNNWeak reinstatement

YYYYYYYYYYCF-reinstatement

NYNYNYYYNNDirectionality

YYYYYYNYYYI-maximality

YYYYYYYNNNAdmissibility

??

??

STA

N

Y

CF2

N

Y

CO

N

Y

GR

Y

Y

STB

Y

N

PR

Y

N

SST

??

??

EAG

NYRes. adequacy

NYSkept. adequacy

IDNA


Roadmap

. . . Properties of semantics (so far)Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related propertiesA richer notion of justification statusSkepticism-related criteria for semanticsSatisfying all criteria


An embarassing finding?

None of the semantics we considered satisfies allprinciples

A limit of literature semantics?

An inherent incompatibility of the proposedprinciples?


From principles to design Trying to define a new semantics letting the principles drive the designFirst step: a parametric (i.e. defined in terms of another arbitrary semantics S) family of semantics(called resolution-based)

Intuitively: compute all resolutions of AF, apply S toall of them, among the resulting extensions selectthe minimal (wrt inclusion) ones


From principles to design Any resolution-based semantics satisfies:» I-maximality» skepticism-adequacy» resolution-adequacy

Putting relatively mild requirements on S we obtainalso:» admissibility» reinstatement (all forms of)


From principles to design Second step: identify an instance of this family ableto satisfy the only missing principle (directionality)The result has been achieved using the traditionalgrounded semantics as S i.e. with “resolution-basedgrounded semantics”


A laboratory monster … We initially called this family of semantics“synthetic” since it was obtained in a sort of “unnatural” way

The definition is quite complicated and looks like the product of a purely theoretical exercise …

which promises in particular an abnormouscomputational complexity


… or ugly duckling?A reasonably efficient implementation is possiblewithout following the definition literallyComputational complexity analysis shows that “resolution-based grounded semantics” is better than other multiple-status semantics from thisviewpoint

Principles can be useful not only for a posteriori evaluation but also for designThere is still a lot of work to do


Roadmap

. . . Taking topology seriouslyDifferences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related propertiesA richer notion of justification statusSkepticism-related criteria for semanticsSatisfying all criteriaComputational issues


Computational problems

Several “generic” (wrt semantics) computationalproblems can be identified in abstractargumentationGiven a semantics S, there are problems whoseinstances are represented by:» an argumentation framework» an argumentation framework and an argument» an argumentation framework and a set of arguments


Complexity analysis

Complexity analysis has been carried out for mostsemanticsGrounded semantics is the only fully tractablesemanticsTractable classes (e.g. bipartite) of argumentationframeworks have also been investigated


Decision problems (AF):existence of extensions

EXS : Is there at least an extension prescribed by S for AF?This problem is non trivial only for stable semantics

EXSTB is NP-complete


Decision problems (AF):non emptyness of extensions

NES : Is there at least a non empty extensionprescribed by S for AF?

P

NA

P

STA

P

CF2

NP-co

CO

P

GR

NP-co

STB

NP-co

PR

NP-co

SST

NP-h

ID

P

R-BGR

P

STA2

NES


Decision problems (AF + set):verification

VERS : Is this set an extension of S for this AF?

P

NA

coNP-c

STA

P

CF2

P

CO

P

GR

P

STB

coNP-c

PR

NP-co

SST

NP-h

ID

P

R-BGR

P

STA2

VERS


Decision problems (AF + arg):skeptical acceptance

SAS : Is this argument in all extensions of S for thisAF?

P

NA

Π2p-

c

STA

coNP-c

CF2

P

CO

P

GR

coNP-c

STB

Π2p-

c

PR

Π2p-

c

SST

coNP-h

ID

coNP-c

R-BGR

Π2p-

c

STA2

SAS


Decision problems (AF + arg):credulous acceptance

CAS : Is this argument in one extension of S for thisAF?

P

NA

Σ2p-c

STA

NP-c

CF2

NP-c

CO

P

GR

NP-c

STB

Π2p-

c

PR

Σ2p-c

SST

coNP-h

ID

NP-c

R-BGR

Σ2p-c

STA2

CAS


Algorithms for abstractargumentation

Devising efficient approaches for decision (and extension construction) problems in abstractargumentation is a hot research topicBoth “direct” algorithms and encoding methods(ASP, SAT) can be found in the literatureSystematic comparisons and benchmarks are stillmissing


Roadmap

. . . Differences vs commonalities:semantics agreementComparing argumentation frameworksSkepticism and related propertiesA richer notion of justification statusSkepticism-related criteria for semanticsSatisfying all criteriaComputational issuesBeyond Dung’s framework


Extended frameworks

Several extensions/variations of Dung’s frameworkhave been considered in the literatureThey can be classified according to distinct (but notnecessarily disjoint) lines:

» more articulated notions of attack» representation of additional notions» quantitative evaluations


Collective attacks

The binary relation of attack can not express situations of non-binary incompatibilityConsider a situation where you can choosearbitrarily two out of three items (e.g. three people A,B,C, wanting to have a ride on a tandem)Clearly it impossible that the three facts (A,on), (B,on) and (C,on) hold together but they are notpairwise incompatibleIf you need to capture this kind of situations youneed collective attacks


Vreeswijk’s abstractargumentation systems

Vreeswijk’s abstract argumentation systems are notas abstract as Dung’s argumentation frameworkThey include argument structure and inference rulesIn this context a set of arguments (rather than a single argument) may defeat an argument


Nielsen and Parsons’ sets of attacking arguments

Dung’s framework is extended to encompass attacksfrom sets of arguments

The basic semantics notions of Dung’s frameworkare extended to the case of attacking sets in a niceand rather direct wayFurther developments in this direction wouldrepresent a significant advancement in Dung-styleargumentation


Attacks to attacks

May attack be attacked in turn?Basic idea: if there is a reason for an argumentbeing attacked by another one, this reason may bedefeasibleAttacks as part of the reasoning process rather thana sort of syntactic-automatic-non revisable notionMeta-argumentation: reasoning about reasoning, attacks at a lower level are “special arguments” at a meta-level


Modgil’s EAFFocused on reasoning about preferencesGiven two conflicting arguments A and B let say you have a reason to prefer A wrt B. This can be expressed as anargument C attacking (suppressing) an attack from B to A.The reason to prefer A can be defeasible and so on …


Modgil’s EAFArguments can attack attacks between arguments

Two attack levels are explicitly distinguishedOnly one level of “attack recursion” is allowedThe underlying interpretation in terms of preferencesmotivates the fact that attacks to attacks can not be attackedin turn and the final condition (opposite preferences shouldattack each other)


Unlimited attack recursion: AFRA

AFRA: arguments can attack any attack

The definition explicitly encompasses two sorts of entities (like Dung’s one) Unlimited attack recursion levels are allowedNo domain dependencies


More variations: HLAF

Higher Level Argumentation Frames have beenproposed in several flavours:» Unlimited recursive attacks like in AFRA» Joint and disjunctive attacks from sets of

arguments» Attacks may also arise from attacks


Semantics notionswith attack to attacks

Traditional semantics notions have been recasted in frameworks with attacks to attacksBasic intuition 1: attacks which are defeated do notcount as conflicts anymoreBasic intuition 2: you can translate (flatten) anextended framework into a traditional one and thendo semantics evaluation at the flattened levelFormalising these intuitions is not immediate: it hasbeen done differently in the context of the citedapproaches


Additional notions:preference-based AFs

Basic idea: the attack relation may fail to capture the fact that arguments may have different “quality”


Additional notions:preference-based AFs

The “quality” of arguments can be represent through a preference relation

Attacks are “effective” only if the attacked argument is not preferred to the attacking one

A grounded-semantics-like notion of acceptability isdefined accordingly


Additional notions:value-based AFs

A more articulated way to assess the “extra-conflict” merits of argumentsArguments are associated with valuesThere is no unique ordering of values, since they may count differently in different contextsA specific value ordering is called an audienceFor instance, in a political debate different audiences may assess differently argumentspromoting public welfare wrt arguments promotingfinancial stability


Additional notions:value-based AFs

Defeat (and hence any semantics evaluation) becomes an audience-specific notion

In fact, an audience induces a specificargumentation framework


Objective acceptance: a stronger skepticaljustificationAn argument is objectively acceptable if it isskeptically justified in any audience-specific AF (i.e. for any audience)Subjective acceptance: a weaker credulous justificationAn argument is subjectively acceptable if it is credulously justified in an audience-specific AF (i.e. for at least one audience)

Acceptance invalue-based AFs


Additional notions: support

Basic intuition: arguments are not just “conflictables” they may also “help” or “support” each otherA relation of support should parallel the one of attackIs support as “abstractable” as conflict?(It seems that) we can define general conflict management mechanisms independently of the underlying meaning of conflictIs this analogously possible independently of the underlying meaning of support?


Different meanings of support

Support as defense: A supports B if A defends B from an attackSupport as derivation (subargument relation): A supports B if A is used in the construction of BSupport as necessity: A supports B if A is necessary for B

Possible mixing between the notions of argument and of conclusion of an argument in the informal intuitions about support


Bipolar argumentationframeworks

It is required that the two relations have no intersection (an argument can not attack and support another one at the same time)A path of supports gives rise to indirect supportA path of supports behind an attack gives rise to indirect attack


Bipolar argumentation frameworks: set-based relations

Also called set-defeats and set-supports


Bipolar argumentation frameworks: redefining traditional semanticsConflict-freeness: a set S is +conflict free if thereare no arguments a and b in S such that {a} set-defeats bSafeness: a set S is safe if there is no argument b such that S set-attacks b and (b ∈ S or S set-supports b)Different notions of admissibility (then of semantics):» traditional= +conflict-freeness and defense» safe= safeness and defense» closed= closure wrt support and +conflict-freeness and

defense


Flatten them all …The extensions we considered provide an explicitrepresentation of some further notionsThis does not mean that they are “more expressive” than the original framework: “flattening” procedureshave been investigated to translate extendedframeworks back to the original oneTypically flattening means to introduce additional(meta-)arguments and attacks corresponding to the additional notions of the extended frameworkCorrespondences can be drawn between semanticsnotions at the extended and flattened level


Flatten them all …

Several perspectives:» interpretation: the flattened representation may provide

hints of various kinds on the extended one» reuse: existing results are applicable effortless to the

flattened representation» correspondence/verification: a meaningful mapping

should exist between semantics notions in the two levels


… with some concern

A suitable flattening procedure shows that anextended framework is not more expressive thanthe traditional AFOne may argue that extended frameworks arise from modeling carelessness (or modeling indolence) while the “right” modeling is in the flattened versionSynthetic modeling is appropriate for knowledge and reasoning representation: a sort of high-level language to be compiled into AF


Examples of flattening:recursive attacks

Α Β

C

α

β Α Β

C

X Y

C

Α Βα

β

EAF

AFRA


Quantitative evaluation of arguments

Dung’s framework encompasses a qualitative assessment of argument justification status (basedon the binary notion of set membership)The use of numbers in abstract argumentation hasbeen considered in several contexts and withvarious flavors


Argument “strength”According to Pollock, the strength of an argument is“the degree of justification it would confer on its conclusion” in absence of defeatersHere strength is seen as an a priori property of the argument, determined at the moment of its constructionWeakest link principle: an argument can not be stronger than its subargumentsInitial argument strengths are used to determine a final numerical evaluation of arguments, on the basis of the attack (and possibly support) relations


Argument “strength”

The idea of argument strength has been consideredin several variants (often at a non abstract level) in the literatureNo standard reference modelMany open questions (with non univocal answer): suitability of a probabilistic-style treatment, combination operators, the role of accrual ...


Equational approach

In the recent equational approach to argumentationnetworks, an argumentation framework induces a system of equations which are parametric wrt a function f


Equational approach

Several choices for f can be considered

In case of cycles the solutions are fixed points of the function fNo notion of initial (nor final) argument strengthinvolved, rather a form of “numerical semantics” reflecting the structure of the framework


Equational approach:numbers but not for strength


Weighted argumentationframeworks

In weighted argumentation frameworks weights are on attacks rather than on arguments

The weights represent the “strength of the attack” or the amount of inconsistency it carriesAn “inconsistency budget” β defines the amount of inconsistency one is prepared to tolerate



The idea is that one can “ignore” any set of attacks whosetotal weight is not greater than the inconsistency budget β and define β–compatible extensions accordingly



Weighted argumentation frameworks can be seenas a generalization of the approaches where attackscan be suppressed (or resolved) for variousreasons like:» Preference-based AFs» Value-Based AFs» AFs with attacks to attacks» Resolution-based semantics


Even more abstract:abstract dialectical frameworks

Even the nature of the relation between “arguments” is not specified: links of different nature (attack, support, others? …) all belong to the relation LAll the meaning is embedded into the acceptanceconditions (one for each node: heterogeneoussituations may occur)


A non-Dung semantics:“unanimity of attacks”

αΙΝ γ

INβOUT

αOUT γ

INβIN

αΙΝ γ

OUTβIN

αOUT γ

INβOUT


… that can be expresses in Dung’s AF through additional arguments

α

Α3β

γ

Α1

Α2


A new big bang?ADFs represent an alternative perspective wherethe only embedded principle seems the one of directionality (rather than conflict-free)Large variety of “semantics”, actually of acceptancefunctions, even inside the same frameworkADFs use ony in/out labels, but can be extendedSemantics evaluation principles and skepticism comparisons to be revisited/redefined in this more general formal contextA new unexplored universe for lovers of abstract argumentation semantics


Conclusions

A rich and very active research area based on “almost nothing”Conflict (rather than argument) is the key notion ensuring wide scope both in theory and in practice Large corpus of both basic and specific/advanced results with many “research avenues” and “rethinking opportunities” still openMany (at least in principle) reusable and (hopefully) stimulating concepts for other research fields where conflict management plays a key role


Thank you for your patience!

Any argument?


Some referencesThe big bang

P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logicprogramming and n-person games. Artificial Intelligence, 77:321–357, 1995.

Argumentation in AIG. R. Simari, I. Rahwan (Eds.), Argumentation in Artificial Intelligence, Springer Science+BusinessMedia, 2009T. J. M. Bench-Capon, P. E. Dunne: Argumentation in artificial intelligence. Artificial Intelligence 171(10-15): 619-641 (2007)

Introductory material on abstract argumentation semanticsP. Baroni, M. Caminada, M. Giacomin, An introduction to argumentation semantics, Knowledge Engineering Review, 26(4), 2011, 365-410P. Baroni, M. Giacomin, Semantics of abstract argument systems, In G. R. Simari, Iyad Rahwan (Eds.), Argumentation in Artificial Intelligence, Springer Science+Business Media, 2009, 25-44

Semantics evaluation criteriaP. Baroni, M. Giacomin, On principle-based evaluation of extension-based argumentation semantics, Artificial Intelligence, 171(10/15), 2007, 675-700

Non-traditional semanticsB. Verheij, Two approaches to dialectical argumentation: admissible sets and argumentation stages. In Proceedings of the 8th Dutch Conference on Artificial Intelligence (NAIC’96), Meyer, J.-J. & van derGaag, L. (eds). Utrecht University, 357–368. 1996.


Some referencesNon-traditional semantics (continued)

M. W. A. Caminada, Semi-stable semantics. In Proceedings of the 1st International Conf. on Computational Models of Argument (COMMA 2006), 121–130.P. M. Dung, P. Mancarella, F. Toni, Computing ideal sceptical argumentation. Artificial Intelligence 171(10–15), 642–674, 2007.M. W. A.Caminada, Comparing two unique extension semantics for formal argumentation: ideal and eager. In Proceedings of the 19th Belgian-Dutch Conf. on Artificial Intelligence (BNAIC 2007), 81–87.

SCC-recursiveness, CF2 semantics, stage2 semanticsP. Baroni, M. Giacomin, Solving semantic problems with odd-length cycles in argumentation, Proc. of ECSQARU 2003, 7th European Conf. on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Aalborg, DK, 2003, 440-451P. Baroni, M. Giacomin, G. Guida, SCC-recursiveness: a general schema for argumentation semantics, Artificial Intelligence, 168(1-2), 2005, 162-210S. A. Gaggl, S. Woltran, The cf2 argumentation semantics revisited. Journal of Logic and Computation, in press

Semantics agreementP. Baroni, M. Giacomin, Characterizing defeat graphs where argumentation semantics agree, Proc. of ArgNMR, Workshop on Argumentation and Non-Monotonic Reasoning, 2007, 33-48 P. Baroni, M. Giacomin, A Systematic Classification of Argumentation Frameworks where Semantics Agree, Proc. of COMMA 08, 2nd International Conf. on Computational Models of Argument, Toulouse, F, 2008, 37-48


Some referencesComparing argumentation frameworks

E. Oikarinen, S. Woltran, Characterizing strong equivalence for argumentation frameworks, Artificial Intelligence, 175, 2011, 1985-2009P. Baroni, G. Boella, F. Cerutti, M. Giacomin, L. W. N. van der Torre, S. Villata, On Input/Output Argumentation Frameworks, Proc. of COMMA 2012, 4th International Conference on Computational Models of Argument, Vienna, A, 2012, 358-365

Skepticism relations and richer justification statesP. Baroni, M. Giacomin, Skepticism relations for comparing argumentation semantics, International Journal of Approximate Reasoning, 50(6), 2009, 854-866. Y. Wu, M. W. A. Caminada, A labelling-based justification status of arguments. Studies in Logic, 3(4),2010, 12–29. P. Baroni, M. Giacomin, G. Guida, Towards a formalization of skepticism in extension-based argumentation semantics, Proc. of CMNA 2004, 4th Workshop on Computational Models of Natural Argument, Valencia, E, 2004, 47-52

Resolution-based semanticsP. Baroni, P.E. Dunne, M. Giacomin, On the resolution-based family of abstract argumentation semantics and its grounded instance, Artificial Intelligence, 175(3-4), 2011, 791-813

Computational issuesP.E. Dunne, M. Wooldridge, Complexity of abstract argumentation, in G. R. Simari, I. Rahwan (Eds.), Argumentation in Artificial Intelligence, Springer Science+Business Media, 2009, 85-104


Some referencesComputational issues (continued)

P. E. Dunne: Computational properties of argument systems satisfying graph-theoretic constraints. Artif. Intell. 171(10-15): 701-729 (2007)W. Dvorak, Computational Aspects of Abstract Argumentation. Ph.D Thesis. TU Wien. 2012.

Beyond Dung’s frameworkSystems with collective attacksG. Vreeswijk: Abstract Argumentation Systems. Artif. Intell. 90(1-2): 225-279 (1997)P. Baroni, M. Giacomin, G. Guida: Extending abstract argumentation systems theory. Artif. Intell. 120(2): 251-270 (2000)S. H. Nielsen, S. Parsons: Computing Preferred Extensions for Argumentation Systems with Sets of Attacking Arguments. Proc. 1st Int. Conf. on Computational Models of Argument (COMMA 2006) 97-108Attacks to attacksS. Modgil, Reasoning about preferences in argumentation frameworks. Artif. Intell. 173(9-10): 901-934 (2009)P. Baroni, F. Cerutti, M. Giacomin, Giovanni Guida: AFRA: Argumentation framework with recursive attacks. Int. J. Approx. Reasoning 52(1): 19-37 (2011)H. Barringer, D. M. Gabbay, J. V. Woods, Temporal dynamics of support and attack networks: From argumentation to zoology, in: D. Hutter, W. Stephan (Eds.), Mechanizing Mathematical Reasoning, Springer, 2005, 59–98. D. Gabbay, Semantics for higher level attacks in extended argumentation frames Part 1: Overview, Studia Logica 93 (2/3) (2009) 357–381.


Some referencesBeyond Dung’s framework (continued)

Preference-based argumentation frameworksL. Amgoud, C. Cayrol: A Reasoning Model Based on the Production of Acceptable Arguments. Ann. Math. Artif. Intell. 34(1-3): 197-215 (2002) Value-based argumentation frameworksT. J. M. Bench-Capon: Persuasion in Practical Argument Using Value-based Argumentation Frameworks. J. Log. Comput. 13(3): 429-448 (2003)Bipolar argumentation frameworksL. Amgoud, C. Cayrol, M.-C. Lagasquie-Schiex, P. Livet: On bipolarity in argumentation frameworks. Int.

J. Intell. Syst. 23(10): 1062-1093 (2008)Quantitative evaluationsJ. L. Pollock: Defeasible reasoning with variable degrees of justification. Artif. Intell. 133(1-2): 233-282 (2001)D.M. Gabbay, Equational approach to argumentation networks. Argument and Computation. 3(2–3), 87–142 P. E. Dunne, A. Hunter, P. McBurney, S. Parsons, M. Wooldridge: Weighted argument systems: Basic definitions, algorithms, and complexity results. Artif. Intell. 175(2): 457-486 (2011)Abstract dialectical frameworksG. Brewka, S. Woltran, Abstract dialectical frameworks, Proc. of the 12th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2010), 102-111

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