An introduction to Abstract Argumentation - EPCL · An introduction to Abstract Argumentation ......
Transcript of An introduction to Abstract Argumentation - EPCL · An introduction to Abstract Argumentation ......
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An introduction toAbstract Argumentation
Pietro BaroniDII - Dip. di Ingegneria dell’Informazione
University of Brescia (Italy)
EPCL Basic Training Camp 2013
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words ?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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).
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Abstract (for bad)
Poorly related (or totally unrelated) with realproblems(MW: insufficiently factual, difficult to understand, abstruse)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words?From the Big Bang to now
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words?From the Big Bang to nowDung’s framework
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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)
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Dung’s framework is(almost) nothing
α β
β
α γ
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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
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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
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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”
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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?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An example:the stable marriage problem
Steps 1 and 2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An example:the stable marriage problem
Step 3
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An example:the stable marriage problem
Steps 4 and 5
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An example:the stable marriage problem
Step 6
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An example:the stable marriage problem
Step 6
The notion of preferred extension provides a naturalsolution to a non traditional version of the marriageproblem
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Sets of extensions
α β E1 = {{α},{β}}
β
α γ
E1 = {{α},{β},{γ}}E2 = {∅}
E2 = {∅}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labelling-based semantics
αΙΝ
βOUT
αOUT
βIN
L1
αUND
βUND
L2
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labelling-based semantics
βOUT
αIN
γUND
βIN
αUND
γOUT
βUND
αOUT
γIN
L1
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labelling-based semantics
βUND
αUND
γUND
L2
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
αΙΝ β
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Labellings vs. extensions
αΙΝ
βOUT
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
αΙΝ
βOUT
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
αΙΝ
βOUT
αOUT
βIN
L1
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
αUND
βUND
L2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
αΙΝ
βOUT
αOUT
βIN
αUND
βUND
L1
L2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
βOUT
αIN
γUND
βIN
αUND
γOUT
βUND
αOUT
γIN
L1
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Labellings vs. extensions
βUND
αUND
γUND
L2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics properties
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Conflict freeness
α β
α β
α β
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Conflict freeness
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Admissibility
α β
α β
α β
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Admissibility
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Strong defense
α β
α β
α β
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Strong defense
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
β
α γ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Strong defense
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Strong defense
αΙΝ
βOUT
γIN
δUND
εUND
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Reinstatement
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Defense related principles:weak reinstatement
A weaker notion of reinstatement is obtainedconsidering strong defense instead of defense
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Defense related principles:CF reinstatement
Reinstatement does not require explicitly conflictfreeness with the defended argument. Adding thisgives rise to CF reinstatement
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
α
ε
α
ε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
α β γ
δε
α β γ
δε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
α β
γε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
α β
γε
α β
γε
α β
γε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Implicit equal treatment
If there are reasons to prefer α to β, they should bereflected in the topology
α β
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Roadmap
What’s in the words?From the Big Bang to nowDung’s frameworkA “conflict calculus”: argumentation semantics“Basic” semantics propertiesA catalogue of semantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Naive semantics
α β γ
α β γ
α β γ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β γ
α β γ not a stageextension
α β γ
α β γ
A bit less naive: stage semantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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!
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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} }
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Admissibility and completeness
α β γ
δε
{α,γ} is admissible but not complete
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β γ
δε
Admissibility and completeness
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} }
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β γ
δε
Admissibility and completeness
{α,γ} is admissible and complete
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β γ
δε
Admissibility and completeness
{α,γ,δ} is also admissible and complete
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
Grounded semantics:unquestionable=strong defense
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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(∅)…)
Grounded semantics:unquestionable=strong defense
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Computing the grounded extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Computing the grounded extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Computing the grounded extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Computing the grounded extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A stable extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Another stable extension
α β γ
δεζ
θη
ι
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Preferred vs. stable semantics
α β
δ
γ ε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Preferred vs. stable semantics
α β
δ
γ ε
α β
δ
γ ε
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A less cautious single statusβ
α
γ δ
β
α
γ δ
The ideal/eagerextension is {δ}, while the groundedextension is empty
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
“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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
“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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Directionality
Disproving directionality is immediate for stable, semi-stable and eager semanticsProving directionality is less immediate (butpossible) for the other complete-based semantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A synopsis
YYYYYYYYYLanguage indip.
N
Y
N
N
N
N
Y
Y
STA
Y
Y
Y
Y
N
Y
N
Y
CO
Y
Y
Y
Y
Y
Y
Y
Y
GR
N
Y
Y
Y
N
Y
Y
Y
STB
Y
Y
Y
Y
N
Y
Y
Y
PR
N
Y
Y
Y
N
Y
Y
Y
SST
NYNDirectionality
YYNWeak reinstatement
Y
Y
N
Y
Y
Y
EAG
YYCF-principle
YYCF-reinstatement
YNReinstatement
NNStrong defense
YNAdmissibility
YYI-maximality
IDNA
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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?)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β
δ
γ ε
Strongly Connected Components
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β
δ
γ ε
Strongly Connected Components
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
α β
δ
γ ε
Strongly Connected Components
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A partial order for evaluation
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A partial order for evaluation
S1
S2
S3
S5
S6
S7
S4
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A partial order for evaluation: level 1
S1
S2
S3
S5
S6
S7
S4
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
S1
S2
S3
S5
S6
S7
S4
A partial order for evaluation: level 2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
S1
S2
S3
S5
S6
S7
S4
A partial order for evaluation: level 3
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
S1
S2
S3
S5
S6
S7
S4
A partial order for evaluation: level 4
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
4. For any possible choice of E ∩ S, every subsequent SCC S’ ispartitioned into three subsets
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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.
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
S1
S3
S5
S6
S7
S4
S2
SCC-recursiveness at work: propagating choices
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
S1
S3
S5
S6
S7
S4
or ...
S7
S7
S2
SCC-recursiveness at work: reaching final SCCs
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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.
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A motivation for symmetric treatment of cycles
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Unreliabilityclaims amongwitnesses
Consonantwitnesses vs.claims
A dissonantclaim
A motivation for symmetric treatment of cycles
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Unreliabilityclaims amongwitnesses
Consonantwitnesses vs.claims
A dissonantclaim
A motivation for symmetric treatment of cycles
The dissonant claim is skeptically acceptedwith three witnesses
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A motivation for symmetric treatment of cycles
The dissonant claim is NOT skeptically acceptedwith four witnesses
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
CF2 semantics
First choice in the first SCC
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Three extensions based on the first choice in the first SCC
CF2 semantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A maximal CF set whichis not a CF2 extension
CF2 semanticsSecond choice in the first SCC Only one extension based
on the second choice
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Evaluating CF2 semantics
N
Y
N
N
N
N
Y
Y
Y
STA
YYYYYYYYYLanguage indip.
Y
Y
Y
N
N
N
Y
Y
CF2
Y
Y
Y
Y
N
Y
N
Y
CO
Y
Y
Y
Y
Y
Y
Y
Y
GR
N
Y
Y
Y
N
Y
Y
Y
STB
Y
Y
Y
Y
N
Y
Y
Y
PR
N
Y
Y
Y
N
Y
Y
Y
SST
NYNDirectionality
YYNWeak reinstatement
Y
Y
N
Y
Y
Y
EAG
YYCF-principle
YYCF-reinstatement
YNReinstatement
NNStrong defense
YNAdmissibility
YYI-maximality
IDNA
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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”
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Stage2 vs. CF2Stage2 and CF2 extensions Other CF2 extensions
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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”
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Obvious commonalities …
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 …
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Obvious commonalities …
“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
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Obvious commonalities …
Acyclic argumentation framework
δ γ β α
ε
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
Initial acyclic
S1 S2
S3S’1
S’2
A topological characterization of determined arg. frameworks
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An argumentation framework AF is determined if and only if itis initial acyclic, namely is 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
Initial acyclic Not initial acyclic
S’1S’1
S’2
A topological characterization of determined arg. frameworks
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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
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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
Single-status agreement
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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*
Single-status agreement
*to be introduced later
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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
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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
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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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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 ...
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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SCC-symmetric vs. limited controversial
SCC-symmetricand limited controversial
SCC-symmetricnot limited controversial
Limited controversialnot SCC-symmetric
Not SCC-symmetricnot limited controversial
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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
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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
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Basic properties
E.g.
It may be (and it is) the case that for some different sets of semantics and it holds that
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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)
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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})
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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
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Several kinds of inclusion relations between (sets of) extensions:for any argumentation framework AF
•
•
•
•
•
•
Agreement classes: which (known) properties can be exploited?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several kinds of inclusion relations between (sets of) extensions:for any argumentation framework AF
•
•
•
•
•
•
Agreement classes: which (known) properties can be exploited?
The groundedextension is includedin many kinds of extensions
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several kinds of inclusion relations between (sets of) extensions:for any argumentation framework AF
•
•
•
•
•
•
Agreement classes: which (known) properties can be exploited?
Any extension of one kindis included in an extensionof another kind
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several properties based on the inclusion relationships can be derived
Agreement classes: further properties we can prove
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several lemmata based on the inclusion relationships can be derived
Agreement classes: which kind of properties we prove?
Implications of cardinality
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several lemmata based on the inclusion relationships can be derived
Agreement classes: which kind of properties we prove?
Implications of inclusionin the set of extensions
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several lemmata based on the inclusion relationships can be derived
Agreement classes: which kind of properties we prove?
Some agreementsimply others
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Several lemmata based on the inclusion relationships can be derived
Agreement classes: which kind of properties we prove?
Implications of extensionproperties
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Coincidence of agreement classes follows (almost directly) from the lemmata. Examples are:
Agreement classes: coincidence
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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=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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
GR=ST=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
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
Agreement classes: GR unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
GR=ST=PR=CF2=SST=ID=CO
AF1
AF2
AF3
Σ1
Σ3
GR=PR=CF2=SST=ID=COΣ2
Agreement classes: GR unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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=
GR=ST=PR=CF2=SST=ID=CO
AF1
AF2
AF3
Σ1
Σ3
GR=PR=CF2=SST=ID=COΣ2
β
αγ
Agreement classes: GR unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
GR=ST=PR=CF2=SST=ID=CO
AF1
AF2
AF3
Σ1
Σ3
GR=PR=CF2=SST=ID=COΣ2
PR=CF2=SST=ID
PR=CF2=ST=SST=IDPR=ST=SST=ID
PR=SST=ID
Agreement classes: GR unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
GR=PR=CF2=SST=ID=COΣ2
PR=CF2=SST=ID
PR=CF2=ST=SST=IDPR=ST=SST=ID
PR=SST=ID
AF4
Σ4
GR=ST=PR=CF2=SST=ID=CO
AF1Σ1
α β
GR=PR=SST =ID=CO
Agreement classes: ID unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
GR=PR=CF2=SST=ID=COΣ2
GR=ST=PR=CF2=SST=ID=CO
AF1Σ1
PR=ST=SST=ID
γ
Agreement classes: ID unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
GR=PR=CF2=SST=ID=COΣ2
AF6
Σ6
PR=CF2=ST=SST=ID
AF4
Σ4
GR=ST=PR=CF2=SST=ID=CO
AF1Σ1
GR=PR=SST =ID=CO
PR=CF2=SST=ID
PR=ST=SST=ID
β
αγ
Agreement classes: ID unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
GR=PR=CF2=SST=ID=COΣ2
GR=ST=PR=CF2=SST=ID=CO
AF1Σ1
AF7
AF5Σ5
Agreement classes: ID unique-status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
prevents the existence of stable extensions
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
prevents the existence of stable extensions
makes a difference between GR and ID
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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'
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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}}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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}makes a
difference with CF2
GE(AF11) = ID(AF11) = ∅EPR(AF11) = EST(AF11) = {{α},{δ}}ECF2(AF11) = {{α},{β},{γ},{δ}}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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'
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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”
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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''
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
(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
The Σ14 class – Regions of interest
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Σ14\ (Σ8∪Σ12 ∪Σ13) ≠∅ as it includes AF14
(Σ14 ∩ Σ8) \ (Σ12 ∪Σ13) ≠∅ as it includes AF14’’
α βγ δ ε ζ
α βγ δ
The Σ14 class – Regions of interest
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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
Agreement classes: multiple status behavior
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
There are argumentation frameworks where GR and IDagree while all other semantics disagree (and ST is undefined)
Σ8\ (Σ12∪Σ13 ∪Σ14) ≠∅ as it includes AF8
α βγε
δ η
ζθ
Σ8 and nothing else
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
GR=ST=PR=CF2=SST=ID=CO
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
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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
GR=ST=PR=CF2=SST=ID=CO
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
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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
GR=ST=PR=CF2=SST=ID=CO
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
Unique-statusagreement including ID not GR
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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
GR=ST=PR=CF2=SST=ID=CO
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
Unique-statusagreement including ID not GR
Multiple-status agreement
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Skepticism relations for AFs
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Skepticism relations for AFs
α β α β
More skeptical(less committed)No clear winner
Less skeptical(more committed)A clear winner
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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”)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Comparing single extensions/labellings
αΙΝ
βOUT
αOUT
βIN
αUND
βUND
not comparable
less committedwrt both
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Comparing sets of extensions/labellings
α β α β
α β
emptyintersectionfor bothsets
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Comparing sets of extensions/labellings
α β α β
α β
not the sameoutcomewith union
x
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
emptyintersectionagain
Comparing sets of extensions/labellings
β
α
γ
β
α
γ
β
α
γ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
xnot emptywith one more argument
Comparing sets of extensions/labellings
β
α
γ δ
β
α
γ δ
β
α
γ δ
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
The relation
E1’’
E2’’’
can include unrelated extensions like E1*
E1’
E1*
E2’
E2’’
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
The relation
E1’’
E2*
can include unrelated extensions like E2*
E1’ E2
’
E2’’
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A stronger relation
Imposing both conditions is a sort of bidirectionalconstraint
E1’
E1’’
E2’
E2’’
E2’’’
E1’’’
E2’’’’
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Implication ordering
The relations we have presented are progressivelystronger according to the intersection or union perspectiveThe bidirectional relation implies all the other ones
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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.
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Comparing semantics wrtskepticism: ∪ based relations
The skepticism comparison gives the same results also forthe three “union-based” relations
All AFsAFs withstable ext.
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Comparing semantics wrtskepticism: the strong relation
All AFs AFs withstable ext.
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An exercise
Naive and eager semantics are not included in the skepticism comparisons available in the literatureTry to add them
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Using directly the seven labels…
“Non standard” outcomes are possible
αCONT
βCONT
γCONT
δCONT
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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…
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Ordering labelsacceptance
rejection
IN
OUT
UNDEC
{OUT}
{IN}
{UNDEC} {IN, OUT, UNDEC}{IN, OUT}
{OUT, UNDEC}
{IN, UNDEC}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Commitment ordering
commitment
abstention
IN OUT
UNDEC
{OUT}{IN}
{UNDEC}
{IN, OUT, UNDEC}
{IN, OUT}
{OUT, UNDEC}{IN, UNDEC}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Another commitment ordering
commitment
abstention
IN OUT
UNDEC
{OUT}{IN}
{UNDEC} {IN, OUT, UNDEC}
{IN, OUT}
{OUT, UNDEC}{IN, UNDEC}
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
commitment
abstention
IN OUT
UNDEC
{OUT}{IN}
{UNDEC} {IN, OUT, UNDEC}
{IN, OUT}
{OUT, UNDEC}{IN, UNDEC}
A simpler commitment ordering
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Skepticism adequacy
Basic idea: the skepticism relation of argumentationframeworks is reflected by the skepticism relation between sets of extensions prescribed by a givensemantics
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Resolutions of anargumentation framework
AF1
Resolutions of AF1
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
An embarassing finding?
None of the semantics we considered satisfies allprinciples
A limit of literature semantics?
An inherent incompatibility of the proposedprinciples?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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”
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
… 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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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 …
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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Additional notions:preference-based AFs
Basic idea: the attack relation may fail to capture the fact that arguments may have different “quality”
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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Additional notions:value-based AFs
Defeat (and hence any semantics evaluation) becomes an audience-specific notion
In fact, an audience induces a specificargumentation framework
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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?
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Bipolar argumentation frameworks: set-based relations
Also called set-defeats and set-supports
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
… 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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Examples of flattening:recursive attacks
Α Β
C
α
β Α Β
C
X Y
C
Α Βα
β
EAF
AFRA
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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 ...
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Equational approach
In the recent equational approach to argumentationnetworks, an argumentation framework induces a system of equations which are parametric wrt a function f
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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Equational approach:numbers but not for strength
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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Weighted argumentationframeworks
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
Weighted argumentationframeworks
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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)
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
A non-Dung semantics:“unanimity of attacks”
αΙΝ γ
INβOUT
αOUT γ
INβIN
αΙΝ γ
OUTβIN
αOUT γ
INβOUT
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… that can be expresses in Dung’s AF through additional arguments
α
Α3β
γ
Α1
Α2
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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Thank you for your patience!
Any argument?
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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.
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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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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.
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EPCL BTC 2013 – Abstract Argumentation – P. Baroni
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