A Comprehensive Analysis of the cf2 Argumentation Semantics:

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A Comprehensive Analysis of the cf2 Argumentation Semantics:

From Characterization to Implementation

Sarah Alice Gaggl

Institute of Informationsystems, Vienna University of Technology

Vienna — March 4, 2013

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Motivation

Argumentation is one of the major fields inArtificial Intelligence (AI).

Applicationsin diverse domains (legal reasoning, multi-agent systems, social networks, e-government, decision support).

Concept ofabstract Argumentation Frameworks (AFs)[Dung, 1995]

is one of the most popular approaches.

Argumentsand a binaryattackrelation between them, denoting conflicts, are the only components.

Numeroussemanticsto solve the inherent conflicts by selecting acceptable sets of argument.

Admissible-basedversusnaive-basedsemantics.

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Motivation ctd.

cf2

Semantics

is based ondecompositionof the framework along itsstrongly connected components (SCCs)[Baroni et al., 2005];

does not require to defend arguments against attacks;

allows totreat cyclesin a moresensitive waythan other semantics;

isnot well studied, due to quite complicated definition.

Sarah A. Gaggl, TU Vienna Comprehensive Analysis of cf2 Semantics 2

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Motivation ctd.

cf2

Semantics

is based ondecompositionof the framework along itsstrongly connected components (SCCs)[Baroni et al., 2005];

does not require to defend arguments against attacks;

allows totreat cyclesin a moresensitive waythan other semantics;

isnot well studied, due to quite complicated definition.

Goals of the Thesis

Answer-set programming encodingsforcf2. Alternative characterization.

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Outline

1 Background on abstract argumentation frameworks and semantics

2 Alternative characterization ofcf2

3 Combiningcf2and stage semantics

4 Redundancies and strong equivalence

5 Computational complexity

6 Implementations

7 Conclusion

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Argumentation Framework

Abstract Argumentation Framework [Dung, 1995]

Anabstract argumentation framework (AF)is a pairF= (A,R), whereA is a finite set of arguments andR⊆A×A. Then(a,b)∈Rifaattacksb. Argumenta∈AisdefendedbyS⊆A(inF) iff, for eachb∈Awith (b,a)∈R,Sattacksb.

Example

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Semantics

Semantics for AFs

LetF= (A,R)andS⊆A, we say

Sisconflict-freeinF, i.e.S∈cf(F), if∀a,b∈S: (a,b)6∈R;

S∈cf(F)is maximal conflict-free ornaive(inF), i.e. S∈naive(F), if

∀T ∈cf(F),S6⊂T.

Example

naive(F) ={{a,d,g},{a,c,e},{a,c,g}}.

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Semantics ctd.

Naive-based Semantics

LetF= (A,R)andS⊆A. LetS⁺_R =S∪ {b| ∃a∈S, s. t.(a,b)∈R}be therangeofS. Then, a setS∈cf(F)is

astableextension (ofF), i.e. S∈stable(F), ifS⁺_R =A; stageinF, i.e. S∈stage(F), if for eachT ∈cf(F),S⁺_R 6⊂T_R⁺.

Example

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Semantics ctd.

Admissible-based Semantics

Then,S∈cf(F)is

admissibleinF, i.e.S∈adm(F), if eacha∈Sis defended byS; apreferredextension (ofF), i.e.S∈pref(F), ifS∈adm(F)and for eachT ∈adm(F),S6⊂T.

Example

adm(F) ={∅,{a},{c},{d},{a,c},{a,d}},pref(F) ={{a,c},{a,d}}.

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cf2 Semantics

One of the SCC-recursive semantics introduced in [Baroni et al., 2005].

Naive-basedsemantics.

Handlesodd- andeven-lengthcyclesin auniformway.

Can accept arguments out of odd-length cycles.

Can accept arguments attacked byself-attackingarguments.

Satisfies most of theevaluation criteriaproposed in [Baroni and Giacomin, 2007].

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cf2 Semantics

One of the SCC-recursive semantics introduced in [Baroni et al., 2005].

Naive-basedsemantics.

Handlesodd- andeven-lengthcyclesin auniformway.

Can accept arguments out of odd-length cycles.

Can accept arguments attacked byself-attackingarguments.

Satisfies most of theevaluation criteriaproposed in [Baroni and Giacomin, 2007].

Further Notations, letF= (A,R)

SCCs(F): set ofstrongly connected componentsofF, C_F(a): the unique setC∈SCCs(F), s.t.a∈C,

F|S= ((A∩S),R∩(S×S)): sub-frameworkofFw.r.t.S, F|_S−S⁰ =F|_S\S0,F−S=F|_A\S.

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cf2 Semantics ctd.

Definition (D

F(S))

LetF= (A,R)be an AF andS⊆A. An argumentb∈Ais

component-defeatedbyS(inF), if there exists ana∈S, such that (a,b)∈Randa∈/CF(b). The set of arguments component-defeated by SinFis denoted byDF(S).

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cf2 Semantics ctd.

Definition (D

F(S))

LetF= (A,R)be an AF andS⊆A. An argumentb∈Ais

component-defeatedbyS(inF), if there exists ana∈S, such that (a,b)∈Randa∈/C_F(b). The set of arguments component-defeated by SinFis denoted byD_F(S).

cf2

Extensions [Baroni et al., 2005]

LetF= (A,R)be an argumentation framework andSa set of arguments.

Then,Sis acf2extensionofF, i.e.S∈cf2(F), iff S∈naive(F), in case|SCCs(F)|=1;

otherwise,∀C∈SCCs(F),(S∩C)∈cf2(F|C−DF(S)).

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cf2 Semantics ctd.

S∈cf2(F)

iff,

S∈naive(F), in case|SCCs(F)|=1;

Example

S={a,d,e,g,i},S∈cf2(F)?

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cf2 Semantics ctd.

S∈cf2(F)

iff,

otherwise,∀C∈SCCs(F),(S∩C)∈cf2(F|_C−DF(S)).

Example

S={a,d,e,g,i},S∈cf2(F)? C₁={a,b,c},C₂={d}, C₃={e,f,g,h,i}andD_F(S) ={f}.

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cf2 Semantics ctd.

S∈cf2(F)

iff,

otherwise,∀C∈SCCs(F),(S∩C)∈cf2(F|_C−D_F(S)).

Example

S={a,d,e,g,i},S∈cf2(F)? C₁ ={a,b,c},C₂={d}, C₃={e,f,g,h,i}andDF(S) ={f}.

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cf2 Semantics ctd.

S∈cf2(F)

iff,

Example

S={a,d,e,g,i},S∈cf2(F)? C₄={e},C₅={g},C₆={h},C₇={i}

andD_F|_{e,g,h,i}({e,g,i}) ={h}.

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Alt. Characterization of cf2

Original definition ofcf2is rathercumbersometo be directly encoded inASPdue to therecursive computationof different sub-frameworks.

Inalternative characterizationwe shift therecursionto a certainset of arguments.

This enables to directly guessa setS;

checkwhetherSis anaive extensionof a certaininstanceofF.

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Alt. Characterization of cf2 ctd.

Separation

An AFF= (A,R)is calledseparatedif for each(a,b)∈R, there exists a path frombtoa. We define[[F]] =S

C∈SCCs(F)F|_C and call[[F]]the separationofF.

Example

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Alt. Characterization of cf2 ctd.

Separation

An AFF= (A,R)is calledseparatedif for each(a,b)∈R, there exists a path frombtoa. We define[[F]] =S

C∈SCCs(F)F|C and call[[F]]the separationofF.

Example

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Alt. Characterization of cf2 ctd.

Reachability

LetF= (A,R)be an AF,Ba set of arguments, anda,b∈A. We say that bisreachableinFfromamoduloB, in symbolsa⇒^B_F b, if there exists a path fromatobinF|_B.

Definition (∆

_F,S

)

For an AFF= (A,R),D⊆A, and a setSof arguments,

∆F,S(D) ={a∈A| ∃b∈S:b6=a,(b,a)∈R,a6⇒^A\D_F b},

and∆_F,S be the least fixed-point of∆_F,S(∅).

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Alt. Characterization of cf2 ctd.

Reachability

LetF= (A,R)be an AF,Ba set of arguments, anda,b∈A. We say that bisreachableinFfromamoduloB, in symbolsa⇒^B_F b, if there exists a path fromatobinF|_B.

Definition (∆

_F,S

)

For an AFF= (A,R),D⊆A, and a setSof arguments,

∆F,S(D) ={a∈A| ∃b∈S:b6=a,(b,a)∈R,a6⇒^A\D_F b},

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Given an AFF= (A,R).

cf2(F) ={S|S∈naive(F)∩naive([[F−∆_F,S]])}.

Example

S={a,d,e,g,i},S∈naive(F).

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

S={a,d,e,g,i},S∈naive(F).

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

S={a,d,e,g,i}, S∈naive(F),∆_F,S(∅) ={f}.

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

S={a,d,e,g,i}, S∈naive(F), ∆_F,S({f}) ={f,h}.

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

S={a,d,e,g,i}, S∈naive(F), ∆_F,S({f,h}) ={f,h}.

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

S={a,d,e,g,i}, S∈naive(F), ∆_F,S={f,h},S∈naive([[F−∆_F,S]]), thusS∈cf2(F).

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Alt. Characterization of cf2 ctd.

cf2

Extensions [Gaggl and Woltran, 2012]

Example

cf2(F) ={{a,d,e,g,i},{c,d,e,g,i},{b,f,h},{b,g,i}}.

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Shortcomings of cf2

cf2produces questionable results on AFs with cycles of length≥6.

Example

cf2(F) =naive(F) ={{a,d},{b,e},{c,f},{a,c,e},{b,d,f}};

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Shortcomings of cf2 and Stage

cf2produces questionable results on AFs with cycles of length≥6. The grounded extension is not necessarily contained in every stage extension.

Stage semantics does not satisfy directionality.

Example

stage(F) ={{a},{b}}butcf2(F) =ground(F) ={{a}}.

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Combining cf2 and stage

Wecombinecf2and stage semantics [Dvorák and Gaggl, 2012a], by using theSCC-recursive schema of thecf2semantics and instantiate thebase casewithstagesemantics.

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Combining cf2 and stage

Wecombinecf2and stage semantics [Dvorák and Gaggl, 2012a], by using theSCC-recursive schema of thecf2semantics and instantiate thebase casewithstagesemantics.

stage2

Extensions

For any AFF,

stage2(F) ={S|S∈naive(F)∩stage([[F−∆F,S]])}.

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stage2 Examples

For any AFF,stage2(F) ={S|S∈naive(F)∩stage([[F−∆_F,S]])}.

Example

stage2(F) =cf2(F) ={{a}}, wherestage(F) ={{a},{b}}.

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stage2 Examples

For any AFF,stage2(F) ={S|S∈naive(F)∩stage([[F−∆_F,S]])}.

Example

stage2(F) =cf2(F) ={{a}}, wherestage(F) ={{a},{b}}.

stage2(G) =stage(G) ={{a,c,e},{b,d,f}}, but cf2(G) =naive(F) ={{a,d},{b,e},{c,f},{a,c,e},{b,d,f}}.

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Relations between Semantics

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Redundancies

Argumentation is adynamic reasoning process.

Whicheffecthasadditional informationw.r.t. a semantics?

Which informationdoes not contribute to resultsno matter which changes are performed?

Identification ofkernelsto removeredundant attacks [Oikarinen and Woltran, 2011].

Definition

Two AFsFandGarestrongly equivalentto each other w.r.t. a semantics σ, in symbolsF≡^σ_s G, iff for each AFH,σ(F∪H) =σ(G∪H).

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Strong Equivalence w.r.t. cf2

Theorem

For any AFsFandG,F≡^cf2s GiffF =G.

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Strong Equivalence w.r.t. cf2

Theorem

For any AFsFandG,F≡^cf2_s GiffF =G.

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Strong Equivalence w.r.t. cf2

Theorem

H = (A∪ {d,x,y,z},

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Strong Equivalence w.r.t. cf2

Theorem

LetE={d,x,z},E∈cf2(F∪H)butE6∈cf2(G∪H).

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Strong Equivalence w.r.t. cf2

Theorem

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Strong Equivalence w.r.t. cf2

Theorem

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Strong Equivalence w.r.t. cf2

Theorem

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Strong Equivalence w.r.t. cf2

Theorem

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Strong Equivalence w.r.t. cf2

Theorem

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Strong Equivalence w.r.t. cf2

Theorem

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SE w.r.t. cf2 and stage2

Theorem

No matter which AFsF6=G, one can always construct anHs.t.

cf2(F∪H)6=cf2(G∪H);

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SE w.r.t. cf2 and stage2

Theorem

Forstage2semantics alsostrong equivalence coincides with syntactic equivalence.

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SE w.r.t. cf2 and stage2

Forstage2semantics alsostrong equivalence coincides with syntactic equivalence.

Succinctness Property

An argumentation semanticsσsatisfies thesuccinctness propertyor is maximal succinctiff no AF contains a redundant attack w.r.t.σ.

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Comparing Semantics w.r.t. SE

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Complexity Analysis

Ver Cred Skept Exists^¬∅

naive inP inP inP inP stable inP NP-c coNP-c NP-c cf2 inP NP-c coNP-c inP stage coNP-c Σ^P₂-c Π^P₂-c inP stage2 coNP-c Σ^P₂-c Π^P₂-c inP Table:Computational complexity of naive-based semantics.

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Implementation

Reduction-based Approach

Answer-set Programming (ASP) encodings forcf2andstage2. Saturation vs.metaspencodings forstage2.

All encodings incorporated in the system ASPARTIX[Egly et al., 2010].

Direct Approach

Labeling-basedalgorithms forcf2andstage2.

Web-Application

http://rull.dbai.tuwien.ac.at:8080/ASPARTIX

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Conclusion

Alternative characterizationforcf2to avoid the recursive computation of sub-frameworks.

stage2semantics overcomes problems ofcf2.

Strong equivalencew.r.t.cf2(resp.stage2) coincides with syntactic equivalence.

Provided the missingcomplexity resultsforcf2(resp.stage2).

Implementationin terms ofASPandlabeling-based algorithms.

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Future Work

Further relations to other semantics likeintertranslatability.

Optimizationsof ASP encodings.

Development ofappropriate instantiation methodsfor naive-based semantics.

Other combinationsof semantics in the alternative characterization, likesem(F) ={S|σ(F)∩τ([[F−∆_F,S]])}.

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Baroni, P. and Giacomin, M. (2007).

On principle-based evaluation of extension-based argumentation semantics.

Artif. Intell., 171(10-15):675–700, 2007.

Baroni, P., Giacomin, M., and Guida, G. (2005).

Scc-recursiveness: A general schema for argumentation semantics.

Artif. Intell., 168(1-2):162–210.

Dung, P. M. (1995).

On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.

Artif. Intell., 77(2):321–358.

Dvorák, W. and Gaggl, S. A. (2012)

Incorporating stage semantics in the scc-recursive schema for argumentation semantics.

In Proceedings of the 14th International Workshop on Non-Monotonic Reasoning(NMR 2012), 2012.

Dvorák, W. and Gaggl, S. A. (2012)

Computational aspects of cf2 and stage2 argumentation semantics.

In Bart Verheij, Stefan Szeider, and Stefan Woltran, editors, Proceedings of the 4th International Conference on

Computational Models of Argument(COMMA 2012), volume 245 of Frontiers in Artificial Intelligence and Applications, pages 273–284. IOS Press, 2012.

Egly, U., Gaggl, S. A. and Woltran, S. (2010)

Answer-set programming encodings for argumentation frameworks.

Argument and Computation, 1(2):144–177, 2010.

Gaggl, S. A. and Woltran, S. (2012).

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Tractable Fragments

cf2 stage2 stable stage

Cred_σ^acycl inP inP P-c P-c

Skept_σâcycl inP inP P-c P-c Cred_σêven−free NP-c coNP-h P-c Σ^P₂-c Skept_σêven−free coNP-c coNP-h P-c Π^P₂-c

Cred_σ^bipart inP inP P-c P-c

Skept_σ^bipart inP inP P-c P-c Cred_σ^sym inP inP/Σ^P₂∗ inP inP/Σ^P₂∗ Skept_σ^sym inP inP/Π^P₂∗ inP inP/Π^P₂∗ Table:Complexity results for special AFs (^∗with self-attacking arguments).