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Computational Aspects of cf2 and stage2
Argumentation Semantics
COMMA 2012 (Vienna)
Wolfgang Dvo°ák1, Sarah Alice Gaggl2
1Theory and Applications of Algorithms Group, University of Vienna2Institute of Information Systems, Vienna University of Technology
Sept. 11th, 2012
Supported by the Vienna Science and Technology Fund (WWTF) through project ICT08-028.
Computational Aspects of cf2 and stage2 Semantics Slide 1
1. Motivation
Motivation
cf2 -Semantics:
Uniform treatment of odd- and even-length cycles.
Ful�lls most evaluation criteria proposed in [Baroni & Giacomin 07].
But produces questionable results on several frameworks.↪→ �xed by stage2 semantics
stage2 -Semantics:
Instantiates the SCC-recursive schema of cf2 semantics with stagesemantics.
Satis�es the evaluation criteria.
Coincides with stable semantics in the absence of odd-length cycles.
Computational Aspects of cf2 and stage2 Semantics Slide 2
1. Motivation
Motivation
cf2 -Semantics:
Uniform treatment of odd- and even-length cycles.
Ful�lls most evaluation criteria proposed in [Baroni & Giacomin 07].
But produces questionable results on several frameworks.↪→ �xed by stage2 semantics
stage2 -Semantics:
Instantiates the SCC-recursive schema of cf2 semantics with stagesemantics.
Satis�es the evaluation criteria.
Coincides with stable semantics in the absence of odd-length cycles.
Computational Aspects of cf2 and stage2 Semantics Slide 2
1. Motivation
Motivation
Computational Issues:
For cf2 and stage2 typical reasoning tasks are computationally hard,i.e. NP/coNP-hard.
But this is worst case complexity � there might be classes ofinstances that show milder complexity ⇒ tractable fragments.
The analysis of computational complexity and identifying tractablecases are indispensable for building e�cient systems.
So far only the general complexity of the main reasoning tasks wasstudied [Gaggl & Woltran 12; Dvo°ák & Gaggl 12].
Computational Aspects of cf2 and stage2 Semantics Slide 3
1. Motivation
Contributions
Complexity analysis of cf2 and stage2 semantics:
We consider four graph classes for being tractable fragments and
provide either polynomial time algorithms orhardness results.
We discuss possible parameters for �xed-parameter tractability.
Discussion of implementation methods:
A labeling based algorithm for cf2 and stage2 semantics.
Computational Aspects of cf2 and stage2 Semantics Slide 4
2. Background
Argumentation Frameworks
Abstract Argumentation Framework [Dung 95]
An argumentation framework (AF ) is a pair F = (A,R), where A is a�nite set of arguments and R ⊆ A× A a attack relation.
Example
F = (A,R), A = {a, b, c}, R = {(a, b), (b, c), (c, b), (c, c)}.
Computational Aspects of cf2 and stage2 Semantics Slide 5
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
cf2 -Semantics
cf2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a maximal con�ict free set S of C and add it to theextension.
3 Delete C and all arguments attacked by S .
a
b
c d
e
f
g h
ij
g h
ij
Computational Aspects of cf2 and stage2 Semantics Slide 6
2. Background
stage2 -Semantics
stage2 -extensions are given as follows:
While the AF is non-empty
1 Pick a minimal strongly connected component C .
2 Compute a stage extension S of C and add it to the extension.
3 Delete C and all arguments attacked by S .
Computational Aspects of cf2 and stage2 Semantics Slide 7
2. Background
Reasoning Problems
Credulous Acceptance
Credσ: Given AF F = (A,R) and a ∈ A; is a contained in at least oneσ-extension of F?
Skeptical Acceptance
Skeptσ: Given AF F = (A,R) and a ∈ A; is a contained in everyσ-extension of F?
Verifying an extension
Verσ: Given AF F = (A,R) and S ⊆ A; is S a σ-extension of F?
naive stable stage cf2 stage2
Credσ in P NP-c ΣP2 -c NP-c Σ
P2 -c
Skeptσ in P coNP-c ΠP2 -c coNP-c Π
P2 -c
Verσ in P in P coNP-c in P coNP-c
Computational Aspects of cf2 and stage2 Semantics Slide 8
2. Background
Reasoning Problems
Credulous Acceptance
Credσ: Given AF F = (A,R) and a ∈ A; is a contained in at least oneσ-extension of F?
Skeptical Acceptance
Skeptσ: Given AF F = (A,R) and a ∈ A; is a contained in everyσ-extension of F?
Verifying an extension
Verσ: Given AF F = (A,R) and S ⊆ A; is S a σ-extension of F?
naive stable stage cf2 stage2
Credσ in P NP-c ΣP2 -c NP-c Σ
P2 -c
Skeptσ in P coNP-c ΠP2 -c coNP-c Π
P2 -c
Verσ in P in P coNP-c in P coNP-c
Computational Aspects of cf2 and stage2 Semantics Slide 8
3. Complexity Analysis
Acyclic AFs
On Acyclic AFs both semantics coincide with grounded semantics.↪→ complexity classi�cation follows immediate from grounded semantics.
Theorem
For acyclic AFs and σ ∈ {cf2 , stage2} the problems Credσ and Skeptσare in P.
Computational Aspects of cf2 and stage2 Semantics Slide 9
3. Complexity Analysis
Even Cycle Free Argumentation FrameworksBy [Dunne & Bench-Capon 01], reasoning with admissible-basedsemantics in AFs without even-length cycles is tractable.
But as cf2 and stage2 treat cycles uniformly this result does not extend.
Theorem
For AFs without even-length cycles: Credcf2 is NP-complete, Skeptcf2 is
coNP-complete, Cred stage2 is NP-hard, and Skeptstage2 is coNP-hard.
Proof.
Computational Aspects of cf2 and stage2 Semantics Slide 10
3. Complexity Analysis
Even Cycle Free Argumentation FrameworksBy [Dunne & Bench-Capon 01], reasoning with admissible-basedsemantics in AFs without even-length cycles is tractable.
But as cf2 and stage2 treat cycles uniformly this result does not extend.
Theorem
For AFs without even-length cycles: Credcf2 is NP-complete, Skeptcf2 is
coNP-complete, Cred stage2 is NP-hard, and Skeptstage2 is coNP-hard.
Proof.
Computational Aspects of cf2 and stage2 Semantics Slide 10
3. Complexity Analysis
Bipartite Argumentation Frameworks
Bipartite AFs have been shown to be tractable for admissible-basedsemantics in [Dunne 07]. We show that on bipartite AFs the credulously(skeptically) accepted arguments w.r.t. cf2 are exactly those credulously(skeptically) accepted w.r.t. stable semantics.
Theorem
For bipartite AFs the problems Credcf2 and Skeptcf2 are in P.
Moreover on bipartite AFs stage2 and stable semantics coincides.
Theorem
For bipartite AFs the problems Cred stage2 and Skeptstage2 are in P.
Computational Aspects of cf2 and stage2 Semantics Slide 11
3. Complexity Analysis
Symmetric AFs
In symmetric AFs there are no attacks between di�erent stronglyconnected components. Hence,
cf2 coincides with naive semantics and;
stage2 coincides with stage semantics.
Theorem
For symmetric AFs the problems Credcf2 and Skeptcf2 are in P.
Theorem
For symmetric, irre�exive AFs the problems Cred stage2 and Skeptstage2are in P.
But for symmetric AFs with self-attacks the problems Cred stage2 andSkeptstage2 are still Σ
P2 / Π
P2 complete.
Computational Aspects of cf2 and stage2 Semantics Slide 12
3. Complexity Analysis
Fixed-Parameter Tractability
Fixed parameter tractability w.r.t. parameters Tree-Width /Clique-Width follows from MSO1 characterizations in [Dvo°ák,Szeider, Woltran 12].
Backdoor approach [Ordyniak & Szeider 11]:
Negative Results for stage semantics extends to stage2 semantics.
Negative result for bipartite AFs extends also to cf2 semantics.
Open: backdoors for cf2 and acyclic or symmetric AFs.
Computational Aspects of cf2 and stage2 Semantics Slide 13
3. Complexity Analysis
Fixed-Parameter Tractability
Fixed parameter tractability w.r.t. parameters Tree-Width /Clique-Width follows from MSO1 characterizations in [Dvo°ák,Szeider, Woltran 12].
Backdoor approach [Ordyniak & Szeider 11]:
Negative Results for stage semantics extends to stage2 semantics.
Negative result for bipartite AFs extends also to cf2 semantics.
Open: backdoors for cf2 and acyclic or symmetric AFs.
Computational Aspects of cf2 and stage2 Semantics Slide 13
4. A Labelling based Algorithm
A Labelling based Algorithm
Require: AF F = (A,R), labeling L = (Lin,Lout ,Lundec );Ensure: Return all cf2 labelings of F ;X = {a ∈ Lundec | att(a) ⊆ Lout};Y = {a ∈ Lundec | ∃b ∈ Lin, (b, a) ∈ R, a 6⇒
A\LoutF b};
while (X ∪ Y ) 6= ∅ doLin = Lin ∪ X ,Lout = Lout ∪ Y ,Lundec = Lundec \ (X ∪ Y );update X and Y ;
end whileB = {a ∈ Lundec | Lin ∪ {a} ∈ cf (F )};if B 6= ∅ then
C = {a ∈ B |6 ∃b ∈ B : b ⇒A\LoutF a, a 6⇒A\LoutF b};
E = ∅;for all L′ ∈ naiveL(F |C ) do
update L with L′;E = E ∪ cf2L(F ,L);
end forreturn E;
elsereturn {(Lin,Lout ,Lundec )};
end if
Computational Aspects of cf2 and stage2 Semantics Slide 14
5. Conclusion
Conclusion
Summary
Complexity analysis of reasoning with cf2 and stage2 semantics:
consider four graph classes for being tractable fragments;
discuss possibilities for �xed-parameter tractability.
Discussion of implementation methods:
provide a labelling based algorithm.
Future Work
Considering characterizations via the equational approach for furthertractable fragments.
Computational Aspects of cf2 and stage2 Semantics Slide 15
5. Conclusion
Conclusion
Summary
Complexity analysis of reasoning with cf2 and stage2 semantics:
consider four graph classes for being tractable fragments;
discuss possibilities for �xed-parameter tractability.
Discussion of implementation methods:
provide a labelling based algorithm.
Future Work
Considering characterizations via the equational approach for furthertractable fragments.
Computational Aspects of cf2 and stage2 Semantics Slide 15
MotivationBackgroundComplexity AnalysisA Labelling based AlgorithmConclusion
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