Aggregating Alternative Extensions of AAFs: Preservation Results for Quota Rules Weiwei Chen Sun Yat-sen University, China [Joint work with Ulle Endriss]
Objectives and Outline Motivatd by the challenge of modelling collective argumentation, we consider the problem of aggregating extensions of AAFs and study the preservation results for quota rules. We make use of results from two fields: • argumentation integrity constraints for semantics • binary aggregation with integrity constraints I will present: • the problem of preservation when aggregating extensions • examples for preservation results, sketching some of our techniques 1/10
Aggregation of Extensions Fix an AF = 〈 Arg , � 〉 . Suppose each agent supplies us with an extension reflecting her individual views of what constitutes an acceptable set of arguments in the context of AF . We would like to aggregate this information by making use of quota rules . Terminology : The quota rule F q with quota q is defined as F q ( ∆ ) = { A ∈ Arg | #{ i ∈ N | A ∈ Δ i } � q } where n is the number of agents and ∆ = ( Δ 1 ,..., Δ n ) is a profile of extensions. Related work : Rahwan and Tohmé, Caminada and Pigozzi 2/10
An Example Suppose three agents evaluate the following AF: E D C B A They report the extensions { A , C } , { A , D } , and { A , E } , respectively, all of which are admissible. But applying the majority rule (i.e., the quota rule F q with q = ⌈ n 2 ⌉ ) yields { A } , which is not admissible! Research Question: Which properties are preserved by which quota rules? 3/10
Integrity Constraints for Semantics Let AF = 〈 Arg , � 〉 be an AF and let Δ ⊆ Arg be an extension. Then Δ is conflict-free (self-defending, reinstating) iff: ⋀︂ Δ | = IC CF = ( ¬ A ∨¬ B ) IC CF where A , B ∈ Arg A � B Δ is self-defending iff: ⋀︂ ⋀︂ ⋁︂ Δ | = IC SD = [ C → A ] IC SD where C ∈ Arg B ∈ Arg A ∈ Arg B � C A � B Δ is admissible iff Δ | = IC CF ∧ IC SD , Terminology : Δ is self-defending if Δ ⊆ { C | Δ defends C } . P. Besnard and S. Doutre. Checking the acceptability of a set of arguments. In Proc. of N.M.R , 2004. 4/10
Extension Aggregation with Integrity Constraints Given an intrgrity constraint φ = p 1 ∨ ,..., ∨ p n with k 1 positive literals and k 2 negative literals, a quota rule F q with quota q ∈ { 1 ,..., n } preserves the property Mod ( φ ) if and only if: q · ( k 2 − k 1 ) n · ( k 2 − 1 ) − k 1 > If F preserves both Mod ( φ 1 ) and Mod ( φ 2 ) . Then F also preserves Mod ( φ 1 ∧ φ 2 ) (Grandi and Endriss, 2013). 2 preserves ¬ A ∨¬ B , the quota Example : the quota rule with q > n rule with q > 2 · n 3 preserves ¬ C ∨¬ D ∨¬ E , then the quota rule with q > 2 · n 3 preserves ( ¬ A ∨¬ B ) ∧ ( ¬ C ∨¬ D ∨¬ E ) 5/10
Preserving Conflict-Freeness A quota rule F q for n agents preserves conflict-freeness for AF if and only if q > n 2 : • The integrity constraint for CF is ⋀︁ A � B ( ¬ A ∨¬ B ) (Besnard and A , B ∈ Arg Doutre, 2004) A B C • For any quota q > n 2 , F q preserves the The IC for CF for the clauses of the form ¬ A ∨¬ B above AF is ( ¬ A ∨ • Thus, F q preserves the conjunction of ¬ B ) ∧ ( ¬ B ∨¬ C ) . clauses of the form ¬ A ∨¬ B , namely preserves IC CF 6/10
Preserving Self-defense A quota rule F q for n agents preserves self-defense for an AF if q · ( MaxDef ( AF ) − 1 ) < MaxDef ( AF ) : • The integrity constraint for SD is ⋀︁ C ∈ Arg [ C → ⋀︁ ⋁︁ A � B A ] B ∈ Arg A ∈ Arg B D C B � C • C → ⋀︁ ⋁︁ B ∈ Arg A ∈ Arg A � B A can be rewrite as B � C ⋀︁ B � C ( ¬ C ∨ ⋁︁ A � B A ) , and F A B ∈ Arg A ∈ Arg preserves it iff q · ( k C , B − 1 ) < k C , B The IC for SD for the • the largest value of k C , B is MaxDef ( AF ) above AF is D → ( A ∨ B ) , rewritten • we satisfy all inequalities in case as ¬ D ∨ A ∨ B . q · ( MaxDef ( AF ) − 1 ) < MaxDef ( AF ) Terminology : MaxDef ( AF ) denotes the maximum number of attackers of an argument that itself is the source of an attack. 7/10
Preserving Admissibility • The nomination rule preserves the property of self-defense for all argumentation frameworks. • Every quota rule F q for n agents with a quota q > n 2 preserves admissibility for all argumentation frameworks AF with MaxDef ( AF ) � 1. • No quota rule preserves admissibility for all argumentation frameworks. 8/10
Preservation Results Property Constraint(s) Uniform Quota Rule(s) q > n Conflict-freeness 2 q · ( MaxDef ( AF ) − 1 ) < MaxDef ( AF ) Self-defending Self-defending Nomination rule MaxDef ( AF ) � 1 q > n Admissibility 2 Admissibility None q · ( MaxAtt ( AF ) − 1 ) > n · ( MaxAtt ( AF ) − 1 ) − 1 Being Reinstating Being Reinstating Unanimity rule MaxDef ( AF ) � 1 q > n Completeness 2 Completeness None | σ | � 2 and n is even I-Maximal property σ None | σ | � 2 and n is odd I-Maximal property σ No quota rule different from the majority rule | σ | = 2 Property σ Majority rule 9/10
Summary and Furture Works We have seen: • encoding of argumentation semantics in propositional logic along with prior work in judgment aggregation establish positive results in extension aggregation. • social choice theory can be fruitfully applied to the analysis of scenarios of collective argumentation . Future work: further properties of extensions, other aggregation rules besides the quota rules, other types of argumentation formalisms, ... 10/10
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