Bridges between Abstract Argumentation and Belief Revision Sylvie Coste-Marquis S´ ebastien Konieczny Jean-Guy Mailly Pierre Marquis Centre de Recherche en Informatique de Lens Universit´ e d’Artois – CNRS UMR 8188 2 nd Madeira Workshop on Belief Revision and Argumentation February 9 th – February 13 th 1/18 AMANDE
Outline Introduction Abstract Argumentation Belief Revision Overview of our Contributions Adapting AGM to Abstract Argumentation Using AGM to Revise Abstract AF Translation-based Revision of Argumentation Frameworks Encoding AF and their Semantics Distance-based Operators and Minimal Change Conclusion and Future Work 2/18 AMANDE
Outline Introduction Abstract Argumentation Belief Revision Overview of our Contributions Adapting AGM to Abstract Argumentation Using AGM to Revise Abstract AF Translation-based Revision of Argumentation Frameworks Encoding AF and their Semantics Distance-based Operators and Minimal Change Conclusion and Future Work 2/18 AMANDE
Abstract Argumentation [Dung 1995] ◮ An abstract argumentation framework is a pair �A , R� with R ⊆ A × A : a c b ◮ An extension is a set of arguments that can be accepted together ◮ Different semantics to define the extensions: complete, stable, preferred, grounded, etc. ◮ The aim is to know whether an argument is accepted or not w.r.t. the chosen semantics σ ◮ An argument a ∈ A is (skeptically) accepted iff it belongs to every extension of the AF w.r.t. the considered semantics σ : � F | ∼ σ a ⇔ a ∈ Ext σ ( F ) 3/18 AMANDE
AGM Framework for Belief Revision ◮ AGM Framework [Alchourr´ on, G¨ ardenfors and Makinson 1985] ◮ Adaptation for propositional logic [Katsuno and Mendelzon 1991] ◮ Incorporate a new piece of information α in the agent’s beliefs ϕ wrt some notion of plausibility p : Mods ( ϕ ◦ α ) = min( Mods ( α ) , ≤ p ) 4/18 AMANDE
AF Revision ◮ Aim: Incorporation of a new piece of information about the attack relation and/or the acceptance statuses of arguments ◮ Two kind of minimal change: Attack � = Acceptance 5/18 AMANDE
Outline Introduction Abstract Argumentation Belief Revision Overview of our Contributions Adapting AGM to Abstract Argumentation Using AGM to Revise Abstract AF Translation-based Revision of Argumentation Frameworks Encoding AF and their Semantics Distance-based Operators and Minimal Change Conclusion and Future Work 5/18 AMANDE
Adapting AGM to Abstract Argumentation ◮ A Two-step Process ϕ AF revised extensions AFs 6/18 AMANDE
Summary of this Contribution ◮ New piece of information: formula about acceptance statuses ex: ϕ = ( a 1 ∨ a 2 ) ∧ ¬ a 3 ◮ First minimality criterion: minimal change of arguments statuses ◮ Other (less important) minimality criterion: minimal change of the attack relation, minimality of the output’s size ◮ More details: Coste-Marquis, Konieczny, Mailly, Marquis, On the Revision of Argumentation Systems: Minimal Change of Arguments Statuses , KR 2014 7/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ F ⋆ ϕ 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ F ⋆ ϕ σ -encoding F encoded 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ F ⋆ ϕ σ -encoding F encoded Revision of F encoded ◦ 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ F ⋆ ϕ σ -encoding F encoded Revision of F encoded ◦ AGM Revision 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F F , ϕ F ⋆ ϕ σ -encoding σ -decoding F encoded Revision of F encoded ◦ AGM Revision 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F ⋆ F , ϕ F ⋆ ϕ σ -encoding σ -decoding F encoded Revision of F encoded ◦ AGM Revision 8/18 AMANDE
Using AGM to Revise Abstract AF ◮ σ : a semantics to define acceptable arguments ◮ F : an argumentation framework ◮ ϕ : a propositional formula indicating how to revise F AF Revision ⋆ F , ϕ F ⋆ ϕ σ -encoding σ -decoding F encoded Revision of F encoded ◦ AGM Revision 8/18 AMANDE
Outline Introduction Abstract Argumentation Belief Revision Overview of our Contributions Adapting AGM to Abstract Argumentation Using AGM to Revise Abstract AF Translation-based Revision of Argumentation Frameworks Encoding AF and their Semantics Distance-based Operators and Minimal Change Conclusion and Future Work 8/18 AMANDE
Propositional Language ◮ ∀ x ∈ A , acc ( x ) = “ x is skeptically accepted by F ” ◮ ∀ x , y ∈ A , att ( x , y ) = “ x attacks y in F ” ◮ Prop A = { acc ( x ) | x ∈ A } ∪ { att ( x , y ) | x , y ∈ A } ◮ L A is the propositional language built on the set of variables Prop A and the connectives ¬ , ∨ , ∧ 9/18 AMANDE
Encoding an AF σ -formula of F Given an AF F = � A , R � and a semantics σ , the σ -formula of F is � � f σ ( F ) = att ( x , y ) ∧ ¬ att ( x , y ) ( x , y ) ∈ R ( x , y ) �∈ R 10/18 AMANDE
Encoding an AF σ -formula of F Given an AF F = � A , R � and a semantics σ , the σ -formula of F is � � f σ ( F ) = att ( x , y ) ∧ ¬ att ( x , y ) ∧ th σ ( A ) ( x , y ) ∈ R ( x , y ) �∈ R where the σ -theory of A th σ ( A ) is a formula which encodes the semantics σ . 10/18 AMANDE
Encoding the Stable Semantics (1) Stable extensions of an AF F = � A , R � [Besnard and Doutre 2004] � � ( a ⇔ ¬ b ) a ∈ A b :( b , a ) ∈ R Example F 1 = a c a ∧ [ b ⇔ ( ¬ a ∧ ¬ c )] b d ∧ [ c ⇔ ¬ b ] ∧ [ d ⇔ ¬ c ] One single model / stable extension: { a , c } 11/18 AMANDE
Encoding the Stable Semantics (1) Stable extensions of an AF F = � A , R � [Besnard and Doutre 2004] � � ( a ⇔ ¬ b ) a ∈ A b :( b , a ) ∈ R Example F 1 = a c a ∧ [ b ⇔ ( ¬ a ∧ ¬ c )] b d ∧ [ c ⇔ ¬ b ] ∧ [ d ⇔ ¬ c ] One single model / stable extension: { a , c } 11/18 AMANDE
Encoding the Stable Semantics (2) From � � ( a ⇔ ¬ b ) a ∈ A b :( b , a ) ∈ R to . . . Stable theory of the set A th st ( A ) = � a i ∈ A ( acc ( a i ) ⇔ ∀ a 1 , . . . , a n , ( � a ∈ A ( a ⇔ � b ∈ A ( att ( b , a ) ⇒ ¬ b )) ⇒ a i )) 12/18 AMANDE
Encoding the Stable Semantics (2) From � � ( a ⇔ ¬ b ) a ∈ A b :( b , a ) ∈ R to . . . Stable theory of the set A th st ( A ) = � a i ∈ A ( acc ( a i ) ⇔ ∀ a 1 , . . . , a n , ( � a ∈ A ( a ⇔ � b ∈ A ( att ( b , a ) ⇒ ¬ b )) ⇒ a i )) 12/18 AMANDE
Encoding the Stable Semantics (2) From � � ( a ⇔ ¬ b ) a ∈ A b :( b , a ) ∈ R to . . . Stable theory of the set A th st ( A ) = � a i ∈ A ( acc ( a i ) ⇔ ∀ a 1 , . . . , a n , ( � a ∈ A ( a ⇔ � b ∈ A ( att ( b , a ) ⇒ ¬ b )) ⇒ a i )) 12/18 AMANDE
Decoding Tools ◮ Proj att (Φ): projection of the models of Φ on the variables att ( x , y ) ◮ arg ( Mods att ): generation of AFs from models projected on att ( x , y ) Example of decoding With A = { a , b } , the revised models could be: Mod (Φ) = {{ acc ( a ) , ¬ acc ( b ) , ¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} . So, Proj att (Φ) = {{¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} and arg ( Proj att (Φ)) = { F } with F the AF below: a b 13/18 AMANDE
Decoding Tools ◮ Proj att (Φ): projection of the models of Φ on the variables att ( x , y ) ◮ arg ( Mods att ): generation of AFs from models projected on att ( x , y ) Example of decoding With A = { a , b } , the revised models could be: Mod (Φ) = {{ acc ( a ) , ¬ acc ( b ) , ¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} . So, Proj att (Φ) = {{¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} and arg ( Proj att (Φ)) = { F } with F the AF below: a b 13/18 AMANDE
Decoding Tools ◮ Proj att (Φ): projection of the models of Φ on the variables att ( x , y ) ◮ arg ( Mods att ): generation of AFs from models projected on att ( x , y ) Example of decoding With A = { a , b } , the revised models could be: Mod (Φ) = {{ acc ( a ) , ¬ acc ( b ) , ¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} . So, Proj att (Φ) = {{¬ att ( a , a ) , att ( a , b ) , ¬ att ( b , a ) , ¬ att ( b , b ) }} and arg ( Proj att (Φ)) = { F } with F the AF below: a b 13/18 AMANDE
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