A TOOL FOR RANKING ARGUMENTS THROUGH VOTING-GAMES POWER INDEXES Stefano Bistarelli, Francesco Faloci, Francesco Santini and Carlo Taticchi
INDEX ▸ Argumentation Theory ▸ Argumentation Semantics ▸ Power Indexes ▸ Definitions ▸ SV-based semantics ▸ Description + Example ▸ Conclusion ConArg Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 2
ABSTRACT ARGUMENTATION FRAMEWORKS 1 a b c d e 1 Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–357. Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 3
EXTENSION-BASED SEMANTICS a b c d e CF = {{}, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, d}} ADM = {{}, {a}, {c}, {d}, {a, c}, {a, d}} COM = {{a}, {a, c}, {a, d}} STA = {{a,d}} Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 4
LABELLING-BASED SEMANTICS 2 a b c d e 2 Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123. Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 5
LABELLING-BASED SEMANTICS 2 a b c d e IN if it is attacked only by OUT arguments OUT if it is attacked by at least an IN argument UNDEC otherwise 2 Martin Caminada. On the Issue of Reinstatement in Argumentation. JELIA 2006: 111-123. Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 6
LABELLING-BASED SEMANTICS 2 a b c d e WHICH IS THE BEST? Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 7
RANKING-BASED SEMANTICS 3 ▸ Transforms an Argumentation Framework into a ranking ▸ Criteria: direct attacks, lengths of the incoming paths, rewards ▸ Good properties a d ≻ c ≻ e ≻ b ≻ 3 Leila Amgoud, Jonathan Ben-Naim. Ranking-Based Semantics for Argumentation Frameworks. SUM 2013: 134-147. Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 8
RANKING-BASED SEMANTICS - PROPERTIES 4 ▸ Abstraction ▸ Quality Precedence ▸ Independence ▸ Counter-Transitivity ▸ Void Precedence ▸ Strict Counter-Transitivity ▸ Self-contradiction ▸ Defense Precedence ▸ Cardinality Precedence ▸ Non-attacked Equivalence 4 Elise Bonzon, Jérôme Delobelle, Sébastien Konieczny, Nicolas Maudet. A Comparative Study of Ranking-Based Semantics for Abstract Argumentation. AAAI 2016: 914-920. Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 9
POWER INDEXES ▸ Shapley Value ▸ Banzhaf Index ▸ Deegan-Packel ▸ Johnstone Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 10
POWER INDEXES ▸ Shapley Value s !( n − 1 − s )! ∑ ϕ i ( v ) = ( v ( S − i ∪ { i }) − v ( S − i )) n ! S − i ⊆ G ∖ { i } ▸ Banzhaf Index 1 ∑ β i ( v ) = ( v ( S ∪ { i }) − v ( S )) 2 | N | − 1 S ⊆ N ∖ { i } Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 11
RANKING FUNCTIONS σ , F ( S ) = { 1, if S ∈ in ( L σ ) v I if otherwise 0, a b c d e Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 12
RANKING FUNCTIONS In the event of a tie σ , F ( S ) = { σ , F ( S ) = { 1, if S ∈ in ( L σ ) if S ∈ out ( L σ ) 1, v I v O if otherwise if otherwise 0, 0, a b c d e Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 13
ConArg www.dmi.unipg.it/conarg
SV-BASED SEMANTICS - PROPERTIES Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 15
SV-BASED SEMANTICS - PROPERTIES Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 16
SV-BASED SEMANTICS - PROPERTIES Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 17
SV-BASED SEMANTICS - PROPERTIES Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 18
SV-BASED SEMANTICS - PROPERTIES Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 19
CONCLUSION ▸ Ranking-based semantics • Shapley Value • Banzhaf Index • Deegan-Packel • Johnstone ▸ Tool available at www.dmi.unipg.it/conarg Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes � 20
CONCLUSION NEXT ▸ Ranking-based semantics ▸ Understand the differences between Power Indexes • Shapley Value ▸ Capture global properties: • Banzhaf Index • Deegan-Packel • structure of the AF • Johnstone • full attacking paths ▸ Tool available at www.dmi.unipg.it/conarg Carlo Taticchi — June 20, 2019 A Tool For Ranking Arguments Through Voting-Games Power Indexes 21 �
A TOOL FOR RANKING ARGUMENTS THROUGH VOTING-GAMES POWER INDEXES Thanks for your attention!
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