Hunt for the Collapse of Semantics in Infinite Abstract Argumentation Frameworks 1 Christof Spanring Department of Computer Science, University of Liverpool, UK Institute of Information Systems, TU Wien, Austria ICCSW15, September 25, 2015 1 This research has been supported by FWF (project I1102).
Fact Check I a some argument Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 1 / 16
Fact Check II some attack Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 2 / 16
Fact Check III a 0 , 0 a 1 , 0 a 2 , 0 · · · a 0 , 1 ¯ ¯ a 1 , 1 a 2 , 1 ¯ a 0 , 1 a 1 , 1 a 2 , 1 b 0 , 0 b 1 , 0 b 2 , 0 ¯ ¯ ¯ b 0 , 1 b 1 , 1 b 2 , 1 · · · b 0 , 1 b 1 , 1 b 2 , 1 ¯ ¯ ¯ b 0 , 2 b 1 , 2 b 2 , 2 b 0 , 2 b 1 , 2 b 2 , 2 c 0 , 0 c 1 , 0 c 2 , 0 ¯ ¯ ¯ c 0 , 1 c 1 , 1 c 2 , 1 c 0 , 1 c 1 , 1 c 2 , 1 · · · ¯ ¯ ¯ c 0 , 2 c 1 , 2 c 2 , 2 c 0 , 2 c 1 , 2 c 2 , 2 ¯ c 0 , 3 c 1 , 3 ¯ ¯ c 2 , 3 c 0 , 3 c 1 , 3 c 2 , 3 · · · · · · · · · · · · some infinite argumentation framework Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 3 / 16
Let’s get to the real stuff Theorem ([Baumann and Spanring, 2015, Weydert, 2011]) Any finitary (no argument with infinitely many attackers) argumentation framework provides semi-stable and stage extensions. Theorem (Not yet published) For any framework-property that is subframework-valid and guarantees existence of stage extensions, we can have any finite amount of arguments violating this property without loosing the guarantee for the existence of stage extensions. Corollary (Conjecture from this paper) If for some argumentation framework there is no stage extension, then there is an infinite amount of arguments with infinitely many attackers. Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 4 / 16
Outline Introduction 1 Fact Checks Real Stuff ;) Background 2 Examples Definitions 3 Real Real Stuff More Examples Theorems Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 5 / 16
Stable, Stage and Semi-Stable Semantics a a c c a a c c b b stb : {{ b }} stb : ∅ sem : {{ b }} sem : {{ a }} stg : {{ b }} stg : {{ a } , { b }} a a c c a a c c b b stb : ∅ stb : ∅ sem : {∅} sem : {{ a }} stg : {{ b }} stg : {{ a }} Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 6 / 16
Stable, Stage and Semi-Stable Semantics ctd. a a a b b b stb : {{ a }} stb : ∅ stb : ∅ sem : {{ a }} sem : {∅} sem : {{ a }} stg : {{ a }} stg : {∅} stg : {{ a }} Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 7 / 16
Stage and Semi-Stable Semantics c c a a b d d sem : {{ a }} stg : {{ b }} Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 8 / 16
Definitions Definition ([Dung, 1995]) An argumentation framework is a pair F = ( A , R ) of arguments A and attacks R ⊆ A × A . The range of a set of arguments S is given as S + = S ∪ { a ∈ A , S a } . Definition ([Verheij, 2003, Caminada and Verheij, 2010]) A set S ⊆ A is called conflict conflict-free, S ∈ cf ( F ) , if S × S ∩ R = ∅ . S ∈ cf ( F ) is called admissible, S ∈ adm ( F ) , if a S implies S a ; a stable extension, S ∈ stb ( F ) , if S + = A ; a stage extension, S ∈ stg ( F ) , if it is maximal in range. An set S ∈ adm ( A ) is called a semi-stable extension, S ∈ sem ( F ) , if it is maximal in range. Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 9 / 16
Outline Introduction 1 Fact Checks Real Stuff ;) Background 2 Examples Definitions 3 Real Real Stuff More Examples Theorems Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 10 / 16
Collapse of Stage Semantics [Verheij, 2003] · · · p 0 p 0 p 1 p 2 p 3 p 4 p 5 p 5 Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 11 / 16
Collapse of Semi-Stable and Stage Semantics [Verheij, 2003] r 5 r 0 r 1 r 2 r 3 r 4 r 5 p 0 p 1 p 2 p 3 p 4 p 5 · · · q 0 q 0 q 1 q 2 q 3 q 4 q 5 Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 12 / 16
Collapse of Semi-Stable Semantics r 5 s r 0 r 1 r 2 r 3 r 4 r 5 p 0 p 1 p 2 p 3 p 4 p 5 · · · q 0 q 0 q 1 q 2 q 3 q 4 q 5 Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 13 / 16
Collapse of Semi-Stable and Stage Semantics r 4 r 0 r 1 r 2 r 3 r 4 p 0 p 1 p 2 p 3 p 4 · · · s 0 s 1 s 2 s 3 s 4 q 0 q 0 q 1 q 2 q 3 q 4 Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 14 / 16
Collapse of Semi-Stable Semantics y 0 y 1 y 2 y 3 r 5 z 0 z 3 · · · z 1 z 2 x 0 x 1 x 2 x 3 q 0 Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 15 / 16
Insights Theorem ([Baumann and Spanring, 2015, Weydert, 2011]) Any finitary (no argument with infinitely many attackers) argumentation framework provides semi-stable and stage extensions. Theorem (Not yet published) For any framework-property that is subframework-valid and guarantees existence of stage extensions, we can have any finite amount of arguments violating this property without loosing the guarantee for the existence of stage extensions. Corollary (Conjecture from this paper) If for some argumentation framework there is no stage extension, then there is an infinite amount of arguments with infinitely many attackers. Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 16 / 16
References Baroni, P . and Giacomin, M. (2009). Semantics of abstract argument systems. In Rahwan, I. and Simari, G. R., editors, Argumentation in Artificial Intelligence , chapter 2, pages 25–44. Springer. Baumann, R. and Spanring, C. (2015). Infinite argumentation frameworks. Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation , pages 281–295. Caminada, M. W. A. and Verheij, B. (2010). On the existence of semi-stable extensions. In Procs. of the 22nd Benelux Conference on Artificial Intelligence (BNAIC’10) . 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. Verheij, B. (2003). Deflog: on the logical interpretation of prima facie justified assumptions. J. Log. Comput. , 13(3):319–346. Weydert, E. (2011). Semi-stable extensions for infinite frameworks. In Procs. of the 23rd Benelux Conference on Artificial Intelligence (BNAIC’11) , pages 336–343. Christof Spanring, ICCSW15 Collapses in Infinite Argumentation 16 / 16
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