Simple Problems. . . Example a 0 a 1 a 2 b 0 b 1 b 2 Question What - PowerPoint PPT Presentation

From Intractability to Inconceivability 1 Christof Spanring Institute of Information Systems, TU Wien, Austria Workshop on New Trends in Formal Argumentation 2017 1 This research has been supported by FWF (project I1102).

Simple Problems. . . Example a 0 a 1 a 2 b 0 b 1 b 2 Question What is some preferred extension? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 1 / 14

Simple Problems. . . Example a 0 a 1 a 2 b 0 b 1 b 2 Question What is some preferred extension? Answer The set { a 0 , a 1 , a 2 } is a preferred extension. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 1 / 14

Simple Problems. . . Example a 2 b 0 b 3 a 1 b 1 a 3 a 0 b 2 Question What is some preferred extension? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 2 / 14

Simple Problems. . . Example a 2 b 0 b 3 a 1 b 1 a 3 a 0 b 2 Question What is some preferred extension? Answer The set { a 0 , a 1 , a 2 , a 3 } is a preferred extension. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 2 / 14

Simple Problems. . . , . . . Example a 70 b 50 b 20 b 00 b 10 b 30 b 40 b 60 a 60 a 40 a 10 a 30 a 00 a 20 a 50 b 70 a 71 b 11 a 51 b 01 b 41 a 31 b 21 a 61 b 61 a 21 b 31 a 01 a 41 b 51 a 11 b 71 a 02 a 12 b 72 b 42 b 22 a 32 a 62 b 52 a 52 b 62 a 22 b 32 a 42 a 72 b 02 b 12 Question What is some preferred extension? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 3 / 14

Simple Problems. . . , . . . Example a 60 a 70 b 30 b 00 b 40 b 10 b 20 b 50 a 50 a 20 a 10 a 40 a 00 a 30 b 60 b 70 a 21 b 31 a 51 a 71 b 41 b 11 a 61 a 01 b 01 b 61 a 11 a 41 a 31 b 71 b 21 b 51 a 12 b 32 a 02 a 22 a 72 b 62 a 42 a 52 b 42 b 52 a 62 b 72 b 02 b 22 a 32 b 12 Question What is some preferred extension? Answer The set { a ij : i ∈ { 0 , 1 , . . . , 7 } , j ∈ { 0 , 1 , 2 }} is a preferred extension. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 3 / 14

Simple Problems. . . , . . . , ? Example Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 4 / 14

Simple Problems. . . , . . . , ? Example Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 4 / 14

Tractable vs. Intractable pr sm st sg na c2 s2 Ver σ - - � - � � - Cred σ - - - - � - - Skept σ - - - - � - - EX σ � � - � � � � NEX σ - - - � � � � Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 5 / 14

Approaches to the infinite case ASPIC Variants [Modgil and Prakken, 2014] Automata [Baroni et al., 2013] Logic Programming [García and Simari, 2004] Structured Argumentation . . . Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 6 / 14

Approaches to the infinite case ASPIC Variants [Modgil and Prakken, 2014] Automata [Baroni et al., 2013] Logic Programming [García and Simari, 2004] Structured Argumentation . . . Set Theoretic Approach for Arbitrary Infinities Zermelo-Fraenkel Set Theory Axiom of Choice, Zorn’s Lemma, Well-Ordering Theorem Transfinite Induction Bourbaki-Witt Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 6 / 14

Collapse I Example p 0 p 1 p 2 p 3 p 4 p 5 · · · · · · 0 0 0 1 0 2 0 3 0 4 0 5 · · · 1 0 1 1 1 2 1 3 1 4 1 5 2 0 2 1 2 2 2 3 2 4 2 5 · · · 3 0 3 1 3 2 3 3 3 4 3 5 · · · · · · · · · · · · · · · · · · Collapse of stable, semi-stable, stage, cf2, stage2 semantics in ZFC. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 7 / 14

Simple Problems? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 8 / 14

Simple Problems? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 8 / 14

Collapse II Example · · · Possible collapse of stable, semi-stable, stage, cf2, stage2, preferred, naive semantics in ZF , i.e. models of ZF where AC does not hold. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 9 / 14

Collapse and Perfection co na pr st sg sm c2 s2 gr id eg well-founded � � � � � � � � � � � bipartite � � � � � � � � � � � finite � � � - � � � � � � � limited controversial � AC AC AC AC AC AC AC � AC AC symmetric loop-free � AC AC AC AC AC AC AC � AC AC finitary � AC AC - AC AC ? ? � AC AC symmetric � AC AC - - - AC - � AC AC planar � AC AC - ? - ? ? � AC AC finitely superseded � AC AC - - - - - � AC AC finitarily superseded � AC AC - - - - - � AC AC arbitrary � AC AC - - - - - � AC AC Table: Perfection results. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 10 / 14

Expressiveness Question In the infinite case: Computational Complexity Intertranslatability Signatures Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 11 / 14

Expressiveness Question In the infinite case: Computational Complexity Intertranslatability Signatures Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 11 / 14

Expressiveness Question In the infinite case: Computational Complexity Intertranslatability Signatures Example Admissibility based semantics widely yield the same comparability, regardless of ZF or ZFC; , given extension set { 0 , 1 } ω we can give an AF with matching In ZF semantic evaluation; In ZF , a collection of pairs of arguments with symmetric conflicts might not provide maximal extensions; How do cf-based semantics compare? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 11 / 14

Facilitating Collapse for Translations Definition B ( x ) y ⇒ y x x Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

Facilitating Collapse for Translations Definition B ( x ) y ⇒ y x x Example B ( 1 ) B ( 2 ) B ( 3 ) B ( 4 ) 3 1 2 4 Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

Facilitating Collapse for Translations Definition B ( x ) y ⇒ y x x Example B ( 1 ) B ( 2 ) B ( 3 ) B ( 4 ) 3 1 2 4 Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

Facilitating Collapse for Translations Definition B ( x ) y ⇒ y x x Theorem In ZFC stable, stage, cf2 and stage2 semantics provide the same expressiveness. In ZF without AC even naive, stable, stage, cf2 and stage2 are comparable. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

Expressiveness st finite pr , sm na sg , s2 c2 Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

Expressiveness st finite pr , sm na sg , s2 c2 pr ZFC na c2 , sg , s2 , st sm Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

Expressiveness finite st pr , sm na sg , s2 c2 pr ZFC na c2 , sg , s2 , st sm ZF pr , sm na , c2 , sg , s2 , st Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

Conclusions The possibility of collapse can be considered a valuable tool. Inconceivable (i.e. collapsing) subframeworks can enforce other extensions. Similarly, can we make use of intractability for expressiveness in terms of tractable extensions? For general (finite) AFs, can we tractably detect intractability of subframeworks? Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 14 / 14