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

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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. . .

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SLIDE 1

From Intractability to Inconceivability 1

Christof Spanring

Institute of Information Systems, TU Wien, Austria

Workshop on New Trends in Formal Argumentation 2017

1This research has been supported by FWF (project I1102).

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SLIDE 2

Simple Problems. . .

Example a0 b0 a1 b1 a2 b2 Question

What is some preferred extension?

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 1 / 14

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SLIDE 3

Simple Problems. . .

Example a0 b0 a1 b1 a2 b2 Question

What is some preferred extension?

Answer

The set {a0, a1, a2} is a preferred extension.

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 1 / 14

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SLIDE 4

Simple Problems. . .

Example a0 b0 a1 b1 a2 b2 a3 b3 Question

What is some preferred extension?

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 2 / 14

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SLIDE 5

Simple Problems. . .

Example a0 b0 a1 b1 a2 b2 a3 b3 Question

What is some preferred extension?

Answer

The set {a0, a1, a2, a3} is a preferred extension.

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 2 / 14

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SLIDE 6

Simple Problems. . . , . . .

Example

a00 b00 a01 b01 a02 b02 a10 b10 a11 b11 a12 b12 a20 b20 a21 b21 a22 b22 a30 b30 a31 b31 a32 b32 a40 b40 a41 b41 a42 b42 a50 b50 a51 b51 a52 b52 a60 b60 a61 b61 a62 b62 a70 b70 a71 b71 a72 b72

Question

What is some preferred extension?

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 3 / 14

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SLIDE 7

Simple Problems. . . , . . .

Example

a00 b00 a01 b01 a02 b02 a10 b10 a11 b11 a12 b12 a20 b20 a21 b21 a22 b22 a30 b30 a31 b31 a32 b32 a40 b40 a41 b41 a42 b42 a50 b50 a51 b51 a52 b52 a60 b60 a61 b61 a62 b62 a70 b70 a71 b71 a72 b72

Question

What is some preferred extension?

Answer

The set {aij : 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

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SLIDE 8

Simple Problems. . . , . . . , ?

Example

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 4 / 14

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SLIDE 9

Simple Problems. . . , . . . , ?

Example

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 4 / 14

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SLIDE 10

Tractable vs. Intractable

pr sm st sg na c2 s2

Verσ

  • Credσ
  • Skeptσ
  • EXσ
  • NEXσ
  • Christof Spanring, Workshop on New Trends in Formal Argumentation 2017

From Intractability to Inconceivability 5 / 14

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SLIDE 11

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

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SLIDE 12

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

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SLIDE 13

Collapse I

Example · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 30 20 10 00 p0 31 21 11 01 p1 32 22 12 02 p2 33 23 13 03 p3 34 24 14 04 p4 35 25 15 05 p5

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

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SLIDE 14

Simple Problems?

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

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SLIDE 15

Simple Problems?

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

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SLIDE 16

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

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SLIDE 17

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

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SLIDE 18

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

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SLIDE 19

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

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SLIDE 20

Expressiveness

Question

In the infinite case: Computational Complexity Intertranslatability Signatures

Example

Admissibility based semantics widely yield the same comparability, regardless of ZF or ZFC; In ZF , given extension set {0, 1}ω we can give an AF with matching 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

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SLIDE 21

Facilitating Collapse for Translations

Definition x y ⇒ x y B(x)

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

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SLIDE 22

Facilitating Collapse for Translations

Definition x y ⇒ x y B(x) Example 1 B(1) 2 B(2) 3 B(3) 4 B(4)

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

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SLIDE 23

Facilitating Collapse for Translations

Definition x y ⇒ x y B(x) Example 1 B(1) 2 B(2) 3 B(3) 4 B(4)

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 12 / 14

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SLIDE 24

Facilitating Collapse for Translations

Definition x y ⇒ x y B(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

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Expressiveness

na

finite

c2 sg,s2 st pr,sm

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

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SLIDE 26

Expressiveness

na

finite

c2 sg,s2 st pr,sm na

ZFC

c2,sg,s2,st pr sm

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

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SLIDE 27

Expressiveness

na

finite

c2 sg,s2 st pr,sm na

ZFC

c2,sg,s2,st pr sm na,c2,sg,s2,st

ZF

pr,sm

Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 13 / 14

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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

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References

Baroni, P ., Cerutti, F., Dunne, P . E., and Giacomin, M. (2013). Automata for infinite argumentation structures.

  • Artif. Intell., 203:104–150.

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):321–357.

García, A. J. and Simari, G. R. (2004). Defeasible logic programming: An argumentative approach. TPLP, 4(1-2):95–138. Jech, T. (2006). Set Theory. Springer, 3rd edition. Modgil, S. and Prakken, H. (2014). The ASPIC+ framework for structured argumentation: a tutorial. Argument & Computation, 5(1):31–62. Spanring, C. (2015). Hunt for the Collapse of Semantics in Infinite Abstract Argumentation Frameworks. In OASIcs-OpenAccess Series in Informatics, volume 49. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Christof Spanring, Workshop on New Trends in Formal Argumentation 2017 From Intractability to Inconceivability 14 / 14