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Commonsense Reasoning and Argumentation 18/19 HC 7 Abstract Argumentation Proof theory

Commonsense Reasoning and Argumentation 18/19 HC 7 Abstract Argumentation Proof theory. Henry Prakken (with contributions by Liz Black) March 1, 2019. Recap. Grounded, stable and preferred semantics. Semantics define sets of arguments ( extensions ) that it is reasonable to accept.

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Commonsense Reasoning and Argumentation 18/19 HC 7 Abstract Argumentation Proof theory

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  1. Commonsense Reasoning and Argumentation 18/19HC 7Abstract ArgumentationProof theory Henry Prakken (with contributions by Liz Black) March 1, 2019

  2. Recap Grounded, stable and preferred semantics. Semantics define sets of arguments (extensions) that it is reasonable to accept. Two approaches to defining extensions Dung’s set based approach Labelling approach Today: procedure to determine whether a given argument is a member of an extension

  3. Proof theory for abstract argumentation Argument games between proponentP and opponent O : Proponent starts with an argument Then each party replies with a suitable defeater A winning criterion E.g. the other player cannot move Acceptability status corresponds to existence of a winning strategy.

  4. Strategies A strategy for player p is a partial game tree: Every branch is a dispute (sequence of allowable moves) The tree only branches after moves by p The children of p’s moves are all the legal moves by the other player P: A O: C O: B P: E P: D O: G O: F P: H

  5. Strategies A strategy S for player p is winning iff p wins all disputes in S Let S be an argument game: A is S-provable iff P has a winning strategy in an S-dispute that begins with A P: A O: C O: B P: E P: D O: G O: F P: I P: H

  6. Rules of the game: choice options • The appropriate rules of the game and winning criterion depend on the semantics: • May players repeat their own arguments? • May players repeat each other’s arguments? • May players use weakly defeating arguments? • May players backtrack?

  7. The G-game for grounded semantics: A sound and complete game: Each move must reply to the previous move Proponent cannot repeat his moves Proponent moves strict defeaters, opponent moves defeaters A player wins iff the other player cannot move Proposition: A is in the grounded extension iff A is G-provable

  8. The G-game for grounded semantics: A sound and complete game: Each move must reply to the previous move Proponent cannot repeat his moves Proponent moves strict defeaters, opponent moves defeaters A player wins iff the other player cannot move Result: A is in the grounded extension iff A is G-provable

  9. A defeat graph A F B C E D

  10. A game tree move P: A A F B C E D

  11. A game tree move P: A A F O: F B C E D

  12. A game tree P: A A F O: F B P: E C move E D

  13. A game tree P: A A F O: B move O: F B P: E C E D

  14. A game tree P: A A F O: B O: F B P: E P: C C E move D

  15. A game tree P: A A F O: B O: F B P: E P: C C E O: D move D

  16. A game tree P: A A F O: B O: F B P: E P: C P: E C move E O: D D

  17. Proponent’s winning strategy P: A A F O: B O: F B P: E P: E C move E D

  18. Exercise F A B E C D • Draw the complete game tree for D. • How many strategies are there for P? • How many strategies are there for O? • Who has a winning strategy?

  19. Exercise F A B P: D O: B E C O: C D P: A P: A P: E O: F

  20. The P-game for credulous preferred semantics Credulous, so testing for membership of some preferred extension. Idea: A preferred extension is a maximal admissible set. Each admissible set is contained in a maximal admissible set. Try to construct admissible set around the argument in question.

  21. Rules of the game: choice options The appropriate rules of the game and winning criterion depend on the semantics: May players repeat their own arguments? May players repeat each other’s arguments? May players use weakly defeating arguments? May players backtrack?

  22. P-game: May P defeat weakly? A B C A B C A B C

  23. P-game: May P defeat weakly? A B C P: A

  24. P-game: May P defeat weakly? A B C P: A O: B

  25. P-game: May P defeat weakly? A B C P: A O: B P: C

  26. P-game: Must O defeat strictly? See Example 5.3.6 Reader (and page 52 below)

  27. P-game: May P repeat P? A B A B A B

  28. P-game: May P repeat P? A B P: A

  29. P-game: May P repeat P? A B P: A O: B

  30. P-game: May P repeat P? A B P: A O: B P: A

  31. P-game: May O repeat O? A B P: A O: B P: A

  32. P-game: May P repeat O? A B C D

  33. P-game: May P repeat O? A B C D P: A

  34. P-game: May P repeat O? A B C D P: A O: B

  35. P-game: May P repeat O? A B C D P: A O: B P: C

  36. P-game: May P repeat O? A B C D P: A O: B P: C O: D

  37. P-game: May O repeat P? A B C P: A

  38. P-game: May O repeat P? A B C P: A O: B

  39. P-game: May O repeat P? A B C P: A O: B P: C

  40. P-game: May O repeat P? A B C P: A O: B P: C O: A

  41. P-game: May O backtrack? A B C A B A B C C

  42. P-game: May O backtrack? A B C P: A

  43. P-game: May O backtrack? A B C P: A O: B

  44. P-game: May O backtrack? A B C P: A O: B P: C

  45. P-game: May O backtrack? A B C P: A O: B P: C

  46. P-game: May O backtrack? A B C P: A O: C O: B P: C

  47. P-game: May O backtrack? A B C P: A P: A O: B O: C O: C O: B P: B P: C

  48. Single games vs. Game tree P: A P: A P: A O: B O: C O: B O: C O: C O: B P: C P: B P: B P: C O: B O: C

  49. Two notions for the P-game A dispute line is a sequence of moves each replying to the previous move: An eo ipso move is a move that repeats a move of the other player

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