1 / 26

Argumentation Logics Lecture 4: Games for abstract argumentation

Argumentation Logics Lecture 4: Games for abstract argumentation. Henry Prakken Chongqing June 1, 2010. Contents. Summary of lecture 3 Abstract argumentation: proof theory as argument games Game for grounded semantics Prakken & Sartor (1997) Game for preferred semantics

Download Presentation

Argumentation Logics Lecture 4: Games for abstract argumentation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Argumentation LogicsLecture 4:Games for abstract argumentation Henry Prakken Chongqing June 1, 2010

  2. Contents • Summary of lecture 3 • Abstract argumentation: proof theory as argument games • Game for grounded semantics • Prakken & Sartor (1997) • Game for preferred semantics • Vreeswijk & Prakken (2000)

  3. Semantics of abstract argumentation INPUT: an abstract argumentation theoryAAT = Args,Defeat OUTPUT: A division of Args into justified, overruled and defensible arguments Labelling-based definitions Extension-based definitions

  4. Labelling-based definitions:status assignments A status assignment assigns to zero or more members of Args either the status In or Out (but not both) such that: 1. An argument is In iff all arguments defeating it are Out. 2. An argument is Out iff it is defeated by an argument that is In. Let Undecided = Args / (InOut): A status assignment is stable if Undecided = . A status assignment is preferred if Undecided is -minimal. A status assignment is grounded if Undecided is -maximal.

  5. Extension-based definitions S is conflict-free if no member of S defeats a member of S S is admissible if S is conflict-free and all its members are defended by S S is a stable extension if it is conflict-free and defeats all arguments outside it S is a preferred extension if it is a -maximally admissible set S is the grounded extension if S is the endpoint of the following sequence: S0: the empty set Si+1: Si + all arguments in Args that are defended by Si Propositions: S is the In set of a stable/preferred/grounded status assignment iff S is a stable/preferred/grounded extension

  6. Semantic status of arguments Grounded semantics: A is justified if A is in the grounded extension So if A is In in the grounded s.a. A is overruled if A is not justified and A is defeated by an argument that is justified So if A is Outin the grounded s.a. A is defensible otherwise (so if it is not justified and not overruled) So if A is undecided in the grounded s.a. Stable/preferred semantics: A is justified if A is in all stable/preferred extensions So if A is Inin all s./p.s.a. A is overruled if A is in no stable/preferred extensions So if A is Out or undecided in all s./p.s.a. A is defensible if A is in some but not all stable/preferred extension So if A is Inin some but not all s./p.s.a.

  7. Proof theory of abstract argumentation Argument games between proponent (P) 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 Semantic status corresponds to existence of a winning strategy for P.

  8. Strategies A dispute is a single game played by the players A strategy for player p (p {P,O}) is a partial game tree: Every branch is a dispute The tree only branches after moves by p The children of p’s moves are all legal moves by the other player 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

  9. Rules of the game: choice options • The 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?

  10. The G-game for grounded semantics: A sound and complete game: Each move replies to the previous move (Proponent does not repeat moves) Proponent moves (strict) defeaters, opponent moves defeaters A player wins iff the other player cannot make a legal move Theorem: A is in the grounded extension iff A is G-provable

  11. A defeat graph A F B C E D

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

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

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

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

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

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

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

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

  20. The G-game for grounded semantics: A sound and complete game: Each move replies to the previous move (Proponent does not repeat moves) Proponent moves (strict) defeaters, opponent moves defeaters A player wins iff the other player cannot make a legal move Theorem: A is in the grounded extension iff A is G-provable

  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. 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

  23. The P-game for preferred semantics A move is legal iff: P repeats no move of O O repeats no own move in the same dispute line P replies to the previous move O replies to some earlier move New replies to the same move are different The winner is P iff: O cannot make a legal move, or The dispute is infinite The winner is O iff: P cannot make a legal move, or O does an eo ipso move

  24. Soundness and completeness Theorem:A is in some preferred extension iff A is P-provable Also: If all preferred extensions are stable, then A is in all preferred extensions iff A is P-provable and none of A’s defeaters are P-provable

More Related