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

2. Contents Abstract argumentation: proof theory as argument games Game for grounded semantics Game for preferred semantics

3. Proof theory 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 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 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

5. 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?

6. The G-game for grounded semantics: A sound and complete game: Each move replies to previous move (Proponent does not repeat 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

7. A defeat graph A B C D E F

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

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

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

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

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

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

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

17. 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?

18. 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

19. 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 has run out of legal moves, or The dispute is infinite The winner is O iff: P has run out of legal moves, or O does an eo ipso move

20. Soundness and completeness Proposition 6.3.8: 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