1. Argumentation Logics Lecture 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 theory AAT =  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 Out in 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 In in 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 In in 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 B C D E F

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

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

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

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

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

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

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

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

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

22. 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?

23. 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

24. 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

25. 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