Argumentation Logics Lecture 4 Games for abstract argumentation

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

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

Proof theory n Argument games between proponent (P) and opponent (O): n n n Proponent starts with an argument Then each party replies with a suitable defeater A winning criterion n n E. g. the other player cannot move Acceptability status corresponds to existence of a winning strategy.

Strategies n A strategy for player p is a partial game tree: n n n 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 Sdispute that begins with A

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

The G-game for grounded semantics: n A sound and complete game: n n n 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

A defeat graph A F B C D E

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

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

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

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

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

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

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

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

Two notions for the P-game n n 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

The P-game for preferred semantics n A move is legal iff: n 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 n New replies to the same move are different n n n The winner is P iff: n O has run out of legal moves, or n The dispute is infinite The winner is O iff: n n P has run out of legal moves, or O does an eo ipso move

Soundness and completeness n n 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 Pprovable and none of A’s defeaters are Pprovable