Argumentation LogicsLecture 4:

Games for abstract argumentation

Henry PrakkenChongqing

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

Vreeswijk & Prakken (2000)

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

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.

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

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.

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.

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

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?

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

A defeat graph

A

B

C

D

E

F

A game tree

P: AA

B

C

D

E

F

move

A game tree

P: AA

B

C

D

E

F

O: F

move

A game tree

P: AA

B

C

D

E

F

O: F

P: E

move

A game tree

P: A

O: B

A

B

C

D

E

F

O: F

P: E

move

A game tree

P: A

O: B

P: C

A

B

C

D

E

F

O: F

P: E

move

A game tree

P: A

O: B

P: C

O: D

A

B

C

D

E

F

O: F

P: E

move

A game tree

P: A

O: B

P: C P: E

O: D

A

B

C

D

E

F

O: F

P: E

move

Proponent’s winning strategy

P: A

O: B

P: E

A

B

C

D

E

F

O: F

P: E

move

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

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?

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

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

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