argumentation logics lecture 4: games for abstract argumentation henry prakken chongqing june 1,...
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Argumentation LogicsLecture 4:
Games for abstract argumentation
Henry PrakkenChongqing
June 1, 2010
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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)
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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
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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.
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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
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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.
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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.
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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
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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?
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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
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A defeat graph
A
B
C
D
E
F
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A game tree
P: AA
B
C
D
E
F
move
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A game tree
P: AA
B
C
D
E
F
O: F
move
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A game tree
P: AA
B
C
D
E
F
O: F
P: E
move
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A game tree
P: A
O: B
A
B
C
D
E
F
O: F
P: E
move
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A game tree
P: A
O: B
P: C
A
B
C
D
E
F
O: F
P: E
move
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A game tree
P: A
O: B
P: C
O: D
A
B
C
D
E
F
O: F
P: E
move
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A game tree
P: A
O: B
P: C P: E
O: D
A
B
C
D
E
F
O: F
P: E
move
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Proponent’s winning strategy
P: A
O: B
P: E
A
B
C
D
E
F
O: F
P: E
move
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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
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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?
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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
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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
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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
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