Argumentation Logics
Lecture 4:
Games for abstract
argumentation
Henry Prakken
Chongqing
June 1, 2010
Contents
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Summary of lecture 3
Abstract argumentation: proof theory as
argument games
Game for grounded semantics
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Prakken & Sartor (1997)
Game for preferred semantics
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Vreeswijk & Prakken (2000)
Semantics of abstract argumentation
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INPUT: an abstract argumentation
theory AAT = Args,Defeat
OUTPUT: A division of Args into
justified, overruled and defensible
arguments
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Labelling-based definitions
Extension-based definitions
Labelling-based definitions:
status assignments
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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.
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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
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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:
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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
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Grounded semantics:
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A is justified if A is in the grounded extension
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A is overruled if A is not justified and A is defeated by an argument
that is justified
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So if A is Out in the grounded s.a.
A is defensible otherwise (so if it is not justified and not overruled)
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So if A is In in the grounded s.a.
So if A is undecided in the grounded s.a.
Stable/preferred semantics:
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A is justified if A is in all stable/preferred extensions
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A is overruled if A is in no stable/preferred extensions
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So if A is In in all s./p.s.a.
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
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So if A is In in some but not all s./p.s.a.
Proof theory of abstract
argumentation
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Argument games between proponent (P) and
opponent (O):
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Proponent starts with an argument
Then each party replies with a suitable defeater
A winning criterion
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E.g. the other player cannot move
Semantic status corresponds to existence of a
winning strategy for P.
Strategies
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A dispute is a single game played by the players
A strategy for player p (p  {P,O}) is a partial game
tree:
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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
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The rules of the game and winning
criterion depend on the semantics:
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May players
May players
arguments?
May players
arguments?
May players
repeat their own arguments?
repeat each other’s
use weakly defeating
backtrack?
The G-game for grounded semantics:
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A sound and complete game:
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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
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
Proponent’s winning strategy
A
P: A
F
O: F
O: B
B
P: E
C
D
move
E
P: E
The G-game for grounded semantics:
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A sound and complete game:
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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
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The appropriate rules of the game and
winning criterion depend on the
semantics:
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May players
May players
arguments?
May players
arguments?
May players
repeat their own arguments?
repeat each other’s
use weakly defeating
backtrack?
Two notions for the P-game
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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
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A move is legal iff:
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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
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New replies to the same move are different
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The winner is P iff:
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O cannot make a legal move, or
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The dispute is infinite
The winner is O iff:
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P cannot make a legal move, or
O does an eo ipso move
Soundness and completeness
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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 Pprovable and none of A’s defeaters are Pprovable