Commonsense Reasoning and
Argumentation 16/17
HC 7
Abstract Argumentation
Proof theory
Henry Prakken
(with contributions by Liz Black)
March 1, 2017
Recap



Grounded, stable and preferred semantics.
Semantics define sets of arguments
(extensions) that it is reasonable to accept.
Two approaches to defining extensions



Dung’s set based approach
Labelling approach
Today: procedure to determine whether a
given argument is a member of an extension
2
Proof theory for 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
Acceptability status corresponds to existence
of a winning strategy.
Strategies

A strategy for player p is a partial game tree:



Every branch is a dispute (sequence of allowable moves)
The tree only branches after moves by p
The children of p’s moves are all the legal moves by the
other player
P: A
O: B
P: E
P: D
O: F
O: C
O: G
P: H
4
Strategies


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
P: A
O: B
O: C
P: E
P: D
O: F
O: G
P: I
P: H
5
Rules of the game: choice options

The appropriate rules of the game and
winning criterion depend on the
semantics:




May players
May players
arguments?
May players
arguments?
May players
repeat their own arguments?
repeat each other’s
use weakly defeating
backtrack?
6
The G-game for grounded semantics:

A sound and complete game:





Each move must reply to the previous move
Proponent cannot repeat his moves
Proponent moves strict defeaters, opponent
moves defeaters
A player wins iff the other player cannot move
Proposition: A is in the grounded extension
iff A is G-provable
7
The G-game for grounded semantics:

A sound and complete game:





Each move must reply to the previous move
Proponent cannot repeat his 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
8
A defeat graph
A
F
B
C
E
D
9
A game tree
move
A
F
P: A
B
C
E
D
10
A game tree
move
A
P: A
F
O: F
B
C
E
D
11
A game tree
A
P: A
F
O: F
B
P: E
C
move
E
D
12
A game tree
A
P: A
F
move
O: F
O: B
B
P: E
C
E
D
13
A game tree
A
P: A
F
O: B
O: F
B
P: E
C
move
P: C
E
D
14
A game tree
A
P: A
F
O: B
O: F
B
P: E
C
P: C
E
O: D
move
D
15
A game tree
A
P: A
F
O: B
O: F
B
P: E
C
move
P: C
P: E
E
O: D
D
16
Proponent’s winning strategy
A
P: A
F
O: F
O: B
B
P: E
C
move
P: E
E
D
17
Exercise
F
E




A
C
B
D
Draw the complete game tree for D.
How many strategies are there for P?
How many strategies are there for O?
Who has a winning strategy?
18
Exercise
F
A
E
C
B
P: D
O: B
O: C
D
P: A
P: E
P: A
O: F
19
20
The P-game for credulous preferred
semantics


Credulous, so testing for membership of
some preferred extension.
Idea:



A preferred extension is a maximal
admissible set.
Each admissible set is contained in a
maximal admissible set.
Try to construct admissible set around the
argument in question.
21
Rules of the game: choice options

The appropriate rules of the game and
winning criterion depend on the
semantics:




May players
May players
arguments?
May players
arguments?
May players
repeat their own arguments?
repeat each other’s
use weakly defeating
backtrack?
22
P-game: May P defeat weakly?
A
A
B
B
C
C
A
B
C
23
P-game: May P defeat weakly?
A
B
C
P: A
24
P-game: May P defeat weakly?
A
B
C
P: A
O: B
25
P-game: May P defeat weakly?
A
B
C
P: A
O: B
P: C
26
P-game: Must O defeat strictly?
See Example 5.3.6 Reader
(and page 52 below)
27
P-game: May P repeat P?
A
A
B
B
A
B
28
P-game: May P repeat P?
A
B
P: A
29
P-game: May P repeat P?
A
B
P: A
O: B
30
P-game: May P repeat P?
A
B
P: A
O: B
P: A
31
P-game: May O repeat O?
A
B
P: A
O: B
P: A
32
P-game: May P repeat O?
A
B
C
D
33
P-game: May P repeat O?
A
B
C
D
P: A
34
P-game: May P repeat O?
A
B
C
D
P: A
O: B
35
P-game: May P repeat O?
A
B
C
D
P: A
O: B
P: C
36
P-game: May P repeat O?
A
B
C
D
P: A
O: B
P: C
O: D
37
P-game: May O repeat P?
A
P: A
B
C
38
P-game: May O repeat P?
A
P: A
B
C
O: B
39
P-game: May O repeat P?
A
P: A
B
C
O: B
P: C
40
P-game: May O repeat P?
A
P: A
B
C
O: B
P: C
O: A
41
P-game: May O backtrack?
A
B
C
A
B
C
A
B
C
42
P-game: May O backtrack?
A
P: A
B
C
43
P-game: May O backtrack?
A
P: A
B
C
O: B
44
P-game: May O backtrack?
A
P: A
B
C
O: B
P: C
45
P-game: May O backtrack?
A
P: A
B
C
O: B
P: C
46
P-game: May O backtrack?
A
C
P: A
O: B
B
O: C
P: C
47
P-game: May O backtrack?
A
C
P: A
O: B
P: C
B
O: C
P: A
O: B
O: C
P: B
48
Single games vs. Game tree
P: A
P: A
P: A
O: B
O: B
O: C
O: B
O: C
O: C
P: C
P: B
P: B
O: C
O: B
P: C
49
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
50
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

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The winner is P iff:

O has run out of legal moves, or

The dispute is infinite
The winner is O iff:


P has run out of legal moves, or
O does an eo ipso move
51
Example
A
C
D
B
E
52
Example
A
C
A
C
D
B
D
B
A
E
C
E
D
B
E
53
Example
A
P: A
C
D
B
E
54
Example
A
P: A
O: D
C
D
B
E
55
Example
A
P: A
O: D
C
D
B
E
P: E
56
Example
A
P: A
O: D
C
D
B
E
P: E
O: B
57
Example
A
P: A
O: D
C
D
B
E
P: E
O: B
P: E
58
Example
A
P: A
O: D
O: C
C
D
B
E
P: E
O: B
P: E
59
Soundness and completeness




Proposition: A is in some preferred extension
iff A is P-provable
Recall: Every admissible set is contained in a
maximal admissible set.
Hence: If A is in an admissible set, then it is
in some preferred extension.
To show: A is in an admissible set iff A is Pprovable
60
Soundness and completeness


To show: A is in an admissible set iff A is Pprovable
: A can be defended in every dispute.
S


A
To show: S is conflict
free
To show: S is
acceptable wrt itself
(i.e. defends all its
members)
61
Soundness and completeness


To show: A is in an admissible set iff A is Pprovable
: A can be defended in every dispute.
S
B

C

A
To show: S is conflict
free
To show: S is
acceptable wrt itself
(i.e. defends all its
members)
62
Soundness and completeness


To show: A is in an admissible set iff A is Pprovable
: A can be defended in every dispute.
B
S

C

A
To show: S is conflict
free
To show: S is
acceptable wrt itself
(i.e. defends all its
members)
63
Soundness and completeness


To show: A is in an admissible set iff A is Pprovable
: S is conflict free and acceptable wrt itself.
S


A
To show: P does not
run out of legal moves
To show: O does not
make an eo ipso move
64
Soundness and completeness


S
To show: A is in an admissible set iff A is Pprovable
: S is conflict free and acceptable wrt itself.
B
D

C

A
To show: P does not
run out of legal moves
To show: O does not
make an eo ipso
65
Soundness and completeness


To show: A is in an admissible set iff A is Pprovable
: S is conflict free and acceptable wrt itself.
S
B

C

A
To show: P does not
run out of legal moves
To show: O does not
make an eo ipso
66
Soundness and completeness



Proposition: 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
Recently proven: P has a winning strategy
for A iff there exists a terminated game for A
won by P (Caminada & Wu 2014)

A game is terminated if no new legal replies can
be added
67
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