Trumpet Reincarnations: Abstract Argumentation1
Christof Spanring
Department of Computer Science, University of Liverpool, UK
Institute of Information Systems, TU Wien, Austria
Porgy & Bess: Lost & FoundApril 24, 2017
1This research has been supported by FWF (project I1102).
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
May all beings be happy.May they live in safety and joy.All living beings,Whether weak or strong,Tall, stout, average or short,Seen or unseen, near or distant,Born or to be born,May they all be happy.
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
happy
joy
safe
weak
strong
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
h
js
wr
in: s,r
out: w,j
undec: h
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
h
js
wr
in: s,r
out: w,j
undec: h
h
js
wr
in: j,r
out: s,w
undec: h
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Introduction
h
js
wr
in: s,r
out: w,j
undec: h
h
js
wr
in: j,r
out: s,w
undec: h
h
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in: w,j
out: s,h,r
undec:
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21
Outline
1 Introduction
2 Stable Extensions and Conflict
3 Variations of Stable Semantics
4 Collapse
5 Perfection
6 References
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 2 / 21
What are the stable extensions?
a b
u
d
cr
p
t1t2t3· · ·
{c, r, b, t1, t2, t3, . . .}{d, b, t1, t2, t3, . . .}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 3 / 21
What are the stable extensions?
a b
u
d
cr
p
t1t2t3· · ·
Stable Extensions:
{c, r, b, t1, t2, t3, . . .}
{d, b, t1, t2, t3, . . .}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 3 / 21
What are the stable extensions?
a b
u
d
cr
p
t1t2t3· · ·
Stable Extensions:
{c, r, b, t1, t2, t3, . . .}{d, b, t1, t2, t3, . . .}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 3 / 21
Relation between arguments c and u?
a b
u
d
cr
p
t1t2t3· · ·
Stable Extensions:
{c, r, b, t1, t2, t3, . . .}{d, b, t1, t2, t3, . . .}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 4 / 21
Relation between arguments c and u?
a b
u
d
cr
p
t1t2t3· · ·
Stable Extensions:
{c, r, b, t1, t2, t3, . . .}{d, b, t1, t2, t3, . . .}
Arguments c and u are implicitly in conflict.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 4 / 21
Relation between arguments c and u?
a b
u
d
cr
p
t1t2t3· · ·
Stable Extensions:
{c, r, b, t1, t2, t3, . . .}{d, b, t1, t2, t3, . . .}
Arguments c and u are implicitly in conflict. An attack (c, u) does notchange the extension sets.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 4 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
{3, 1}{4, 2}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
Stable Extensions:
{3, 1}{4, 2}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
Stable Extensions:
{3, 1}{4, 2}
Only Implicit Conflict:
1 vs. 4
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
Stable Extensions:
{3, 1}{4, 2}
Only Implicit Conflict:
1 vs. 4
Resultion:
(1, 4)
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
Stable Extensions:
{3, 1}{4, 2}
Only Implicit Conflict:
1 vs. 4
Resultion:
(1, 4) or (4, 1)
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts made Explicit in a Simpler Case
12 3
4
Stable Extensions:
{3, 1}{4, 2}
Only Implicit Conflict:
1 vs. 4
Resultion:
(1, 4) or (4, 1) or both
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21
Implicit Conflicts vs. Explicit Conflicts III
x2
y2 b0 x0
v0
a1 y1
u1b2
a0y0
u0
b1x1
v1 a2
Argument set of interest: {a0, a1, y2, u0, u1, v0, v1}
TheoremSome implicit conflicts can not be made explicit for stable semantics.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 6 / 21
Implicit Conflicts vs. Explicit Conflicts III
x2
y2 b0 x0
v0
a1 y1
u1b2
a0y0
u0
b1x1
v1 a2
TheoremSome implicit conflicts can not be made explicit for stable semantics.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 6 / 21
Outline
1 Introduction
2 Stable Extensions and Conflict
3 Variations of Stable Semantics
4 Collapse
5 Perfection
6 References
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 7 / 21
What are the stable extensions?
12
3
4 5
6
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21
What are the stable extensions?
12
3
4 5
6
Stable Extensions:
{1, 3, 5} and {2, 4, 6}
There are none!!!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21
What are the stable extensions?
1
2
3
4
5
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21
What are the stable extensions?
1
2
3
4
5
Stable Extensions:
{1, 3, 5} and {2, 4, 6}
There are none!!!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21
What are the stable extensions?
12
3
4 5
6
1
2
3
4
5
Why is collapse a problem?
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21
Collapse Solutions
1 2
3
4
ExampleCollapse of Stable Semantics.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21
Collapse Solutions
1 2
3
4
Definition (Stage)Consider argument sets whereundecided arguments areminimized.
Stage Extensions:
{2}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21
Collapse Solutions
1 2
3
4
Definition (Stage)Consider argument sets whereundecided arguments areminimized.
Stage Extensions:
{2}
1 2
3
4
Definition (Semi-stable)Consider self-defendingargument sets where undecidedarguments are minimized.
Semi-stable Extensions:
{1}
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21
Collapse Solutions II
TheoremIf there only finitely many arguments then stage and semi-stable nevercollapse.
TheoremIf stage does not collapse for any induced sub-structure, then it does stillnot collapse if we add any finite amount of arguments and induced attacks.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 10 / 21
Outline
1 Introduction
2 Stable Extensions and Conflict
3 Variations of Stable Semantics
4 Collapse
5 Perfection
6 References
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 11 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Stage Collapse
b1 b2 b3 b4 b5 b6. . .
ExampleThere is no biggest natural number. . .
ExampleBut the empty set is a semi-stable extension!
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21
Collapse or not?
· · ·
q0
p0
r0
q1
p1
r1
q2
p2
r2
q3
p3
r3
q4
p4
r4
q5
p5
r5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 13 / 21
Collapse or not?
· · ·
q0
p0
r0
q1
p1
r1
q2
p2
r2
q3
p3
r3
q4
p4
r4
q5
p5
r5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 13 / 21
Collapse or not?
· · ·
p0
r0
p1
r1
p2
r2
p3
r3
p4
r4
p5
r5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 14 / 21
Collapse or not?
· · ·
p0
r0
p1
r1
p2
r2
p3
r3
p4
r4
p5
r5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 14 / 21
Loop-free: collapse or not?
· · ·
q0
s0
p0
r0
q1
s1
p1
r1
q2
s2
p2
r2
q3
s3
p3
r3
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 15 / 21
Loop-free: collapse or not?
· · ·
q0
s0
p0
r0
q1
s1
p1
r1
q2
s2
p2
r2
q3
s3
p3
r3
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 15 / 21
Cycle-free: collapse or not?
· · ·
· · ·
· · ·
· · ····
···
···
···
···
···
30
20
10
00
p0
31
21
11
01
p1
32
22
12
02
p2
33
23
13
03
p3
34
24
14
04
p4
35
25
15
05
p5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 16 / 21
Cycle-free: collapse or not?
· · ·
· · ·
· · ·
· · ····
···
···
···
···
···
30
20
10
00
p0
31
21
11
01
p1
32
22
12
02
p2
33
23
13
03
p3
34
24
14
04
p4
35
25
15
05
p5
ExampleStage and semi-stable collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 16 / 21
Only one argument with finitely many attackers:collapse or not?
· · ·x0
y0
z0
x1
y1
z1
x2
y2
z2
x3
y3
z3
ExampleSemi-stable collapses, stage not.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 17 / 21
Only one argument with finitely many attackers:collapse or not?
· · ·x0
y0
z0
x1
y1
z1
x2
y2
z2
x3
y3
z3
ExampleSemi-stable collapses, stage not.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 17 / 21
Outline
1 Introduction
2 Stable Extensions and Conflict
3 Variations of Stable Semantics
4 Collapse
5 Perfection
6 References
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 18 / 21
Collapse Solutions II
TheoremIf there is no infinite downwards path with no starting point (such as(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse.
TheoremIn ZFC if all attacks are symmetric and not self-attacking, then stage andsemi-stable do not collapse.
TheoremIn ZFC bipartite argumentation frameworks never collapse for stage orsemi-stable.
TheoremIn ZFC if there is no argument with infinitely many attackers then stageand semi-stable do not collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21
Collapse Solutions II
TheoremIf there is no infinite downwards path with no starting point (such as(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse.
TheoremIn ZFC if all attacks are symmetric and not self-attacking, then stage andsemi-stable do not collapse.
TheoremIn ZFC bipartite argumentation frameworks never collapse for stage orsemi-stable.
TheoremIn ZFC if there is no argument with infinitely many attackers then stageand semi-stable do not collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21
Collapse Solutions II
TheoremIf there is no infinite downwards path with no starting point (such as(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse.
TheoremIn ZFC if all attacks are symmetric and not self-attacking, then stage andsemi-stable do not collapse.
TheoremIn ZFC bipartite argumentation frameworks never collapse for stage orsemi-stable.
TheoremIn ZFC if there is no argument with infinitely many attackers then stageand semi-stable do not collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21
Collapse Solutions II
TheoremIf there is no infinite downwards path with no starting point (such as(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse.
TheoremIn ZFC if all attacks are symmetric and not self-attacking, then stage andsemi-stable do not collapse.
TheoremIn ZFC bipartite argumentation frameworks never collapse for stage orsemi-stable.
TheoremIn ZFC if there is no argument with infinitely many attackers then stageand semi-stable do not collapse.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21
What is ZFC and why bother about Choice?
Selecting Nodes/Elements: a choice function
ExampleConsider a bucket of water. You know that there is one O-atom in eachmolecule. But can you select exactly one O-atom from each H20molecule?
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 20 / 21
What is ZFC and why bother about Choice?
Selecting Nodes/Elements: a choice function
ExampleConsider a bucket of water. You know that there is one O-atom in eachmolecule. But can you select exactly one O-atom from each H20molecule?
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 20 / 21
Downsides of Choice: Axiom of Determinacy
B AB A
0
0
000 000
001 001
010 010
011 011
1
1
100 100
101 101
110 110
111 111
00
00
00000000
00010001
00100010
00110011
01
01
01000100
01010101
01100110
01110111
10
10
10001000
10011001
10101010
10111011
11
11
11001100
11011101
11101110
11111111
S0000
S0001
S0010
S0011
S0100
S0101
S0110
S0111
S1010
S1011
S1100
S1101
S1000
S1001
S1110
S1111
0
0
00
00
01
01
1
1
10
10
11
11
000 000
00000000
00010001
001 001
00100010
00110011
010 010
01000100
01010101
011 01101100110
01110111
100 100
10001000
10011001
101 10110101010
10111011 110 110
11001100
11011101
111 111
11101110
11111111
S0000
S0001
S0010
S0011
S0100
S0101
S0110
S0111
S1000
S1001S1010
S1011
S1100
S1101
S1110
S1111
TheoremThe 2ω generalization of this framework does not collapse for stablesemantics if and only if we do not assume Choice.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 21 / 21
References
Baumann, R., Dvorák, W., Linsbichler, T., Spanring, C., Strass, H., and Woltran, S. (2016).On rejected arguments and implicit conflicts: The hidden power of argumentation semantics.Artif. Intell., 241:244–284.
Dung, P. M. (1995).On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-persongames.Artif. Intell., 77(2):321–357.
Gödel, K. and Brown, G. W. (1940).The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory.Princeton University Press.
Jech, T. (2006).Set Theory.Springer, 3rd edition.
Mycielski, J. (1964).On the axiom of determinateness.Fundamenta Mathematicae, 53(2):205–224.
Spanring, C. (2015).Hunt for the Collapse of Semantics in Infinite Abstract Argumentation Frameworks.In OASIcs-OpenAccess Series in Informatics, volume 49. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Walton, D. N. (1984).Logical Dialogue-Games.University Press of America, Lanham, Maryland.
Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 21 / 21
. . . may we all be happy . . .