[PPT] - Introduction May all beings be happy. May they live in safety and PowerPoint Presentation

SLIDE 1

Trumpet Reincarnations: Abstract Argumentation1

Christof Spanring

Department of Computer Science, University of Liverpool, UK Institute of Information Systems, TU Wien, Austria

Porgy & Bess: Lost & Found April 24, 2017

1This research has been supported by FWF (project I1102).

SLIDE 2

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

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

Introduction

happy joy safe weak strong

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 9

Introduction

happy joy safe weak strong

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 10

Introduction

happy joy safe weak strong

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 11

Introduction

happy joy safe weak strong

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 12

Introduction

h j s w r in: s,r

ut: w,j

undec: h

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 13

Introduction

h j s w r in: s,r

ut: w,j

undec: h h j s w r in: j,r

ut: s,w

undec: h

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 14

Introduction

h j s w r in: s,r

ut: w,j

undec: h h j s w r in: j,r

ut: s,w

undec: h h j s w r in: w,j

ut: s,h,r

undec:

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 1 / 21

SLIDE 15

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

SLIDE 16

What are the stable extensions?

a b u d c r p t1 t2 t3 · · ·

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 3 / 21

SLIDE 17

What are the stable extensions?

a b u d c r p t1 t2 t3 · · ·

Stable Extensions:

{c, r, b, t1, t2, t3, . . .}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 3 / 21

SLIDE 18

What are the stable extensions?

a b u d c r p t1 t2 t3 · · ·

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

SLIDE 19

Relation between arguments c and u?

a b u d c r p t1 t2 t3 · · ·

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

SLIDE 20

Relation between arguments c and u?

a b u d c r p t1 t2 t3 · · ·

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

SLIDE 21

Relation between arguments c and u?

a b u d c r p t1 t2 t3 · · ·

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 not change the extension sets.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 4 / 21

SLIDE 22

Implicit Conflicts made Explicit in a Simpler Case

1 2 3 4

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21

SLIDE 23

Implicit Conflicts made Explicit in a Simpler Case

1 2 3 4

Stable Extensions:

{3, 1} {4, 2}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21

SLIDE 24

Implicit Conflicts made Explicit in a Simpler Case

1 2 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

SLIDE 25

Implicit Conflicts made Explicit in a Simpler Case

1 2 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

SLIDE 26

Implicit Conflicts made Explicit in a Simpler Case

1 2 3 4

Stable Extensions:

{3, 1} {4, 2}

Only Implicit Conflict: 1 vs. 4 Resultion:

(1, 4)

r

(4, 1)

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21

SLIDE 27

Implicit Conflicts made Explicit in a Simpler Case

1 2 3 4

Stable Extensions:

{3, 1} {4, 2}

Only Implicit Conflict: 1 vs. 4 Resultion:

(1, 4)

r

(4, 1)

r both

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 5 / 21

SLIDE 28

Implicit Conflicts vs. Explicit Conflicts III

x2 y2 b0 x0 v0 a1 y1 u1 b2 a0 y0 u0 b1 x1 v1 a2

Argument set of interest: {a0, a1, y2, u0, u1, v0, v1}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 6 / 21

SLIDE 29

Implicit Conflicts vs. Explicit Conflicts III

x2 y2 b0 x0 v0 a1 y1 u1 b2 a0 y0 u0 b1 x1 v1 a2 Theorem

Some implicit conflicts can not be made explicit for stable semantics.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 6 / 21

SLIDE 30

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

SLIDE 31

What are the stable extensions?

1 2 3 4 5 6

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21

SLIDE 32

What are the stable extensions?

1 2 3 4 5 6

Stable Extensions:

{1, 3, 5} and {2, 4, 6}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21

SLIDE 33

What are the stable extensions?

1 2 3 4 5

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21

SLIDE 34

What are the stable extensions?

1 2 3 4 5

Stable Extensions: There are none!!!

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 8 / 21

SLIDE 35

What are the stable extensions?

1 2 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

SLIDE 36

Collapse Solutions

1 2 3 4 Example

Collapse of Stable Semantics.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21

SLIDE 37

Collapse Solutions

1 2 3 4 Definition (Stage)

Consider argument sets where undecided arguments are minimized. Stage Extensions:

{2}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21

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Collapse Solutions

1 2 3 4 Definition (Stage)

Consider argument sets where undecided arguments are minimized. Stage Extensions:

{2} 1 2 3 4 Definition (Semi-stable)

Consider self-defending argument sets where undecided arguments are minimized. Semi-stable Extensions:

{1}

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 9 / 21

SLIDE 39

Collapse Solutions II

Theorem

If there only finitely many arguments then stage and semi-stable never collapse.

Theorem

If stage does not collapse for any induced sub-structure, then it does still not collapse if we add any finite amount of arguments and induced attacks.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 10 / 21

SLIDE 40

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

SLIDE 41

Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Example

There is no biggest natural number. . .

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

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Stage Collapse

b1 b2 b3 b4 b5 b6

. . .

Example

There is no biggest natural number. . .

Example

But the empty set is a semi-stable extension!

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 12 / 21

SLIDE 48

Collapse or not?

··· q0 p0 r0 q1 p1 r1 q2 p2 r2 q3 p3 r3 q4 p4 r4 q5 p5 r5

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 13 / 21

SLIDE 49

Collapse or not?

··· q0 p0 r0 q1 p1 r1 q2 p2 r2 q3 p3 r3 q4 p4 r4 q5 p5 r5 Example

Stage and semi-stable collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 13 / 21

SLIDE 50

Collapse or not?

··· p0 r0 p1 r1 p2 r2 p3 r3 p4 r4 p5 r5

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 14 / 21

SLIDE 51

Collapse or not?

··· p0 r0 p1 r1 p2 r2 p3 r3 p4 r4 p5 r5 Example

Stage and semi-stable collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 14 / 21

SLIDE 52

Loop-free: collapse or not?

··· q0 s0 p0 r0 q1 s1 p1 r1 q2 s2 p2 r2 q3 s3 p3 r3

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 15 / 21

SLIDE 53

Loop-free: collapse or not?

··· q0 s0 p0 r0 q1 s1 p1 r1 q2 s2 p2 r2 q3 s3 p3 r3 Example

Stage and semi-stable collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 15 / 21

SLIDE 54

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

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 16 / 21

SLIDE 55

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 Example

Stage and semi-stable collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 16 / 21

SLIDE 56

Only one argument with finitely many attackers: collapse or not?

··· x0 y0 z0 x1 y1 z1 x2 y2 z2 x3 y3 z3

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 17 / 21

SLIDE 57

Only one argument with finitely many attackers: collapse or not?

··· x0 y0 z0 x1 y1 z1 x2 y2 z2 x3 y3 z3 Example

Semi-stable collapses, stage not.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 17 / 21

SLIDE 58

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

SLIDE 59

Collapse Solutions II

Theorem

If there is no infinite downwards path with no starting point (such as

(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21

SLIDE 60

Collapse Solutions II

Theorem

If there is no infinite downwards path with no starting point (such as

(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse. Theorem

In ZFC if all attacks are symmetric and not self-attacking, then stage and semi-stable do not collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21

SLIDE 61

Collapse Solutions II

Theorem

If there is no infinite downwards path with no starting point (such as

(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse. Theorem

In ZFC if all attacks are symmetric and not self-attacking, then stage and semi-stable do not collapse.

Theorem

In ZFC bipartite argumentation frameworks never collapse for stage or semi-stable.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21

SLIDE 62

Collapse Solutions II

Theorem

If there is no infinite downwards path with no starting point (such as

(x1, x0), (x2, x1), . . ., or any cycle) then stable does not collapse. Theorem

In ZFC if all attacks are symmetric and not self-attacking, then stage and semi-stable do not collapse.

Theorem

In ZFC bipartite argumentation frameworks never collapse for stage or semi-stable.

Theorem

In ZFC if there is no argument with infinitely many attackers then stage and semi-stable do not collapse.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 19 / 21

SLIDE 63

What is ZFC and why bother about Choice?

Selecting Nodes/Elements: a choice function

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 20 / 21

SLIDE 64

What is ZFC and why bother about Choice?

Selecting Nodes/Elements: a choice function

Example

Consider a bucket of water. You know that there is one O-atom in each

molecule. But can you select exactly one O-atom from each H20

molecule?

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 20 / 21

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Downsides of Choice: Axiom of Determinacy

B A B A 000 000 001 001 010 010 011 011 1 1 100 100 101 101 110 110 111 111 00 00 0000 0000 0001 0001 0010 0010 0011 0011 01 01 0100 0100 0101 0101 0110 0110 0111 0111 10 10 1000 1000 1001 1001 1010 1010 1011 1011 11 11 1100 1100 1101 1101 1110 1110 1111 1111 S0000 S0001 S0010 S0011 S0100 S0101 S0110 S0111 S1010 S1011 S1100 S1101 S1000 S1001 S1110 S1111 00 00 01 01 1 1 10 10 11 11 000 000 0000 0000 0001 0001 001 001 0010 0010 0011 0011 010 010 0100 0100 0101 0101 011 011 0110 0110 0111 0111 100 100 1000 1000 1001 1001 101 101 1010 1010 1011 1011 110 110 1100 1100 1101 1101 111 111 1110 1110 1111 1111 S0000 S0001 S0010 S0011 S0100 S0101 S0110 S0111 S1000 S1001 S1010 S1011 S1100 S1101 S1110 S1111

Theorem

The 2ω generalization of this framework does not collapse for stable semantics if and only if we do not assume Choice.

Christof Spanring, Porgy & Bess: Lost & Found Trumpet Reincarnations: Abstract Argumentation 21 / 21

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References

Baumann, R., Dvoˇ rák, 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-person games.

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

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