trumpet reincarnations: abstract argumentation=1this ......christof spanring, porgy & bess: lost...

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


Introduction

happy

joy

safe

weak

strong


Introduction

h

js

wr

in: s,r

out: w,j

undec: h


Introduction

h

js

wr

in: s,r

out: w,j

undec: h

h

js

wr

in: j,r

out: s,w

undec: h


Introduction

h

js

wr

in: s,r

out: w,j

undec: h

h

js

wr

in: j,r

out: s,w

undec: h

h

js

wr

in: w,j

out: s,h,r

undec:


Outline

1 Introduction

2 Stable Extensions and Conflict

3 Variations of Stable Semantics

4 Collapse

5 Perfection

6 References


What are the stable extensions?

a b

u

d

cr

p

t1t2t3· · ·

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



a b

u

d

cr

p

t1t2t3· · ·

Stable Extensions:

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

{d, b, t1, t2, t3, . . .}



a b

u

d

cr

p

t1t2t3· · ·

Stable Extensions:

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


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, . . .}



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.



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.


Implicit Conflicts made Explicit in a Simpler Case

12 3

4

{3, 1}{4, 2}



12 3

4

Stable Extensions:

{3, 1}{4, 2}



12 3

4

Stable Extensions:

{3, 1}{4, 2}

Only Implicit Conflict:

1 vs. 4



12 3

4

Stable Extensions:

{3, 1}{4, 2}


1 vs. 4

Resultion:

(1, 4)



12 3

4

Stable Extensions:

{3, 1}{4, 2}


1 vs. 4

Resultion:

(1, 4) or (4, 1)



12 3

4

Stable Extensions:

{3, 1}{4, 2}


1 vs. 4

Resultion:

(1, 4) or (4, 1) or both


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.


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.


Outline

1 Introduction



4 Collapse

5 Perfection

6 References



12

3

4 5

6



12

3

4 5

6

Stable Extensions:

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

There are none!!!



1

2

3

4

5



1

2

3

4

5

Stable Extensions:

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

There are none!!!



12

3

4 5

6

1

2

3

4

5

Why is collapse a problem?


Collapse Solutions

1 2

3

4

ExampleCollapse of Stable Semantics.


Collapse Solutions

1 2

3

4

Definition (Stage)Consider argument sets whereundecided arguments areminimized.

Stage Extensions:

{2}


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}


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.


Outline

1 Introduction



4 Collapse

5 Perfection

6 References


Stage Collapse

b1 b2 b3 b4 b5 b6. . .

ExampleThere is no biggest natural number. . .

ExampleBut the empty set is a semi-stable extension!


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.


Collapse or not?

· · ·

p0

r0

p1

r1

p2

r2

p3

r3

p4

r4

p5

r5



Loop-free: collapse or not?

· · ·

q0

s0

p0

r0

q1

s1

p1

r1

q2

s2

p2

r2

q3

s3

p3

r3



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



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.


Outline

1 Introduction



4 Collapse

5 Perfection

6 References


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.


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?


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.


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.


. . . may we all be happy . . .

trumpet reincarnations: abstract argumentation=1this ......christof spanring, porgy & bess: lost...

Documents