Commonsense Reasoning and Argumentation 14/15

HC 12Dynamics of Argumentation

(1)

Henry PrakkenMarch 23, 2015

Overview Extended argumentation

frameworks Arguing about defeat relations

Expanding abstract argumentation frameworks

Resolving abstract argumentation frameworks

Dynamics in abstract argumentation

Adding or deleting: Attacks/defeats Arguments (plus induced

attacks/defeats)

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Arguing about defeat relations

Standards for determining defeat relations are often: Domain-specific Defeasible and conflicting

So determining these standards is argumentation!

Recently Modgil (AIJ 2009) has extended Dung’s abstract approach Arguments can also attack attack relations

CB

Modgil 2009

Will it rain in Calcutta?

BBC says rain

CNN says sun

C

T

B

Modgil 2009

Will it rain in Calcutta?

BBC says rain

CNN says sun

Trust BBC more than CNN

C

T

B

S

Modgil 2009

Will it rain in Calcutta?

BBC says rain

CNN says sun

Trust BBC more than CNN

Stats say CNN better than BBC

C

T

B

S

Modgil 2009

Will it rain in Calcutta?

BBC says rain

CNN says sun

Trust BBC more than CNN

Stats say CNN better than BBC

R

Stats more rational than trust

Expanding abstract argumentation frameworks

(Baumann 2010)

AF’ is an expansion of AF = (A, C) iff AF’ = (A A’, C C’) for some nonempty A’ disjoint from A such that: If (A,B) C’ then A C’ or B C’

Property: for any non-selfdefeating argument A there exist expansions in which A is justified But assumes that all arguments are

attackable!

9

Resolution semantics (Modgil 2006; Baroni & Giacomin

2007) AF’ = (A,C’ ) is a resolution of AF = (A,C)

iff If (A,B) C and (B,A) not in C then (A,B) C’ If (A,B) C and (B,A) C then (A,B) C’ or

(B,A) C’ If (A,B) C’ then (A,B) C

A resolution is full if it leaves no symmetric attacks in C’

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Resolutions: some properties to be studied

A

BC D

AF

A

BC D

A

BC D

AF1 (A > B) AF2 (B > A)

Grounded semantics satisfies Left to Right. Preferred semantics satisfies Right to Left

X is justified in AF iff X is justified in all full resolutions of AF

Resolution semantics (Modgil 2006; Baroni & Giacomin

2007) AF’ = (A,C) is a resolution of AF = (A,C) iff

If (A,B) C and (B,A) not in C then (A,B) C’ If (A,B) C and (B,A) C then (A,B) C’ or

(B,A) C’ If (A,B) C’ then (A,B) C

So assumes that: Only symmetric attacks can be resolved All attacks are independent from each other All symmetric attacks can be resolved

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13

Resolution of asymmetric attack in ASPIC+

s1: r ¬q Kn = ; Kp = {q,r} r <’ q

q

q

r

s1>

B1A1

B2

B1A1

B2

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Resolution of asymmetric attack in ASPIC+

s1: r ¬q s2: q ¬r Kn = ; Kp = {q,r} r <’ q

q

q

r

s1

r

Constraint on a:If A = B then A ≈ a B

>B1A1

B2A2

A2

s2

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John does not misbehave in the library

John snores when nobody else is in the library

John misbehaves in the library

John snores in the library

John may be removed

R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2R1 < R3, R2 < R3

R1 R2

R3

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John does not misbehave in the library

John snores when nobody else is in the library

John misbehaves in the library

John snores in the library

John may be removed

R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2R1 < R3, R2 < R3

R1 R2

R3

R1 < R2

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John does not misbehave in the library

John snores when nobody else is in the library

John misbehaves in the library

John snores in the library

John may be removed

R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2R1 < R3, R2 < R3

R1 R2

R3

A1

A2

A3

B1

B2

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R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2 so A2 < B2 (with last link)R1 < R3, R2 < R3 so B2 < A3 (with last link)

A1

A2

A3

B1

B2

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R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2 so A2 < B2 (with last link)R1 < R3, R2 < R3 so B2 < A3 (with last link)

A1

A2

A3

B1

B2

20

John does not misbehave in the library

John snores when nobody else is in the library

John misbehaves in the library

John Snores in the library

John may be removed

R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2R1 < R3, R2 < R3

R1 R2

R3

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R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2 so A2 < B2 (with last link)R1 < R3, R2 < R3 so B2 < A3 (with last link)

A1

A2

A3

B1

B2

Resolution semantics does not recognize that B2’s attacks on A2 and A3 are the same

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R1: If you snore, you misbehaveR2: If you snore when nobody else is around, you don’t misbehaveR3: If you misbehave in the library, the librarian may remove you

R1 < R2 so A2 < B2 (with last link)R1 < R3, R2 < R3 so B2 < A3 (with last link)

A1

A2

A3

B1

B2

This is the correct outcome

Preference-based resolutions in ASPIC+ (Modgil & Prakken

2012) SAF’ = (A,C,≤’) preference-extends SAF =

(A,C, ≤) iff ≤ ≤’ X < Y implies X <’ Y

Let D’ and D be the defeat relations of SAF’ and SAF. Then SAF’ is a resolution of SAF iff D’ D.

Resolution SAF’ is a full resolution of SAF iff there exists no resolution SAF’’ of SAF such that D’’ D’

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Properties of preference-based resolutions

Grounded semantics still satisfies LtoR sceptical (but only for finitary Afs)

Preferred now fails RtoL sceptical

Counterexample to RtoL sceptical for preferred

Counter-example to RtoL illustrates failure even when resolving only symmetric attacks

Priority ordering over premises determines preferences over arguments

Example (in Modgil & Prakken 2012) shows that no way to extend priority ordering (and hence preference ordering)

exists so that D asymmetrically defeats B and E asymmetrically defeats C

D

E

B

CA X