Post on 17-Dec-2015
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!
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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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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
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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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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