Commonsense Reasoning and Argumentation 14/15 HC 12 Dynamics of Argumentation (1) Henry Prakken March 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) 3

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’ 10

Resolutions: some properties to be studied A B CD AF A B C D A B C 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 12

13 Resolution of asymmetric attack in ASPIC+ s1: r  ¬q K n =  ; K p = {q,r} r <’ q q qq r s1 > B1 A1 B2 B1 A1 B2

14 Resolution of asymmetric attack in ASPIC+ s1: r  ¬q s2: q  ¬r K n =  ; K p = {q,r} r <’ q q qq r s1 rr Constraint on  a : If A = B   then A ≈ a B > B1 A1 B2 A2 s2

15 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 misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2 R1 < R3, R2 < R3 R1R2 R3

16 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 misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2 R1 < R3, R2 < R3 R1R2 R3 R1 < R2

17 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 misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2 R1 < R3, R2 < R3 R1R2 R3 A1 A2 A3 B1 B2

18 R1: If you snore, you misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2so A2 < B2 (with last link) R1 < R3, R2 < R3so B2 < A3 (with last link) A1 A2 A3 B1 B2

19 R1: If you snore, you misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2so A2 < B2 (with last link) R1 < R3, R2 < R3so 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 misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2 R1 < R3, R2 < R3 R1R2 R3

21 R1: If you snore, you misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2so A2 < B2 (with last link) R1 < R3, R2 < R3so 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

22 R1: If you snore, you misbehave R2: If you snore when nobody else is around, you don’t misbehave R3: If you misbehave in the library, the librarian may remove you R1 < R2so A2 < B2 (with last link) R1 < R3, R2 < R3so 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’ 23

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 C AX