1. Some problems with modelling preferences in abstract argumentation Henry Prakken Luxemburg 2 April 2012

2. 2 Overview The ASPIC+ framework for structured argumentation Preference-based abstract argumentation frameworks (PAFs) Combination with ASPIC+ Abstract resolution semantics Combination with ASPIC+ (Joint work with Sanjay Modgil)

3. 3 ASPIC framework: overview Argument structure: Trees where Nodes are wff of a logical language L Links are applications of inference rules R s = Strict rules (  1,...,  n   ); or R d = Defeasible rules (  1,...,  n   ) Reasoning starts from a knowledge base K  L Defeat: attack on conclusion, premise or inference, + preferences Argument acceptability based on Dung (1995)

4. We should lower taxes Lower taxes increase productivity Increased productivity is good

5. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad

6. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity USA lowered taxes but productivity decreased

7. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … USA lowered taxes but productivity decreased

8. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective USA lowered taxes but productivity decreased

9. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective USA lowered taxes but productivity decreased

10. We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased

11. 11 Argumentation systems An argumentation system is a tuple AS = ( L, -, R, ≤) where: L is a logical language - is a contrariness function from L to 2 L R = R s  R d is a set of strict and defeasible inference rules ≤ is a partial preorder on R d S  L is (directly) consistent iff for no ,   L it holds that   - (  )

12. 12 Knowledge bases A knowledge base in AS = ( L, -, R,≤’) is a pair ( K, ≤’) where K  L and K is a partition K n  K p  K a  K i where: K n = necessary premises K p = ordinary premises K a = assumptions K i = issues (ignored below) Moreover, ≤’ is a partial preorder on K / K n.

13. 13 Structure of arguments An argument A on the basis of ( K,≤’) in ( L, -, R,≤) is:  if   K with Prem(A) = {  }, Conc(A) = , Sub(A) = {  } A 1,..., A n  /   if there is a strict/defeasible inference rule Conc(A 1 ),..., Conc(A n )  /   Prem(A) = Prem(A 1 ) ...  Prem(A n ) Conc(A) =  Sub(A) = Sub(A 1 ) ...  Sub(A n )  {A}

14. 14 R s :R d : p,q  sp  t u,v  ws,r,t  v K n = {q} K p = {p,u} K a = {r} w vu s r t pqp p pnp a u, v  w  R s p, q  s  R s s,r,t  v  R d p  t  R d A1 = pA5 = A1  t A2 = qA6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = uA8 = A7,A4  w

15. 15 Argumentation theories An argumentation theory is a triple AT = (AS,KB,≤ a ) where: AS is an argumentation system KB is a knowledge base in AS ≤ a is an argument ordering on Args AT where Args AT = {A | A is an argument on the basis of KB in AS}

16. 16 Attack and defeat (with - symmetric and K a =  ) A undermines B (on  ) if Conc(A) = -  for some   Prem(B )/ K n ; A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’  Sub(B ) with a defeasible top rule A undercuts B (on B’ ) if Conc(A) = -r ’for some B’  Sub(B ) with defeasible top rule r A defeats B iff for some B’ A undermines B on  and not A < a  ; or A rebuts B on B’ and not A < a B’ ; or A undercuts B on B’ Naming convention implicit Direct vs. subargument attack/defeat Preference-dependent vs. preference-independent attacks

17. 17 R s :R d : p,q  sp  t u,v  ws,r,t  v w vu s r t p q p p pnp a A1 = pA5 = A1  t A2 = qA6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = uA8 = A7,A4  w K n = {q} K p = {p,u} K a = {r}

18. 18 Argument acceptability Dung-style semantics applied to (Args AT, defeat AT )

19. 19 We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased

20. AB C D E

21. AB C D E A’

22. AB C D E P1 P2P3P4 P8P9P7P5P6

23. Rationality postulates (Caminada & Amgoud 2007) Let E be any complete extension, CONC(E) = {  |  = Conc(A) for some A  E }: 1. If A  E and B  Sub(A) then B  E 2. Conc(E) is closed under R S ; consistent.

24. Rationality postulates for ASPIC+ Closure under subarguments always satisfied Strict closure and consistency: without preferences satisfied if R s closed under transposition or AS closed under contraposition Strict closure of K n is consistent AT is `well-formed’ with preferences satisfied if in addition  a is ‘reasonable’

25. Relation with other work Assumption-based argumentation (Dung, Kowalski, Toni...) is special case with Only assumption-type premises Only strict inference rules No preferences Variants of classical argumentation with undermining (Amgoud & Cayrol, Besnard & Hunter) are special case with Only ordinary premises Only strict inference rules (all valid propositional or first-order inferences) - = ¬ Arguments must have classically consistent premises Carneades (Gordon et al.) is a special case If R s corresponds to a Tarskian abstract logic (cf. Amgoud & Besnard), then they are well-behaved wrt the rationality postulates

26. Preference-based abstract argumentation PAF = (Args,attack, ≤) ≤ an argument ordering A defeats B iff A attacks B and not A < B Argument acceptability: Dung-style semantics applied to (Args, defeat) 26

27. What if ASPIC+ semantics is defined by PAFs? No distinction possible between preference-dependent and preference- independent attacks Possibly violations of postulates of subargument closure and consistency 27

28. Counterexample to subargument closure 28 R d : r1: p  r r2: q  -r r3: -r  s K : p,q  : r2 < r1, r1 < r3  a = last link A1 = pA2 = A1  r B1 = qB2 = B1  -rB3 = B2  s attackPAF-defeatASPIC+-defeat

29. Abstract resolution semantics (Modgil 2006, Baroni et al. 2008-2011) AF2 = (Args,attack2) is a resolution of AF1 = (Args,attack1) iff attack2  attack1 If (A,B)  attack1,  attack2, then (B,A)  attack1,  attack2 So partial resolutions turn one or more symmetric attacks into asymmetric ones 29

30. Possible properties of abstract resolution semantics NB: A is sceptically s-justified wrt AF iff A is in all s- extensions of AF L2R-sc: If A is sceptically justified wrt AF, then A is sceptically justified wrt all resolutions of AF Holds for grounded but not for preferred R2L-sc: If A is sceptically justified wrt all resolutions of AF, then A is sceptically justified wrt AF Holds for preferred but not for grounded 30

31. Counterexample R2L-sc for grounded semantics AB C D AB C D AB C D

32. Resolutions in ASPIC+ (Modgil & Prakken 2012) Let ≤ and ≤’ be two partial preorders: ≤’ extends ≤ iff ≤  ≤’; and If x < y then x <’ y AT2 = (AS,KB, ≤ a2 ) is a resolution of AT1 = (AS,KB, ≤ a1 ) iff ≤ a2 extends ≤ a1 ; and defeat AT2  defeat AT1 32

33. Deviations from abstract resolution semantics Some asymmetric attacks can be resolved Some symmetric attacks cannot be resolved Preference-independent attacks A ≈ a1 B Preferences may imply other preferences 33 r1:  -r2 r2:  -r1

34. Results on resolution semantics for ASPIC+ L2R-sc holds for grounded but not for preferred R2L-sc holds for neither grounded nor preferred While it holds for preferred in abstract resolution semantics Special case: R2L-sc holds for preferred for classical instantiations with the KB- ordering a total preorder. 34

35. Methodological message Abstract argumentation approaches are dangerous: Only significant when combined with accounts of the structure of arguments But often implicitly make assumptions that exclude reasonable instantiations While these assumptions often cannot be expressed at the abstract level 35