Some problems with modelling preferences in abstract argumentation

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

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)

ASPIC framework: overview Argument structure: • Trees where • Nodes are wff of a logical language L • Links are applications of inference rules • Rs = Strict rules (1, ..., n  ); or • Rd= 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)

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

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

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

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

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

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

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

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

Structure of arguments • An argumentA on the basis of (K,≤’) in (L, -,R,≤) is: •  if K with • Prem(A) = {}, Conc(A) = , Sub(A) = {} • A1, ..., An/ if there is a strict/defeasible inference rule Conc(A1), ..., Conc(An) / • Prem(A) = Prem(A1)  ...  Prem(An) • Conc(A) =  • Sub(A) = Sub(A1)  ...  Sub(An)  {A}

Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,u} Ka = {r} A1 = p A5 = A1  t A2 = q A6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = u A8 = A7,A4  w w u, v  w Rs p v u p, q  s Rs s,r,t  v Rd a t r s p  t Rd p n p p q p

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 ArgsATwhere ArgsAT= {A | A is an argument on the basis of KB in AS}

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

Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,u} Ka = {r} A1 = p A5 = A1  t A2 = q A6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = u A8 = A7,A4  w w p v u a t r s p n p p q p

Argument acceptability • Dung-style semantics applied to (ArgsAT, defeatAT)

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

A B E D C

A B A’ E D C

P1 P2 P3 P4 A B A’ E D C P5 P6 P7 P8 P9

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

Rationality postulatesfor ASPIC+ Closure under subarguments always satisfied Strict closure and consistency: without preferences satisfied if Rs closed under transposition or AS closed under contraposition Strict closure of Kn is consistent AT is `well-formed’ with preferences satisfied if in additiona is ‘reasonable’

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 Rs corresponds to a Tarskian abstract logic (cf. Amgoud & Besnard), then they are well-behaved wrt the rationality postulates

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)

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

Rd: r1: p  r r2: q  -r r3: -r  s K: p,q : r2 < r1, r1 < r3 a= last link Counterexample to subargument closure A1 = p A2 = A1  r B1 = q B2 = B1  -r B3 = B2  s attack PAF-defeat ASPIC+-defeat

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

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

Counterexample R2L-sc for grounded semantics A A B B A B C C C D D D

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 • defeatAT2 defeatAT1

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 r1:  -r2 r2:  -r1

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.

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

Some problems with modelling preferences in abstract argumentation