some problems with modelling preferences in abstract argumentation henry prakken luxemburg 2 april...

Some problems with modelling preferences in abstract argumentation

Henry PrakkenLuxemburg2 April 2012

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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)

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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 not lower taxes

Lower taxes increase inequality

Increased inequality is bad







Lower taxes do not increase productivity

USA lowered taxes but productivity decreased








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


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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 , L it holds that -()

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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.

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Structure of arguments

An argument A 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}

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Rs: Rd:

p,q s p tu,v w s,r,t v

Kn = {q} Kp = {p,u} Ka = {r}

w

v u

s r t

p q p

p

pnp

a

u, v w Rs

p, q s Rs

s,r,t v Rd

p t Rd

A1 = p A5 = A1 t

A2 = q A6 = A1,A2 s

A3 = r A7 = A5,A3,A6 v

A4 = u A8 = A7,A4 w

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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 ArgsAT where

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

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Attack and defeat(with - symmetric and Ka =

) A undermines B (on ) if

Conc(A) = - for some Prem(B )/ Kn; 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/defeatPreference-dependent vs. preference-independent attacks

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Rs: Rd:

p,q s p tu,v w s,r,t v

w

v u

s r t

p q p

p

pnp

a

A1 = p A5 = A1 t

A2 = q A6 = A1,A2 s

A3 = r A7 = A5,A3,A6 v

A4 = u A8 = A7,A4 w

Kn = {q} Kp = {p,u} Ka = {r}

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Argument acceptability Dung-style semantics applied to

(ArgsAT , defeatAT)

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Increased inequality is good

Increased inequality stimulates competition

Competition is good


A B

C D E

A B

C D E

A’

A B

C D E

A’

P1 P2 P3 P4

P8 P9P7P5 P6

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 E2. 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)

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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

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Counterexample to subargument closure

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Rd:r1: p rr2: q -rr3: -r sK: p,q:r2 < r1, r1 < r3a= last link

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

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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

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Counterexample R2L-sc for grounded semantics

A B

C

D

A B

C

D

A B

C

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

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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

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r1: -r2r2: -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.

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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

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some problems with modelling preferences in abstract argumentation henry prakken luxemburg 2 april...

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