Argumentation Context Systems: A Framework for Abstract Group Argumentation

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Argumentation Context Systems:

A Framework for Abstract Group Argumentation

Gerhard Brewka

Computer Science Institute University of Leipzig brewka@informatik.uni-leipzig.de

joint work with Thomas Eiter

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1. Motivation

• Work based on Dung’s widely used abstract argumentation frameworks (AFs).

• Abstract approach: arguments un-analyzed, attacks represented in digraph; can be instantiated in many different ways.

• Argument accepted unless attacked by an accepted argument.

• Semantics single out appropriate accepted sets of arguments:

• Grounded extension: accept unattacked args, eliminate args attacked by accepted args, continue until fixpoint reached.

• Preferred extension: maximal conflict free set which attacks each of its attackers.

• Stable extension: conflict-free set of arguments which attacks each excluded argument.

• (Value based) preferences captured: modify original AF.

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Limitations

• No distinction between arguments, meta-arguments, sources of arguments etc.

• Our interest: additional structure and modularity

• Benefits:

• A handle on complexity and diversity

• A natural account of multi-agent argumentation

• Explicit means to model meta-argumentation

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Motivating Example: Conference Reviewing

Consider model of the paper review process for a conference

• Hierarchy consisting of PC chair, area chairs, reviewers, authors.

• PC chair determines review criteria.

• Area chairs make sure reviewers make fair judgements and eliminate unjustified arguments from reviews.

• Authors give feedback on reviews. Information flow thus cyclic.

• Reviewers exchange arguments in peer-to-peer discussion.

• Area chairs generate a consistent recommendation.

• PC chair takes recommendations as input for final decision.

Need a flexible framework allowing for cyclic structures encompassing different information integration methods.

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The Short Story

A₁

A (lonely) Dung style argumentation framework.

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The Short Story

Med1

A₁

An argumentation module equipped with a mediator, can “listen" to other modules and “talk" toA₁: sets an argumentation context using a context

definition language; handles inconsistency.

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The Short Story

Med3 Med4

Med1 Med2

A₁ A₂

A₃ A₄

An argumentation context system.

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Outline

1 Motivation (done)

2 Background

3 Context Based Argumentation

4 Mediators

5 Argumentation Context Systems

6 Conclusions

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Background: Inconsistency Handling

Use 4 methods for picking consistent subset of (F₁, . . . ,F_n),F_i set of formulas (details irrelevant)

Preference based Majority based

Credulous sub maj

Skeptical subsk, maj_sk

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Background: Multi-Context Systems

• Model information flow between different local reasoning modules.

(calledcontexts)

• Based onbridge rules; may refer to other modules in their bodies.

• Here only rules referring to a single other module needed⇒ bridge rules ordinary logic programming rules:

s ←p₁, . . . ,p_j,notp_j+1, . . . ,notp_m (1)

headsa context expression (to be defined), body atoms argumentsp_i from a parent argumentation framework.

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3. Context Based Argumentation

First step: a language for representing context:

a,bargs;v,v⁰ values;r ∈ {skep,cred};s∈ {grnd,pref,stab}

arg(a)/arg(a) ais a valid (invalid) argument att(a,b)/att(a,b) (a,b)is a valid (invalid) attack

a>b ais strictly preferred tob val(a,v) the value ofaisv

v >v⁰ valuev is strictly better thanv⁰ mode(r) the reasoning mode isr

sem(s) the chosen semantics iss Context C: set of context expressions.

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Contexts as Modifiers

What are extensions of AFAunder context C?

CtransformsAtoA^Cby (in)validating args and attacks appropriately using new argumentdef:

a b c

d

LetC={arg(a),val(b,v₁),val(d,v₂),v₁>v₂,c >b}.A^Cis:

def

a b c

d

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Acceptable Extensions

• Transformation handles statements exceptmodeandsem.

• These are captured in the following definition:

Definition

Letsem(s)∈C.S⊆ARis anacceptable C-extension forA, if either

1 mode(skep)∈CandS∪ {def}is the intersection of alls- extensions ofA^C, or

2 mode(cred)∈CandS∪ {def}is ans-extension ofA^C. Proposition: Definitions “do the right thing"

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4. Mediators

• Context information may come from parent modules

• Need to “translate" abstract arguments to context statements⇒ use bridge rules

• Also need to guarantee consistency⇒

use consistency method, potentially preferences on parents Definition

A₁andA₂, . . . ,A_k AFs. Amediator forA₁based onA₂, . . . ,A_k is Med = (E₁,R₂, . . . ,R_k,choice)

where

• E₁is a set of context statements forA₁;

• R_i (2≤i ≤k)is a set ofbridge rules forA₁based onA_i;

• choice∈ {sub,sub_sk,,maj,maj_sk}, whereis a strict partial order on{1, . . . ,k}.

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Mediators, ctd.

Mediator determines consistent context based on

• arguments accepted by parents and

• chosen consistency method.

Definition

LetMed = (E₁,R₂, . . . ,R_k,choice)be a mediator forA₁based on

A₂, . . . ,A_k. A contextCforA₁isacceptable wrt. sets of arguments

S₂, . . . ,S_k ofA₂, . . . ,A_k, ifC is achoice-preferred set for (E₁,R₂(S₂), . . . ,R_k(S_k)).

HereR_i(S_i)are the context statements derivable fromS_i underR_i: {h|h←a₁, ...,a_j,notb₁, ...,notb_n ∈R_i, each a_i ∈S_i, each b_m6∈S_i}

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5. The Framework

• Put the pieces together

• Take collection of context based argument systems

• Add mediator to each of them

• Connect them in an arbitrary graph

• Use mediator to generate consistent context

Definition

(Argumentation) module: pairM= (A,Med),Aan AF andMed mediator forAbased on some AFsA₁, . . . ,A_k.

Definition

Argumentation context system (ACS): setF ={M₁, . . . ,M_n}of modulesM_i= (A_i,Med_i)such that eachMed_i is based only on AFs A_i₁, . . . ,A_i_k, wherei_j ∈ {1, . . . ,n}(self-containedness).

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The Module Graph

Definition

Module graphofACS F: digraphG(F) = (F,E)whereM_j → M_i inE iffA_j is among theA_i₁, . . . ,A_i_k on whichMed_i is based.

Med3 Med4

Med1 Med2

A₁ A₂

A₃ A₄

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Acceptable States

• For each module, pick accepted set of arguments and context

• Must fit together: chosen arguments acceptable given context, chosen context acceptable given chosen arguments of parents Definition

State ofF: functionSmapping eachM_i = (A_i,Med_i)to a pair S(M_i) = (Acc_i,C_i), whereAcc_i is a set of arguments ofA_i andC_i a context forA_i.

S acceptable, if

• eachAcc_i is acceptableC_i-extension forA_i, and

• eachC_i is acceptable context forMed_i wrt. allAcc_j for whichG(F) has an arcM_j → M_i.

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Some Results

• Existence of acceptable states

• Not guaranteed, even without stable semantics and default negation

• Guaranteed ifF hierarchic andsem(stab)does not occur in any mediator.

• Complexity

• Reasoning tasks related to acceptable states intractable in general.

• Deciding whetherACS F has some acceptable stateΣ^p₃-complete.

• Has lower complexity depending on the various parameters and graph structure.

• Fhierarchic, modules use grounded semantics and eithersub or maj⇒acceptable state computable in polynomial time.

• Complexity ofC-extensions dominated by underlying argumentation framework.

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6. Conclusions

• Presented flexible, modular framework for abstract argumentation.

• Builds on existing proposals extending them in various respects.

• Argumentation based on contexts described in a native language.

• Comprises preference- and value-based argumentation, direct (in)validation of arguments and attacks, and specification of reasoning mode and semantics.

• Context information integrated by a mediator.

• Easy to integrate other argumentation semantics, consistency methods etc.

• Arbitrary directed module graphs cover wide range of applications involving multi-agent meta-argumentation.

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6. Conclusions

• Arbitrary directed module graphs cover wide range of applications

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6. Conclusions

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6. Conclusions

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6. Conclusions

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6. Conclusions

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6. Conclusions

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Related work

• Instantiation of Brewka/Eiter’s multi-context systems: all reasoners argument systems; BUT: beyond older work in use of mediators.

• Wooldridge, McBurney, Parsons, Meta-Logic of Arguments: higher level defines fundamental notions (provability, argument, etc.);

entirely different focus.

• Thimm, Kern-Isberner, Framework for Distributed Argumentation:

not abstract, less general.

• Binas, McIlraith, Peer-to-peer Query Answering with Inconsistent Knowledge: use argumentation for inconsistency handling in P2P system, focus on integration, not on (meta-) argumentation.

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Related work

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Related work

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Related work

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Closely Related: Modgil 2006/2009

• Framework for meta-argumentation in linear hierarchies.

• Argumentation inA_i about preferences inA_i−1.

• We substantially generalize this by

• including argumentation about values, (in)acceptable arguments, (in)acceptable attacks, reasoning mode, semantics, ...

• considering arbitrary graphs,

• providing preference and majority based integration methods for information from different parents.

Also generalize Modgil’s EAFs (2009): single feedback module

Med

A

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Future Work

• Brewka/Eiter [AAAI-07] discuss groundedness at length:

what’s the relevance of this here?

• should self-justifying cycles be banned?

• if so, can they be banned using the [AAAI-07] techniques?

• can techniques of this paper be used: meta-module observing not just arguments in lower level, but complete argumentation states?

• More general context description languages

• More general bridge rules

• Detailed complexity analysis

• Generalization to arbitrary multi-context systems THANK YOU!

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Future Work

• Brewka/Eiter [AAAI-07] discuss groundedness at length:

what’s the relevance of this here?

• should self-justifying cycles be banned?

• if so, can they be banned using the [AAAI-07] techniques?

• can techniques of this paper be used: meta-module observing not just arguments in lower level, but complete argumentation states?

• More general context description languages

• More general bridge rules

• Detailed complexity analysis

• Generalization to arbitrary multi-context systems