Towards a General Argumentation System Based on Answer-set Programming

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Towards a General Argumentation System Based on Answer-set Programming

Sarah Alice Gaggl

Institute of Informationsystems, Vienna University of Technology

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Legal Reasoning

Sarah A. Gaggl, TU Vienna General Argumentation System Based on ASP 1

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Social Networks

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Decision Support

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Motivation

Argumentation Frameworks (AFs)

AFs provide a formalism for a compactrepresentationand evaluationof such scenarios.

More complex semantics, especially in combination with an increasing amount of data, requires anautomated computationof such solutions.

Most of these problems areintractable, so implementing dedicated systems from the scratch is not the best idea.

Logic Programming (LP), in particularAnswer-set Programming (ASP), turned out to beadequateto solve problems associated to AFs.

We use ASP to design the systemASPARTIXfor the evaluation of several approaches how to deal with AFs.

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Overview

1 Introduction to Abstract Argumentation Frameworks

2 ASP Encoding

3 ASPARTIX - System Demonstration

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Abstract Argumentation Frameworks

First introduced by Phan Minh Dung in 1995.

AFs provide a formal way of dealing with conflicting knowledge.

Represent arguments together with a binary attack relation.

Conflicts are solved via semantics (admissible, preferred, stable).

They can be represented as directed graphs.

More formally

Anargumentation framework(AF) is a pair(A,R), where Ais a set of arguments

R⊆A×Ais a relation representing “attacks” (“defeats”)

b c d e

a

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Semantics

Conflict-free

LetF= (A,R)be an AF. A setS⊆Ais said to beconflict-free (inF), if there arenoa,b∈S, such that(a,b)∈R. We denote the collection of sets which are conflict-free (inF) bycf(F).

Example

b c d e

a

cf(F) = {a,c},

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Semantics

Conflict-free

Example

b c d e

a

cf(F) =

{a,c},{a,d},

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Semantics

Conflict-free

Example

b c d e

a

cf(F) =

{a,c},{a,d},{b,d},

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Semantics

Conflict-free

Example

b c d e

a

cf(F) =

{a,c},{a,d},{b,d},{a},{b},{c},{d},∅

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Semantics cont.

Stable Extension

Given an AF(A,R). A setS⊆AisstableinF, if Sis conflict-free inF

for eacha∈A\S, there exists ab∈S, such that(b,a)∈R.

Example

b c d e

a

stable(F) = {a,c},

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Semantics cont.

Stable Extension

Example

b c d e

a

stable(F) =

{a,c},{a,d},

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Semantics cont.

Stable Extension

Example

b c d e

a

stable(F) =

{a,c},{a,d},{b,d},

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Semantics cont.

Stable Extension

Example

b c d e

a

stable(F) =

{a,c},{a,d},{b,d},{a},{b},{c},{d},∅

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Encodings

Conflict-free Set

Given an AF(A,R).

A setS⊆Aisconflict-freeinF, if, for eacha,b∈S,(a,b)∈/R.

Encoding for

F = (A,R)

Fb ={arg(a)|a∈A} ∪ {att(a,b)|(a,b)∈R}

πcf =







in(X) ← notout(X),arg(X) out(X) ← notin(X),arg(X)

← in(X),in(Y),att(X,Y)





 Result: For each AFF,cf(F)≡ AS(π_cf(bF))

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Encodings cont.

Stable Extension

eacha∈A\S, there exists ab∈S, such that(b,a)∈R.

Encoding

πstable=











in(X) ← notout(X),arg(X) out(X) ← notin(X),arg(X)

← in(X),in(Y),att(X,Y) defeated(X) ← in(Y),att(Y,X)

← out(X),notdefeated(X)









 Result: For each AFF,stable(F)≡ AS(πstable(bF))

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ASPARTIX - System Description

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ASPARTIX - System Description cont.

Semanticsandtypesof AFs incorporated in ASPARTIX:

admissible, complete, stable, preferred, grounded, ideal, stage, semi-stable and cf2;

Preference-based AFs, Value-based AFs, Bipolar AFs, Dynamic AFs and AFs with Recursive Attacks.

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Conclusion

AFs became very important in Artificial Intelligence. They provide a popular tool for modeling and evaluating conflicting knowledge.

Problems associated to AFs are in general intractable, therefore we translate them to ASP.

Web front-end of ASPARTIX is freely available.

http://rull.dbai.tuwien.ac.at:8080/ASPARTIX

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Complexity

Relation between Semantics

pref stable

ground

compl adm

Complexity

stable adm pref comp ground Cred NP-c NP-c NP-c NP-c inP Skept coNP-c (trivial) Π^P₂-c inP inP

[Dimopoulos & Torres 96; Dunne & Bench-Capon 02; Coste-Marquiset al.05]

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Complexity cont.

Complexity of Argumentation

stable adm pref comp ground Cred NP-c NP-c NP-c NP-c inP Skept coNP-c (trivial) Π^P₂-c inP inP

Recall: Data-Complexity of Datalog

stratified programs with negation with neg. and disjunction

|=c P-c NP-c Σ^P₂-c

|=s P-c coNP-c Π^P₂-c

[Dantsin,Eiter,Gottlob,Voronkov 01]

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Performance Tests

Tested Systems:

Grounders: DLV, lparse, GrinGo

Solvers: DLV, smodels, cmodels, clasp, claspD, gnt Testing:

Randomly generated AFs from 90 to 200 arguments with edge density from 10% to 30%.

In total 21303 tests were performed.

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Test Results

Admissible Extensions:

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Test Results cont.

Preferred Extensions:

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Test Results cont.

Stable Extensions: