Synthesis of Argumentation Graphs by Matrix Factorization

SYNTHESIS OF ARGUMENTATIONGRAPHS BY MATRIX FACTORIZATION1ST AI^3 WORKSHOP @ AI*IA 2017

Andrea Pazienza, Stefano Ferilli, Floriana Esposito

16th November 2017 – Bari, Italy

Overview

1. Introduction

2. Principal Argument System

3. Application to a Reddit Thread

4. Conclusions and future works

INTRODUCTION

Abstract Argumentation

Argumentation Framework (AF)# encapsulates arguments as nodes in a digraph

# connects them through a relationship of attack

# defines a calculus of opposition for determiningwhat is acceptable

# allows a range of different semantics

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b

e

cd

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Generalizations of Argumentation Frameworks# Bipolar: add support relation

# Weighted: add weights on attacks# Values, Preferences

# etc.

Extension-based vs Ranking-based Semantics

# extension-based semantics do not fully exploit the weight of relations

# rank arguments from the most to the least acceptable ones

Bipolar Weighted Argumentation Framework

Bipolar Weighted Argumentation Framework (BWAF)

# attack relations with a negativeweight in the interval [−1, 0[

# support relations with a positiveweight in the interval ]0, 1]

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BWAF Ranking-based Semantics by means of Strength Propagation

Argumentation Matrix

An argument graph can be represented with a slightly differentversion of its adiacency matrix.

Let F � 〈A,R〉 be an AF. Let |A| � n, then the Argumentation Matrix ofF is a n × n matrix MF � [Mi j] such that for any two argumentsαi , α j ∈ A it holds that

Mi j �

{−1 if 〈αi , α j〉 ∈ R0 otherwise

BWAF Example:

a b c

de

−0.7 −0.5

0.4 0.6

0.3MG �

0 0.4 0 0 −0.70 0 0.6 0 00 0 0 0 00 0 −0.5 0 0.30 0 0 0 0

PRINCIPAL ARGUMENT SYSTEM

Motivation

In the phase of evaluation of accepted arguments, one may find thatnot all the arguments of discussion are essential when drawingconclusions, especially when the cardinality of the set of arguments ishigh.

Proposal: Produce a synthesized AF in order to:

# decompose huge AFs and build a simplified ones,

# highlight arguments that are extremely useful for the evaluationprocess,

# discard the less relevant arguments,

# preserve the interpretation of the whole discussion.

Principal Argument System

An argument graph can be represented with its adiacency matrix.

Matrix decomposition allow us to deal with the problem of low-rank approximation.

For this purpose, we consider the factorization technique of Singular ValueDecomposition (SVD).

Singular Value Decomposition (SVD)

Let A ∈ Rm×n be a matrix and p � min(m , n), the SVD of A is afactorization in the form

A � UΣVT

where U � (u1 , . . . , um) ∈ Rm×m and V � (v1 , . . . , vn) ∈ Rn×n

are orthogonal, and Σ ∈ Rm×n is a diagonal matrix withelements σ1 ≥ σ2 ≥ . . . ≥ σp ≥ 0.

Each σ1 , . . . , σp is called singular value of A.

Consider r singular values ≥ 0 with r ≤ p, let Ur � (u1 , . . . , ur),Vr � (v1 , . . . , vr) and Σr � diag(σ1 , . . . , σr), it holds that

A � UrΣrVTr �

r∑i�1

aiσi vTi

namely, matrix A has rank r.

Principal Argument System

Given the Argumentation Matrix of the argumentation graphunder consideration (i.e, AF, BAF, WAF, BWAF),

# its low-rank approximation with the truncated SVD# reduced with only the r largest principal components# will ensure that the reconstructed matrix E ∈ Rn×n will be

the best approximation# and at the same time will preserve the meaning of the

discussion

This is tackled with the Kaiser criterion, which defines a ruleto investigate the scree plot of matrix eigenvalues.

Argument System Reconstrunction

How to reconstruct a relation between arguments?

Given a , b ∈ A, there is:

P-AF : an attack relation between them iff Eab < −0.5;P-BAF : an attack relation or a support relation between them iff,

respectively Eab ≤ −0.5 or Eab ≥ 0.5, otherwise any relationis built;

P-WAF : an attack relation between them iff Eab < −0.5 where itsweight is given by approximating the value Eab to thenearest integer number;

P-BWAF : an attack relation or a support relation between themwith weight value equal to Eab with bounds −1 or 1 iff,respectively, −1 < Eab < 0 or 0 < Eab < 1.

APPLICATION TO A REDDIT THREAD

Application to a Reddit Thread

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# We considered a Redditdiscussion of an episode ofBlack Mirror, a popular TVseries

# The produced BWAF is made

up of 70 arguments and 69

weighted relations, of which

◦ 52 attacks and◦ 17 supports.

Application to a Reddit Thread

Scree-plot: elbow at the 25th highesteigenvalue

# According to the Kaiser criterion,we reconstructed the newargument system with 25principal components.

# The P-BWAF has now 59

arguments involved in at least

one relation and 58 weighted

relations, of which◦ 44 attacks and◦ 14 supports

Application to a Reddit Thread

Relations removed:Start End Strength Relationa49 a48 0.05 supporta59 a58 −0.12 attacka60 a58 −0.12 attacka75 a74 −0.1 attacka78 a74 −0.08 attacka80 a74 −0.1 attack

Sub-graphs removed:

0.32 -0.42

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a76a61

a63a75

a62

Application to a Reddit Thread

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Discussion

# The new synthesized P-BWAF is that the global meaningof the discussion has been preserved

# The strongest relations continue to exist in the revisedP-BWAF

# While the weakest relations have been pruned# The SVD has removed the lowest weight relations: the 67%

of relations with weight less than 0.12 in absolute terms# The SVD has removed only the “peripheral” relations# This behavior suggests a possible strategy to determine the

ideal inconsistency budget for WAFs and BWAFs

CONCLUSIONS AND FUTURE WORKS

Conclusions and Future Works

# SVD exploited to extract a synthesized version of an argumentgraph to highlight only the relevant arguments

# We showed its application to a real web-based debate anddiscussed its effectiveness

# Having introduced the basic framework of Principal ArgumentSystems, some important open issues arise:◦ Are extensions affected by the reduction?◦ Are arguments removed those that can be immediately

flagged up as accepted/unaccepted for an extension of agiven semantics?◦ How different semantics are affected?◦ How can the reduced dimension of AFs affect solvers?◦ Is the minimization helpful for the reasoning process?