Presentation from the 1st Workshop on Advances In Argumentation In Artificial Intelligence, co-located with XVI International Conference of the Italian Association for Artificial Intelligence (AI*IA 2017) held in Bari, on November 16-17, 2017.

1. SYNTHESIS OF ARGUMENTATION
GRAPHS BY MATRIX FACTORIZATION
1ST AI^3 WORKSHOP @ AI*IA 2017
Andrea Pazienza, Stefano Ferilli, Floriana Esposito
16th November 2017 – Bari, Italy

2. Overview
1. Introduction
2. Principal Argument System
3. Application to a Reddit Thread
4. Conclusions and future works

3. INTRODUCTION

4. 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 determining
what is acceptable
allows a range of different semantics
a
b
e
c
d
f
g
h
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

5. Bipolar Weighted Argumentation Framework
Bipolar Weighted Argumentation Framework (BWAF)
attack relations with a negative
weight in the interval [−1, 0[
support relations with a positive
weight in the interval ]0, 1]
a
b
0.7
e
-0.7
c0.9
d
-0.4
0.3
f
-0.5 g
-0.3
h
-0.5
-0.1
-0.7
BWAF Ranking-based Semantics by means of Strength Propagation

6. Argumentation Matrix
An argument graph can be represented with a slightly different
version of its adiacency matrix.
Let F A, R be an AF. Let |A| n, then the Argumentation Matrix of
F is a n × n matrix MF [Mij] such that for any two arguments
αi , αj ∈ Ait holds that
Mij
−1 if αi , αj ∈ R
0 otherwise
BWAF Example:
a b c
de
−0.7 −0.5
0.4 0.6
0.3
MG










0 0.4 0 0 −0.7
0 0 0.6 0 0
0 0 0 0 0
0 0 −0.5 0 0.3
0 0 0 0 0











7. PRINCIPAL ARGUMENT SYSTEM

8. Motivation
In the phase of evaluation of accepted arguments, one may find that
not all the arguments of discussion are essential when drawing
conclusions, especially when the cardinality of the set of arguments is
high.
Proposal: Produce a synthesized AF in order to:
decompose huge AFs and build a simplified ones,
highlight arguments that are extremely useful for the evaluation
process,
discard the less relevant arguments,
preserve the interpretation of the whole discussion.

9. 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 Value
Decomposition (SVD).

10. Singular Value Decomposition (SVD)
Let A ∈ Rm×n be a matrix and p min(m, n), the SVD of A is a
factorization 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 with
elements 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 UrΣrVT
r
r
i 1
ai σi vT
i
namely, matrix A has rank r.

11. Principal Argument System
Given the Argumentation Matrix of the argumentation graph
under 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 rule
to investigate the scree plot of matrix eigenvalues.

12. 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 relation
is built;
P-WAF : an attack relation between them iff Eab < −0.5 where its
weight is given by approximating the value Eab to the
nearest integer number;
P-BWAF : an attack relation or a support relation between them
with weight value equal to Eab with bounds −1 or 1 iff,
respectively, −1 < Eab < 0 or 0 < Eab < 1.

13. APPLICATION TO A REDDIT THREAD

14. Application to a Reddit Thread
-0.49
-0.48
0.16
0.32
-0.1
-0.17
-0.1
-0.42
0.16
-0.47
0.16
-0.13
0.5
0.06
0.05
-0.5 0.5
0.45
-0.15
0.08
0.08
-0.5
-0.44
-0.5
-0.5
-0.5
-0.1
-0.08
-0.5
-0.48
-0.46
-0.13
-0.12
-0.12
-0.5
-0.16
-0.32
-0.33
-0.16
0.32
-0.16
0.17
-0.4
-0.5
-0.17
-0.25
0.25
0.24
0.22
-0.25
-0.16
-0.34
-0.48
-0.24
-0.16
-0.45
-0.16
-0.14
-0.48
0.43
-0.45
-0.46
-0.44
-0.47
-0.48
-0.48
-0.48
-0.19
-0.45
a17
a22
a80
a8
a75
a10
a24
a64
a26
a48
a47
a4
a29
a50
a68
a73
a65
a84
a74
a54
a78
a58
a66
a49
a41
a51
a55
a5
a40
a56
a62
a57
a61
a69
a60
a63
a59
a9
a71
a53
a45
a72
a27
a0
a85
a39
a38
a34a37
a36
a35
a28
a33
a43
a31
a42
a32
a13a44
a11
a16
a15
a14
a12
a18
a20
a76
a19
a21
a23
We considered a Reddit
discussion of an episode of
Black Mirror, a popular TV
series
The produced BWAF is made
up of 70 arguments and 69
weighted relations, of which
◦ 52 attacks and
◦ 17 supports.

15. Application to a Reddit Thread
Scree-plot: elbow at the 25th highest
eigenvalue
According to the Kaiser criterion,
we reconstructed the new
argument system with 25
principal 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

16. Application to a Reddit Thread
Relations removed:
Start End Strength Relation
a49 a48 0.05 support
a59 a58 −0.12 attack
a60 a58 −0.12 attack
a75 a74 −0.1 attack
a78 a74 −0.08 attack
a80 a74 −0.1 attack
Sub-graphs removed:
0.32 -0.42
-0.16
-0.16 a60
a76
a61
a63
a75
a62

17. Application to a Reddit Thread
-0.48
-0.19
-0.45
-0.1
-0.5
-0.17
-0.17
0.16
0.32
-0.33
-0.48
0.06
-0.5
-0.13
-0.5
-0.25
-0.5 0.17
0.5
-0.25
-0.4
-0.5
-0.13
-0.5
0.08
-0.5
-0.5
0.25
0.5
-0.44
-0.46
-0.48
-0.46
-0.48
-0.24
-0.45
-0.16
-0.34
-0.45
-0.48
-0.14
0.43
0.24
0.16
0.16
-0.49
-0.47
0.22
-0.47
-0.48
-0.44
-0.16
-0.48
0.45
-0.15
-0.16 -0.16
-0.32
a19
a20
a74
a53
a8
a45
a22
a23
a21
a57
a56
a0
a66
a68
a58
a84
a65
a54
a69
a12
a34
a36
a37
a32
a31
a13
a44
a33
a55
a73
a47
a50
a43
a9
a71
a10
a27
a85
a26
a72 a15
a11
a42
a14
a17
a18
a24
a16
a4 a64
a48 a29
a28
a35a38
a5
a39
a41
a40

18. Discussion
The new synthesized P-BWAF is that the global meaning
of the discussion has been preserved
The strongest relations continue to exist in the revised
P-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

19. CONCLUSIONS AND FUTURE WORKS

20. Conclusions and Future Works
SVD exploited to extract a synthesized version of an argument
graph to highlight only the relevant arguments
We showed its application to a real web-based debate and
discussed its effectiveness
Having introduced the basic framework of Principal Argument
Systems, 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 a
given semantics?
◦ How different semantics are affected?
◦ How can the reduced dimension of AFs affect solvers?
◦ Is the minimization helpful for the reasoning process?