=Paper=
{{Paper
|id=Vol-1769/paper02
|storemode=property
|title=A Study of Robustness in Abstract Argumentation Frameworks
|pdfUrl=https://ceur-ws.org/Vol-1769/paper02.pdf
|volume=Vol-1769
|authors=Carlo Taticchi
|dblpUrl=https://dblp.org/rec/conf/aiia/Taticchi16
}}
==A Study of Robustness in Abstract Argumentation Frameworks==
https://ceur-ws.org/Vol-1769/paper02.pdf
A Study of Robustness in
Abstract Argumentation Frameworks
Carlo Taticchi
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Italy
carlo.taticchi@studenti.dmi.unipg.it
Abstract. In this paper we describe a work-in-progress on the study
of robustness (intended as how many changes can an Argumentation
Framework sustain before its semantics turns into another): in particular
we present a tool able to visualize Argumentation Frameworks as nodes
in a graph and highlight those with certain properties, like a specific
semantics or attacks’ type. This tool will be used to find relations between
an Argumentation Framework and the sets of extensions accepted by
each semantics and eventually, to extract new theorems in order to cope
with some of the open problem related to abstract argumentation.
Key words: Argumentation Frameworks, Robustness, Semantics, Visualization
tools
1 Motivations
The relation between belief revision and argumentation is the subject of several
studies aiming to find a connection between the consistency of beliefs and the
preservation of a certain semantic when the knowledge base changes. However,
in the literature, such a relation is often found for the grounded semantics by
way of comparison, assuming that considerations concerning this semantics have
the same implications of belief revision.
In this work we would like to extend this relationship, disengaging from
the use of (only) the grounded semantics and trying to generalize the already
established concepts; the goal is to focus on the modifications in the semantics
of an Abstract Argumentation Framework (AAF), also called Argumentation
Framework (AF) à la Dung, due to changes in terms of arguments and attacks
between arguments. We will, in this way, investigate the notion of “robustness”
to cope with changes in AFs.
Another goal is to obtain a method to find, given a specific semantics, all
the graphs representing an AF with that semantics. We will extract theorems
and proofs on the possibility or not that semantics with certain properties exist
partitioning the AFs set according to attacks type between arguments. For ex-
ample, one can focus on the properties of the AFs in which the only attacks are
all self attacks or bidirectional attacks, to find relations between AFs containing
the same extensions for a certain semantics. In this way, we could be able to
solve some of the open problem related to abstract argumentation, like those
proposed by Baumann and Strass in [1]. Some example problems could be: given
an AF, can all implicit conflicts be made explicit (by adding one or two attacks
between them)? Or even what is the maximal number of extensions for each
semantics in an AF with n arguments?
In order to better understand problems related to argumentation, we devel-
oped a graphical tool capable of representing different aspects of an AF. This
tool has three panels: in the first one, we find the complete set of the graphs
obtained from a predetermined number of arguments. In the second panel, the
tool displays in detail a particular graph selected from the complete set of them.
The last panel shows a lattice of semantics ordered by inclusion, in which those
corresponding to the selected graphic will be highlighted.
2 Background
In this section we give the basic definitions of AF and extension-based semantics.
Definition 1. An AF [8] is defined as a pair F = hA, Ri where A is a set of
arguments and R is a binary relation on A called attack relation. An argument
a ∈ A is acceptable with respect to S ⊆ A if and only if S defends a, that is
∀b ∈ A such that (b, a) ∈ R, ∃c ∈ S such that (c, b) ∈ R.
The “acceptability” of an argument, defined under different semantics, de-
pends on its membership to some sets, called extensions whose definition is given
in Def. 2.
Definition 2. Let F = hA, Ri be an AF, a set S ⊆ A is i) conflict-free if there
are no attacks between arguments in S, ii) admissible iff it is conflict-free and all
its arguments are acceptable w.r.t. S, iii) complete if it is admissible and every
acceptable argument w.r.t. S belongs to S, iv) grounded only if it is the minimal
element w.r.t. set inclusion among the complete extensions of S.
3 A Visual Tool
Robustness [2] in AF is considered as the property of an argumentation graph
to withstand changes in terms of classical extension-based semantics and is mea-
sured by computing the number of changes needed to change the corresponding
extension. We give two definitions, useful to understand the aim of the tool.
Fig. 1. A screenshot of the tool menu.
Definition 3. Let G = hA, Ri be a graph representing an AF. G is said i)
robust w.r.t. a given semantics if changes in A or R don’t change the semantics,
ii) partially robust w.r.t. a given semantics and an extension if changes in A or
R change the semantics, but preserve the given extension in the new semantics.
Starting from an alpha version (presented in [3]) we have developed Rob 1 , a
tool that allows displaying all the AFs obtainable by any combinations of attacks
between a fixed number of arguments. Each of these AFs is represented as a node
in an oriented graph and each node in this graph is connected by an edge to all
other nodes whose corresponding AF only differs by one attack. Moreover, when
a certain AF is selected by clicking on it, the tool displays the graph relative to
that AF and the sets of extensions obtainable for each semantics. So, the notion
of robustness exploited by Rob is the partial robustness.
The tool’s menu allows selecting the number of arguments that the AFs will
contain and one can choose whether to take in account or not AFs in which
arguments attack themselves (self attacks), defend themselves from every other
attack (symmetric AFs) or attack without direct counterattacks. In Fig. 1, for
example, we chose to consider only all the AFs with 3 arguments and no self
attacks. Whenever the “draw” button is clicked the lattice of all AFs is drawn.
Clicking on a node allows displaying a lattice of extensions for the corresponding
AF. These extensions are computed by ConArg2 , a tool able to solve problems
related to the AFs. Each set is colored according to the less inclusive semantics
to which it belongs. For instance, a set marked as admissible will imply its
belonging to conflict free semantics too. At last, selecting a set of extensions will
show all the AFs on the first panel for which there exists a semantics containing
those extensions and each AF will be colored according to that semantics. The
notion of partial robustness is well represented in Fig. 2: on the left, we can
see the graph in which colored nodes are those representing the AFs allowing
the selected extension {2} and in the right panel we can see that even if the
semantics changes together with the AF, it always contains the extension {2}.
4 Insights
The features introduced in this version of the tool allowed us to get some signif-
icant results. At first, we focused on the set of symmetric AFs, in which every
attack is bidirectional, i.e. if a and b are two arguments of an AF and a attacks
b, then b attacks a and we found out some interesting properties, also suggested
by Coste-Marquis et al. in [6].
For instance, we observed that all the conflict-free extensions are also admis-
sible. Indeed it holds that an extension in a symmetric AF is admissible if and
only if it’s conflict-free. Another property we observed concerns the grounded
extensions: in a symmetric AF the grounded extension is given by the set of ar-
guments which are not attacked. If every argument is attacked, then the empty
1
www.dmi.unipg.it/rob
2
www.dmi.unipg.it/conarg
Fig. 2. On the right, the graph of extensions in which the set {2} is selected. On the
left, the lattice of AF showing in which semantics the set {2} appears.
set is the grounded extension. In Fig. 3 we can see that the argument with label
3 (on the left graph) is the only one which is not attacked. Hence the extension
set containing the only argument 3 (the yellow node in the right graph) is the
grounded extension (and in this example, it is also the only complete extension).
Changing the parameters used for the representation (number of arguments and
attacks type) it is possible to exploit the functionality of the tool in order to
study semantics properties, with respect to the inclusion between AFs, linked to
the notion of robustness.
5 Related Work
Abstract Argumentation has been proved as a simple yet powerful approach to
manage conflicts in reasoning with the purpose to find subsets of “surviving”
arguments. In the following, we condensate some of the work related to what we
intend to do in the future.
Cayrol et al. [5] proposed an Abstract Argumentation system that allows the
addition of a new argument that may interact with previous arguments. The
revision of an AF produces a new framework and a new set of extensions. We
could exploit this system to introduces changes in the set of arguments.
In [11] Stefan Woltran briefly point out two directions for the development
of next-generation argumentation systems: the first one is a review of the Ex-
plicit Conflict Conjecture while the second one discusses how semantics can be
exploited in practice. We could use our tool to cope with the problems here
introduced.
Doyle [7] presents the truth maintenance system, which is a knowledge repre-
sentation method for representing both beliefs and their justifications. The aim
of such systems is to restore consistency when a new justification is added.
Fig. 3. A representation of 3 arguments AFs lattice with a selected AF drawn on the
right side.
Boella et al. [4] try to show a direct relation between argumentation and
belief revision establishing a link between reinstatement (an argument that is
not acceptable become acceptable again) and recovery (AGM postulates). Fur-
thermore in [9] the authors consider some semantics and show how that their
expressive power relies on rejected arguments and implicit conflicts.
In [10] Rotstein et al. introduce an abstract theory that captures the dynam-
ics of an argumentation framework through the application of belief revision
concepts. The presented argument revision techniques allow introducing new
arguments ensuring they will be believed afterwards.
6 Conclusion and Future Work
The work presented in this paper is a first step towards the study of the concept
of robustness in AFs. Several works in the literature concern the connection
between consistency of beliefs and the preservation of semantics in knowledge
base when changes are introduced; however, none of them focuses on studying the
implications of the changes in terms of variations of attacks between arguments.
Due to the inherent complexity in generating the AFs, the current version of the
tool is able to generate all the AFs with a maximum of 3 arguments. By setting
3 as the number of arguments, 32 different attacks can be presented, for a total
of 29 different AFs. In the future, we intend to bring this threshold to 6, by only
considering non-isomorphic AFs in order to reduce the generated nodes. Also, we
would like to study inclusion between extensions of a single AF and between sets
of AF in the graph. At this point, a more detailed study concerning subclasses
of directed graphs (such as symmetric and simple) could be carried out. Finally,
we would like to extend the concept of robustness to coalitions of arguments, by
studying how much a group of arguments derived from partitioning the original
set is more robust than another.
Acknowledgment
I want to thank with gratitude professors Stefano Bistarelli and Francesco San-
tini, for their constant help and their patience in making me learn the basics of
argumentation and belief revision theory. Their endless enthusiasm gave me the
passion for study and research and I hope to carry on the work done together
as well as our collaboration.
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