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 speci c semantics or attacks' type. This tool will be used to nd 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.
The relation between belief revision and argumentation is the subject of several studies aiming to nd 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 modi cations in the semantics of an Abstract Argumentation Framework (AAF), also called Argumentation Framework (AF) a 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 nd, given a speci c 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
example, one can focus on the properties of the AFs in which the only attacks are
all self attacks or bidirectional attacks, to nd 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 [
In order to better understand problems related to argumentation, we developed a graphical tool capable of representing di erent aspects of an AF. This tool has three panels: in the rst one, we nd 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
In this section we give the basic de nitions of AF and extension-based semantics.
De nition 1. An AF [
The \acceptability" of an argument, de ned under di erent semantics, depends on its membership to some sets, called extensions whose de nition is given in Def. 2.
De nition 2. Let F = hA; Ri be an AF, a set S A is i) con ict-free if there are no attacks between arguments in S, ii) admissible i it is con ict-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
Robustness [
De nition 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 [
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 con ict free semantics too. At last, selecting a set of extensions will show all the AFs on the rst 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 f2g and in the right panel we can see that even if the semantics changes together with the AF, it always contains the extension f g 2 . 4
The features introduced in this version of the tool allowed us to get some
significant results. At rst, 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 [
For instance, we observed that all the con ict-free extensions are also admissible. Indeed it holds that an extension in a symmetric AF is admissible if and only if it's con ict-free. Another property we observed concerns the grounded extensions: in a symmetric AF the grounded extension is given by the set of arguments which are not attacked. If every argument is attacked, then the empty 1 www.dmi.unipg.it/rob 2 www.dmi.unipg.it/conarg 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
Abstract Argumentation has been proved as a simple yet powerful approach to manage con icts in reasoning with the purpose to nd 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. [
In [
Doyle [
Boella et al. [
In [
The work presented in this paper is a rst 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 di erent attacks can be presented, for a total of 29 di erent 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. I want to thank with gratitude professors Stefano Bistarelli and Francesco Santini, 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.