A Matrix Approach for Weighted Argumentation Frameworks: a Preliminary Report

The assignment of weights to attacks in a classical Argumentation Framework allows to compute semantics by taking into account the different importance of each argument. We represent a Weighted Argumentation Framework by a non-binary matrix, and we characterize the basic extensions (such as w-admissible, wstable, w-complete) by analysing sub-blocks of this matrix. Also, we show how to reduce the matrix into another one of smaller size, that is equivalent to the original one for the determination of extensions. Furthermore, we provide two algorithms that allow to build incrementally w-grounded and w-preferred extensions starting from a w-admissible extension. Introduction An Abstract Argumentation Framework (AF ) (Dung 1995) is represented by a pair 〈A, R〉 consisting of a set of arguments A and a binary relation of attack R defined between some of them. Given a framework, it is possible to examine the question on which set(s) of arguments can be accepted, hence collectively surviving the conflict defined by R. Answering this question corresponds to define an argumentation semantics. The key idea behind extension-based semantics is to identify some sets of arguments (called extensions) that survive the conflict “together”. A very simple example of AF is 〈{a, b}, {R(a, b), R(b, a)}〉, where two arguments a and b attack each other. In this case, each of the two positions represented by either {a} or {b} can be intuitively valid, since no additional information is provided on which of the two attacks prevails. However, having weights on attacks results in such additional information, which can be fruitfully exploited in this direction. For instance, in case the attack R(a, b) is stronger than (or preferred to) R(b, a), taking the position defined by a may result in a better choice for an intelligent agent, since it can be regarded as more reliable or relevant on the framework. In a recent work, Xu and Cayrol represent an AF by a binary matrix and they give a characterization for stable, admissible and complete extensions by analysing Copyright c © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. sub-blocks of this matrix (Xu and Cayrol 2015). Also, they present the reduced matrix w.r.t. conflict-free subsets, by which the determination of extensions becomes more efficient, and that allows to determine w-grounded and w-preferred extensions. Our aim is to extend the above mentioned results to Weighted Argumentation Frameworks (WAFs) by adopting the paradigm introduced in (Bistarelli, Pirolandi, and Santini 2010; Bistarelli, Rossi, and Santini 2016) for the semiringbased version of classical semantics. In particular, (i) we characterize w-conflict-free, w-admissible, w-stable and w-complete extensions by analysing sub-blocks of a non-binary matrix representing a given WAF, (ii) we show how to reduce this matrix to another one of smaller size that allows to more efficiently determine extensions, and (iii) we provide two algorithms that allow to build incrementally grounded and preferred extensions. This paper is organized as follows: we first recall the basic definitions on AFs and on WAFs, then we give characterizations for weighted extensions by analysing the matrix associated with the given WAF. Finally, we present the matrix reductions of WAFs based on contraction and division of WAFs, and we provide methods for incrementally building w-grounded and w-preferred extensions. Weighted Argumentation Frameworks In this section, we recollect the main definitions at the basis of AFs (Dung 1995), and introduce c-semirings for dealing with attack-weights. We then rephrase some of the classical definitions, with the purpose to parametrise them with the notion of weighted attack and c-semiring. Last, we give definitions about the matrix representation for AFs. Abstract Argumentation FrameworksArgumentation Frameworks In his pioneering work (Dung 1995), Dung proposedAbstract Frameworks for Argumentation, where (as shown in Figure 1) an argument is an abstract entity whose role is solely determined by its relations to other arguments: Definition 1. An Abstract Argumentation Framework (AF) is a pair 〈A, R〉 of a set A of arguments and a binary relation R on A, called attack relation. ∀ai, aj ∈ A, aiRaj (or R(ai, aj)) means that ai attacks aj (R is asymmetric).