Some Supplementaries to the Counting Semantics for Abstract Argumentation

Abstract

Dung's abstract argumentation framework consists of a set of interacting arguments and a series of semantics for evaluating them. Those semantics partition the powerset of the set of arguments into two classes: extensions and nonextensions. In order to reason with a specific semantics, one needs to take a credulous or skeptical approach, i.e. an argument is eventually accepted, if it is accepted in one or all extensions, respectively. In our previous work [1], we have proposed a novel semantics, called counting semantics, which allows for a more fine-grained assessment to arguments by counting the number of their respective attackers and defenders based on argument graph and argument game. In this paper, we continue our previous work by presenting some supplementaries about how to choose the damaging factor for the counting semantics, and what relationships with some existing approaches, such as Dung's classical semantics, generic gradual valuations. Lastly, an axiomatic perspective on the ranking semantics induced by our counting semantics are presented.Definition 1 (Abstract Argumentation Framework). An argumentation framework is a pair AF = X , R where X is a finite set of arguments and R ⊆ X × X is a binary relation on X , also called attack relation. (a, b) ∈ R means that a attacks b, or a is an attacker of b. Often, we write (a, b) ∈ R as aRb.We denote by R − (x) (respectively, R + (x)) the subset of X containing those arguments that attack (respectively, are attacked by) the argument x ∈ X , extending this notation in the natural way to sets of arguments, so that for S ⊆ X , R − (S) {x ∈ X : ∃y ∈ S such that xRy} and R + (S) {x ∈ X : ∃y ∈ S such that yRx}. Now, let us characterise two fundamental notions of conflict-free and defence.Definition 2 (Conflict-free, Defense). Let AF = X , R be an argumentation framework, let S ⊆ X and x ∈ X .

A parametrized ranking-based semantics compatible with persuasion principles

“…In the literature [2,9,17,26], many ranking-based semantics were compared on the basis of the properties recalled in the previous section. However, the relevance of some properties may be very much dependent on the context of application.…”

“…For the purpose of comparison, we also include in this table the results of some existing ranking-based semantics from the literature where the same set of properties has been already checked [9]. Namely, these semantics are: the semantics h-categoriser Cat [6,27], the Discussion-based semantics Dbs and the Burden-based semantics Bbs [2], the α-Burden-based semantics α-Bbs [3], the Counting semantics CS [26,28], the three propagation semantics Propa , Propa 1+ , Propa 1→ [8], the Tuples-based semantics Tuples [13], the semantics M&T [24] and the Iterated Graded Defense semantics IGD [16,17].…”

A Parametrized Ranking-Based Semantics for Persuasion

“…Inspired by these definitions, we formally define this propagation principle, including in addition a damping factor δ which allows to decrease the impact of attackers situated further away along a path (the longer the path length i, the smaller the δ i ). Among other things, such a damping factor will allow to guarantee the convergence of the computation of the arguments' values, as also proposed in [18,20]. Definition 3.…”

Encoding Argumentation Semantics by Boolean Algebra

“…Second, our approach is more intuitive than [19], as our approach is based on Boolean algebra, which has a natural connection with set theory, on which Dung's semantics are founded. Recently, a Boolean matrix approach was proposed to formalize the basic concepts of AFs [8]. In this paper, we enrich their work to encode Dung's acceptability semantics.…”