On the expressive power of collective attacks

Abstract

In this paper, we consider argumentation frameworks with sets of attacking arguments (SETAFs) due to Nielsen and Parsons, an extension of Dung's abstract argumentation frameworks that allow for collective attacks. We first provide a comprehensive analysis of the expressiveness of SETAFs under conflict-free, naive, stable, complete, admissible, preferred, semi-stable, and stage semantics. Our analysis shows that SETAFs are strictly more expressive than Dung AFs. Towards a uniform characterization of SETAFs and Dung AFs we provide general results on expressiveness which take the maximum degree of the collective attacks into account. Our results show that, for each k > 0, SETAFs that allow for collective attacks of k + 1 arguments are more expressive than SETAFs that only allow for collective attacks of at most k arguments. W. Dvořák et al. / On the expressive power of collective attacks PreliminariesWe first introduce formal definitions of argumentation frameworks following [6,16] and then recall the relevant work on signatures. Argumentation frameworks with collective attacksThroughout the paper, we assume a countably infinite domain A of possible arguments.The collection of all SETAFs (resp. k-SETAFs) over A is given as AF A (resp. AF k A ).We shall call 1-SETAFs, i.e. SETAFs that only allow for binary attacks, Dung argumentation frameworks (AFs) as they are equivalent to the AFs introduced in [6]. Definition 2. Given a SETAFthe set {b | S → R b} of argument attacked by S (in R), and define the range of S (w.r.t. R), denoted S ⊕ R , as the set S ∪ S + R . Example 1. Recall the framework from the introduction, with arguments a, b, c where each pair of arguments jointly attacks the remaining argument. This is modeled by the SETAF (A, R) with arguments A = {a, b, c} and attacks R = {({a, b}, c), ({a, c}, b), ({b, c}, a)}. In fact, this SETAF has been already presented in Fig. 1. Note that we have that {a, b} → R c but neither {a} → R c nor {b} → R c for this SETAF. On the other hand {a, b, c} → R c indeed holds.The notions of conflict and defense naturally generalize to SETAFs. Definition 3. Given a SETAFThe notion of defense can be equivalently characterized as follows: an argument a ∈ A is defended by a set S ⊆ A if for each (B, a) ∈ R we have S → R B.Next, we introduce the semantics we study in this work. Besides conflict-free and admissible sets, these are the naive, stable, preferred, complete, grounded, stage, and semi-stable semantics, which we will abbreviate by naive, stb, pref , com, grd, stage, and sem, respectively. All semantics except semistable and stage are defined according to [16], while semi-stable and stage are straight forward generalizations of the according semantics for Dung AFs [4,21], which have been independently proposed in [11,13]. For a given semantics σ , σ (F ) denotes the set of extensions of F under σ .

A comprehensive study of argumentation frameworks with sets of attacking arguments

“…Recent work [16] has shown that this is indeed the case, under the important assumption that the set of arguments is fixed. In other words, [16] showed that, given a set of arguments, one can create a SETAF whose semantics cannot be captured by any AAF (using the same set of arguments).…”

“…• We show how a SETAF can be modelled as an AAF, with an exponential increase in the number of arguments considered, and how the extensions of the two frameworks relate to each other (Section 6). Our approach circumvents the limitation formally proven in [16], where it was shown that SETAFs are strictly more expressive than AAFs when considering a fixed set of arguments.…”

“…Indeed, according to the legislation about voting 2 , you are not allowed to vote in the UK if you are under 18; in this case,"you are under 18" acts as an argument by itself against the eligibility to vote. Moreover, according to the legislation about alcohol and young people 3 , in the UK you are not allowed to drink alcohol in public if you are under 16, or if you are under 18 and you are either not accompanied by an adult or you are not having a meal. In the latter case, "you are under 18" is combined with one of two other statements to form an argument against the eligibility to drink alcohol in public.…”

“…However, to the best of our knowledge, there has been no effort for a complete formal description of SETAF, both in terms of the different kinds of extensions that have been used to describe AAF (apart from the semantics described by Dung himself in [2], which was done in [1]), and in terms of the more fine-grained labelling-based semantics. Also, the relationship between AAF and SETAF has only been recently considered [16] under specific assumptions.…”

Investigating subclasses of abstract dialectical frameworks

“…Secondly, since other generalizations of Dung AFs can be seen as special case of ADFs, results on ADFs carry over to these special cases. We exemplify this aspect in the paper, by deriving new results for argumentation frameworks with collective attacks (SETAFs) [27] which have received increasing interest recently [18,21]. To the best of our knowledge concepts like symmetric SETAFs have not been investigated yet, and we provide first results in this direction.…”