El mal y sus epígonos

We have developed the notion of intervention-based entailment as a formal tool to study the idea that an argumentation framework is a system whose evalua- tion changes as new arguments and attacks are added. A similar perspective is

taken by Baroni et al. [3], who study the so called input/output behaviour of ar- gumentation frameworks. Roughly speaking, the idea is that an argumentation framework is seen as a sort of black box, with both input and output, formed by interactions with arguments outside the argumentation framework. They study two types of properties. The first type of properties concern the decomposability of a semantics, or the question of whether putting together the labellings of different subframeworks under a given semantics (while taking into account the interactions between the subframeworks) results in a set of labellings that coin- cides with the labellings of the whole argumentation framework under the same semantics. The first type of properties concern the replacement properties: the question of whether different argumentation frameworks that behave the same with respect to a given set of input and output arguments are interchange- able in the context of a larger argumentation framework, without affecting the labellings of this larger argumentation framework.

A specific type of decomposability, called SCC-recusiveness, was studied earlier by Baroni et al. [8]. The SCC-recursive scheme exploits the property that an argumentation framework can be decomposed into a set of strongly connected components (SCCs) and that the graph obtained considering SCCs as single nodes is acyclic. This means that this decomposition yields a partial order over SCCs. This partial order can be used for the incremental computation of the extensions of an argumentation framework, by first computing the extensions of the initial SCCs (i.e., those not attacked by other SCCs), and using these results to recursively compute the extensions of the whole argumentation framework. A semantics is called SCC-recursive if its extensions can be computed using this recursive scheme.

We did not, like the work mentioned here, focus on properties related to de- composability or replacement. Nevertheless, the results that we have obtained contribute, like these properties, to a deeper understanding of the behaviour of an argumentation framework when external input is considered.

Liao et al. [65] addressed the question of whether, after modifying an argumen- tation framework, it is possible to partially reuse extensions computed before the modification, when computing the extensions of the modified argumentation framework. They show that, roughly speaking, directionality ensures the eval- uation of the set of arguments not reachable from some argument affected by the modification remains the same. They call this set of unreachable arguments the unaffected part. This is very similar to what the Conditional Directional- ity property states (that is, consequences referring to arguments not reachable from an argument affected by an intervention remain the same). In a follow-up work, Baroni et al. [9] show that the SCC-recursiveness property enables one to reuse the (unchanged) evaluation of the unaffected part when computing the evaluation of the modified argumentation framework.

Sakama [84] studied properties of counterfactual conditionals about the sta- tus of arguments in an argumentation framework. He defines a semantics for counterfactual conditionals in abstract argumentation of the form α 2→ β or α 3→ β, where α and β are single literals of the form in(x) or out(x). The intended meaning of α 2→ β (resp. α 3→ β) is that “if it were the case that α, then it would (resp. might) be the case that β.” The interpretation is similar to ours, i.e., the premise is translated into a modification of the argumentation

framework within which the consequent is evaluated. In particular, acceptance in the premise is translated by removing all attackers pointing towards the ac- cepted argument. A number of properties of these conditionals are studied, and some of them are similar to the properties we considered. These include Reflexivity and a restricted form of Cautious Monotony, as well as properties such as Transitivity and Contraposition. We have extended these results by, for example, demonstrating the failure of Cautious Monotony under the pre- ferred, semi-stable and stable semantics, and proving results with respect to argumentation frameworks containing no (odd-length or even-length) directed cycles.

Cayrol et al. [34] study the impact of modifying an argumentation framework by adding a single new argument that interacts with old arguments. They de- fine a number of properties that describe the impact of such a modification on the extensions of the argumentation framework, such as restrictiveness, con- servativeness and questioning (i.e., modifications that result, respectively, in a smaller, equivalent, and larger set of extensions). They then define necessary and sufficient conditions under which a modification satisfies these properties. While the properties that we studied in this chapter are, like those studied by Cayrol et al., also about the impact of modifying an argumentation framework, the nature of the type of properties used to describe this impact are of a differ- ent nature. For example, all properties used by Cayrol et al. refer to extensions instead of labellings, and hence no distinction is made between out and und labelled arguments. Furthermore, the majority of these properties refer explic- itly to the number of extensions before and after the modification, while this number does not appear in the definition of the properties that we consider.