A novel weighted defence and its relaxation in abstract argumentation

When dealing with Weighted Argumentation, having weights on attacks clearly brings more information. The advantage, for instance, is the possibility to define a different notion of defence, checking also if the weight associated with it is stronger than the attack weight. In this work we study two different relaxations, one related to the new weighted defence we propose, by checking the difference between the composition of inward and outward attack-weights. The second one is related to how much inconsistency we are willing to tolerate inside an extension; such amount is computed by aggregating the costs of the attacks between any two arguments both inside an extension. These two relaxations are strictly linked: allowing a small conflict may lead to have more arguments into an extension, and consequently result in a stronger or weaker defence. Weights are represented by a semiring structure, which can be instantiated to different metrics used in the literature (e.g., costs, probabilities, fuzzy levels).