A relaxation of internal conflict and defence in weighted argumentation frameworks

In Weighted Abstract Argumentation Frameworks (WAAFs), weights on attacks bring more information. An advantage is the possibility to define a different notion of defence, which also checks if the weight associated with defence is compared with the weight of attacks. We study and merge together two different relaxations of classically crisp-concepts in WAAFs: one is related to a new notion of weighted defence (defence can be stronger or weaker at will), while the second one is related to how much inconsistency one is willing to tolerate inside an extension (which can be not totally conflict-free now). These two relaxations are strictly related and influence each other: allowing a small conflict may lead to have more arguments in an extension, and consequently result in a stronger or weaker defence. We model weights with a semiring structure, which can be instantiated to different metrics used in the literature (e.g., fuzzy WAAFs).