Diversity of Extensions in Abstract Argumentation

Assumption-Based Argumentation (ABA) is a well-established formalism for modelling and reasoning over debates, with a wide range of applications. However, the high computational complexity of core reasoning tasks in ABA poses a significant challenge for its applicability. This issue is further aggravated when ABA frameworks (ABAFs) are instantiated into graph-based argumentation formalisms, such as Dung's Argumentation Frameworks (AFs) and Argumentation Frameworks with Collective Attacks (SETAFs). In knowledge representation and reasoning, a key strategy to address computational intractability is to optimise reasoning over a given knowledge base through divide-and-conquer algorithms. A paradigmatic example of this approach is splitting, where extensions of a given framework are computed incrementally, by restricting the search space to sub-frameworks only, and then combining the obtained results. This approach has been successfully applied to AFs, for which also a parametrised version has been introduced under stable semantics. However, the exponential growth produced by the instantiation might undermine the usefulness of splitting on the argument graphs induced by ABAFs. To address this issue, our work investigates the concept of splitting on the knowledge base rather than on its graph-based instantiation. Furthermore, we generalise splitting to its parametrised version for ABAFs.

In Dung-style abstract argumentation, various semantics capture notions of acceptability of arguments. The admissibility semantics capture the notion that an argument can be consistently defended from any potential counterargument. Weak semantics often relax the demands of admissibility by restricting which counterarguments must be taken seriously (e.g., discounting self-defeating or otherwise incoherent attacks). Many prominent proposals for weak semantics remain extension-based in a stronger sense. While these semantics discount attacks from arguments which are considered unreasonable, they still require a uniform defense against all reasonable arguments, even if they are collectively inconsistent. This uniformity can be too demanding when defensibility is inherently strategic, and thus the appropriate reply depends on the opponent's line of attack. We introduce tenability, a family of dialogue-based semantics that formalize when a designated argument (or a set of arguments) can be maintained in debate by a proponent against any conflict-free attack which the opponent may present. The approach is motivated by three natural benchmark patterns: self-defeating attack, floating assignment, and disjunctive reinstatement, on which tenability behaves differently from all weak semantics previously considered in the literature. We define three variants -- static tenability, tenability, and strong tenability -- via monotone commitment games over finite conflict-free moves, differing in the obligations imposed on the disputants. We establish the relative strength of these notions, prove implications and separations with previously studied weak semantics, and we analyze computational complexity on finite frameworks: deciding static tenability is $Π^P_2$-complete, while deciding tenability and strong tenability is PSPACE-complete.

This work proposes novel splitting techniques for argumentation formalisms that incorporate supports between defeasible elements. We base our studies on bipolar set-based argumentation frameworks (BSAFs) which generalize argumentation frameworks with collective attacks (SETAFs), as well as bipolar argumentation frameworks (BAFs), by incorporating both collective attacks and supports. Notably, BSAFs establish a crucial link to structured argumentation as they naturally capture general (potentially non-flat) assumption-based argumentation. The increase in expressiveness calls for diverse forms of splitting. We consider splits over collective attacks (thereby generalizing the recently proposed splitting techniques for SETAFs), splits over collective supports, as well as splits over both collective attacks and supports. We establish suitable splitting schemata and prove their correctness for the most common argumentation semantics.