When Stable and Preferred Semantics Agree: A Path-Based Approach

Abstract argumentation frameworks (AFs) introduced by Dung provide a formal foundation for non-monotonic reasoning in artificial intelligence. While decision problems for general infinite AFs typically reside at high levels of the analytical hierarchy ($Σ_1^1$ or $Π_1^1$), restricting the framework to be computably finitary reduces some of the complexity to the arithmetical hierarchy. In this paper, we present a complexity mapping of grounded and preferred semantics in computably finitary AFs across standard decision problems: credulous acceptance ($\Cred$), skeptical acceptance ($\Skep$), extension existence ($\Ex$), uniqueness ($\Uni$), and non-empty existence ($\NE$). For grounded semantics, credulous and skeptical acceptance are already known to be $Σ_1^0$-complete. We show that non-empty existence is also $Σ_1^0$-complete, whereas existence and uniqueness are trivial. These classifications are understood within the domain of valid computably finitary representations. For preferred semantics, using a computably finitely branching computation tree, $\Cred_{\pref}$ is shown to be in $Π_1^0$-c and $\NE_{\pref}$ is $Σ_2^0$-c. However, it is insufficient to reduce universal quantification and global uniqueness, leaving $\Skep_{\pref}$ in $Π_1^1$ and $\UniPref$ in $Σ_2^1$-c. Our results show the precise boundary where finitarity succeeds to bring reasoning down to the arithmetical hierarchy and where second-order quantification forces problems back into the analytical hierarchy.

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.

Preference-based argumentation frameworks (PAFs) extend Dung's approach to abstract argumentation (AAFs) by encoding preferences over arguments. Such preferences control the transformation of attacks into defeats, and different approaches to doing so result in different reductions from a PAF to an AAF. In this paper we consider a PAF inverse problem which takes an argumentation graph, a labelling and a semantics as an input, and outputs a yes" or no" as to whether there is a preference relation between the arguments which can yield the desired labelling. This inverse problem has applications in areas including preference elicitation and explainability. We consider this problem in the context of the four most widely-used preference based reductions under the complete semantics. We show that in most cases, the problem can be answered in polynomial time.