Relevance for Stability of Verification Status of a Set of Arguments in Incomplete Argumentation Frameworks (with Proofs)
Anshu Xiong, Songmao Zhang
TL;DR
This work extends relevance from stability of justification to stability of verification for a set of arguments in incomplete argumentation frameworks (IAFs) and introduces strong relevance. It formalizes how uncertainties (uncertain arguments/attacks) must be resolved to guarantee the same verification outcome across all completions, and it develops a detailed complexity analysis for relevance and strong relevance across five standard AF semantics. The results establish that relevance for stability is solvable in polynomial time for the ad, st, co, and gr semantics, while the pr semantics incur higher bounds (including $\,\Sigma_2^p$-type and $coNP$-type complexities); strong relevance generally remains within similar or lower complexity bounds, with nuanced differences for pr. Overall, the paper provides tractable methods to identify critical uncertainties, strengthens the theory of stability in IAFs, and lays groundwork for extending these notions to other stability notions and grounded semantics challenges.
Abstract
The notion of relevance was proposed for stability of justification status of a single argument in incomplete argumentation frameworks (IAFs) in 2024 by Odekerken et al. To extend the notion, we study the relevance for stability of verification status of a set of arguments in this paper, i.e., the uncertainties in an IAF that have to be resolved in some situations so that answering whether a given set of arguments is an extension obtains the same result in every completion of the IAF. Further we propose the notion of strong relevance for describing the necessity of resolution in all situations reaching stability. An analysis of complexity reveals that detecting the (strong) relevance for stability of sets of arguments can be accomplished in P time under the most semantics discussed in the paper. We also discuss the difficulty in finding tractable methods for relevance detection under grounded semantics.