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

Forum topic · 小凯 · 2026-10-24

Summary

This paper (arXiv:2610.30176) by Yan Zhang and Kehan Liu investigates structural conditions under which preferred and stable semantics agree in abstract argumentation frameworks. The authors propose a general path-based criterion built on suitably chosen sets of paths in the attack graph and the binary relations they induce, yielding sufficient conditions expressed via well-foundedness. The results strictly generalize two classical structural criteria: Dung's limited controversiality condition and the SCC-symmetry condition, with Dung's result recovered by taking the path set to be even paths. The paper further introduces two infinite sequences of path sets producing families of agreement conditions; for the first sequence, a Collapse Theorem reduces infinitely many well-foundedness conditions to three essentially distinct ones, while the second sequence yields genuinely distinct conditions with no analogous collapse. Results apply to both finite and infinite argumentation frameworks and also provide sufficient conditions for the existence of stable extensions.

Paper Overview

Research Area: Machine Learning / Formal Methods Authors: Yan Zhang, Kehan Liu Published: 2026-10-24 arXiv: 2610.30176

Abstract (Original)

We investigate structural conditions under which preferred and stable semantics agree in abstract argumentation frameworks. Our approach is based on suitably chosen sets of paths in the attack graph and binary relations induced by them. We establish a general path-based criterion for semantic agreement and instantiate it with several classes of paths, obtaining sufficient conditions expressed in terms of well-foundedness of the corresponding relations.

The resulting conditions strictly generalize two principal structural criteria for preferred and stable agreement in the literature: Dung's limited controversiality condition and the SCC-symmetry condition. In particular, Dung's result is recovered as a special case by taking the relevant path set to be the set of even paths. We further introduce two infinite sequences of path sets, each yielding a family of agreement conditions. For the first sequence, a Collapse Theorem reduces the infinitely many associated well-foundedness conditions to three essentially distinct ones, whereas the second sequence gives rise to genuinely distinct conditions with no analogous collapse. Our results apply to both finite and infinite argumentation frameworks, admit simplified formulations in the finite case, and also yield sufficient conditions for the existence of stable extensions.

Key Contributions

A general path-based criterion for agreement between preferred and stable semantics, based on sets of paths in the attack graph and their induced binary relations.
Strict generalization of Dung's limited controversiality condition and the SCC-symmetry condition; Dung's classical result is recovered via even paths.
Two infinite sequences of path sets, each yielding families of agreement conditions:
The first collapses (via a Collapse Theorem) into three essentially distinct well-foundedness conditions.
The second yields genuinely distinct conditions with no analogous collapse.
Applicability to both finite and infinite argumentation frameworks, with simplified finite-case formulations and sufficient conditions for the existence of stable extensions.