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

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.