Abstract: The problem of deciding whether an infinite-state system is self-stabilizing or not is investigated from the decidability viewpoint in this paper. We develop a unified strategy through which checking self-stabilization is shown to be decidable for one-counter machines and conflict-free Petri nets. Our strategy relies on the reachability sets being semilinear, as well as on the capability of extracting periodic behaviors of infinite computations, which, in turn, facilitates the expression of self-stabilization by Presburger Arithmetic. As fairness is frequently used as a qualitative measure to capture the notion of a quantitative measure of 'something happens with probability one,' it is of interest to examine the fair version of the self-stabilization problem, i.e., the problem of asking whether all 'fair' infinite computations eventually become self-stabilizing. We propose a potential method through which the problem is shown to be decidable for conflict-free Petri nets.