Necessary and sufficient conditions for 1-adaptivity

A 1-adaptive self-stabilizing system is a self-stabilizing system that can correct any memory corruption of a single process in one computation step. 1-adaptivity means that if in a legitimate state the memory of a single process is corrupted, then the next system transition will lead to a legitimate state and the system will recover a correct behavior. Thus 1-adaptive self-stabilizing algorithms guarantee the very strong property that a single fault is corrected immediately and consequently that it cannot be propagated. Our aim here is to study necessary and sufficient conditions to obtain that property in order to design such algorithms. In particular we show that this property can be obtained even under the distributed demon and that it can also be applied to probabilistic algorithms. We provide two self-stabilizing 1-adaptive algorithms that demonstrate how the conditions we present here can be used to design and prove 1-adaptive algorithms.