Weak vs. Self vs. Probabilistic Stabilization

Self-stabilization is a strong property, which guarantees that a distributed system always resumes a correct behavior starting from an arbitrary initial state. Since it is a strong property, some problems cannot have self-stabilizing solutions. Weaker guarantees hence have been later introduced to cope with impossibility results, e.g., probabilistic self-stabilization only guarantees probabilistic convergence to a correct behavior, and weak stabilization only guarantees the possibility of convergence. In this paper, we investigate the relative power of self, probabilistic, and weak stabilization, with respect to the set of problems that can be solved. Weak stabilization is by definition stronger than self-stabilization and probabilistic self-stabilization in that precise sense. We first show that weak stabilization allows to solve problems having no self-stabilizing solution. We then show that any finite state deterministic weak stabilizing algorithm to solve a problem under the strongly fair scheduler is always a probabilistic self-stabilizing algorithm to solve the same problem under the randomized scheduler. Unfortunately, this good property does not hold in general for infinite state algorithms. We however show that for some classes of infinite state algorithms, this property holds. These results hint at more practical use of weak stabilizing algorithms, as they are easier to design and prove their correctness than their self-stabilizing and probabilistic self-stabilizing counterparts.