Observing locally self-stabilization in a probabilistic way

A self-stabilizing algorithm cannot detect by itself that stabilization has been reached. For overcoming this drawback Lin and Simon introduced the notion of an external observer: a set of processes, one being located at each node, whose role is to detect stabilization. Furthermore, Beauquier, Pilard and Rozoy introduced the notion of a local observer: a single observing entity located at an unique node. This entity is not allowed to detect false stabilization, must eventually detect that stabilization is reached, and must not interfere with the observed algorithm. We introduce here the notion of probabilistic observer which realizes the conditions above only with probability 1. We show that computing the size of an anonymous ring with a synchronous self-stabilizing algorithm cannot be observed deterministically. We prove that some synchronous self-stabilizing solution to this problem can be observed probabilistically.