Proving convergence of self-stabilizing systems using first-order rewriting and regular languages

In the framework of self-stabilizing systems, the convergence proof is generally done by exhibiting a measure that strictly decreases until a legitimate configuration is reached. The discovery of such a measure is very specific and requires a deep understanding of the studied transition system. In contrast we propose here a simple method for proving convergence, which regards self-stabilizing systems as string rewrite systems, and adapts a procedure initially designed by Dershowitz for proving termination of string rewrite systems. In order to make the method terminate more often, we also propose an adapted procedure that manipulates "schemes", i.e. regular sets of words, and incorporates a process of scheme generalization. The interest of the method is illustrated on several nontrivial examples.