Fair Petri nets and structural induction for rings of processes

We present a structural induction theorem for rings consisting of an arbitrary number of identical components. The components of a ring are modeled using a "fair Petri net," in which the firing of a prespecified set of transitions is assumed to occur fairly, i.e., any of these transitions that becomes firable infinitely often must fire infinitely often. Specifically, we introduce the concept of similarity between rings of different sizes, and give a condition under which the similarity between the rings of sizes two and three guarantees the similarity among the rings of all sizes. So if the given condition is satisfied, then the correctness of a ring of any large size can be inferred from the correctness of a ring having only a few components. The usefulness of the theorem is demonstrated using the examples of token-passing mutual exclusion and a simple producer-consumer system.