On the Undecidability of Deadlock Detection in Families of Nets

In this paper, we are interested in modelization of some aspects of massively parallel computers built by connecting together copies of a unique given pattern. Thus, a natural and important question arises. Is it possible to verify that a property holds, not for one given configuration, but for all the configurations that can be constructed? Of course, the difficulty comes from the fact that the number of distinct configurations is infinite. We focus in this paper on the deadlock problem: can we verify that any net, constructed by connecting an arbitrary number of copies of the pattern, is deadlock free? As in a great number of models, the semantics of the pattern and thus the semantics of a net are given by finite transition systems. The main result of this paper proves that the deadlock problem is surprisingly undecidable.