Co-finiteness and Co-emptiness of Reachability Sets in Vector Addition Systems with States

摘要

The coverability and boundedness problems are well-known exponential-space complete problems for vector addition systems with states (or Petri nets). The boundedness problem asks if the reachability set (for a given initial configuration) is finite. Here we consider a dual problem, the co-finiteness problem that asks if the complement of the reachability set is finite; by restricting the question we get the co-emptiness (or universality) problem that asks if all configurations are reachable. We show that both the co-finiteness problem and the co-emptiness problem are complete for exponential space. While the lower bounds are obtained by a straightforward reduction from coverability, getting the upper bounds is more involved; in particular we use the bounds derived for reversible reachability by Leroux in 2013. The studied problems have been motivated by a recent result for structural liveness of Petri nets; this problem has been shown decidable by Jancar in 2017 but its complexity has not been clarified. The problem is tightly related to a generalization of the co-emptiness problem for non-singleton sets of initial markings, in particular for downward closed sets. We formulate the problems generally for sernilinear sets of initial markings, and in this case we show that the co-emptiness problem is decidable (without giving an upper complexity bound) and we formulate a conjecture under which the co-finiteness problem is also decidable.