Co-finiteness and co-emptiness of reachability sets in vector addition systems with states.

*(English)*Zbl 1428.68200Summary: The boundedness problem is a well-known exponential-space complete problem for vector addition systems with states (or Petri nets); it 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 exponential-space complete. While the lower bounds are obtained by a straightforward reduction from coverability, getting the upper boundsis more involved; in particular we use the bounds derived for reversible reachability by J. Leroux [in: Proceedings of the 2013 28th annual ACM/IEEE symposium on logic in computer science, LICS 2013. Los Alamitos, CA: IEEE Computer Society. 23–32 (2013; Zbl 1366.68209)].

The studied problems were motivated by a result for structural liveness of Petri nets; this problem was shown decidable by P. Jančar [Lect. Notes Comput. Sci. 10139, 91–102 (2017; Zbl 1425.68295)], without clarifying its complexity. The structural liveness problem is tightly related to a generalization of the co-emptiness problem, where the sets of initial configurations are (possibly infinite) downward closed sets instead of just singletons. We formulate the problems even more generally, for semilinear sets of initial configurations; 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.

We show that both the co-finiteness problem and the co-emptiness problem are exponential-space complete. While the lower bounds are obtained by a straightforward reduction from coverability, getting the upper boundsis more involved; in particular we use the bounds derived for reversible reachability by J. Leroux [in: Proceedings of the 2013 28th annual ACM/IEEE symposium on logic in computer science, LICS 2013. Los Alamitos, CA: IEEE Computer Society. 23–32 (2013; Zbl 1366.68209)].

The studied problems were motivated by a result for structural liveness of Petri nets; this problem was shown decidable by P. Jančar [Lect. Notes Comput. Sci. 10139, 91–102 (2017; Zbl 1425.68295)], without clarifying its complexity. The structural liveness problem is tightly related to a generalization of the co-emptiness problem, where the sets of initial configurations are (possibly infinite) downward closed sets instead of just singletons. We formulate the problems even more generally, for semilinear sets of initial configurations; 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.

##### MSC:

68Q85 | Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) |