The Complexity of the Finite Containment Problem for Petri Nets

If the reachability set of a Petri net (or, equivalently, vector addition system) is finite it can be effectively constructed. Furthermore, the finiteness is decidable. Thus, the containment and equality problem for finite reachability sets become solvable. We investigate the complexity of decision procedures for these two problems and show by reducing a bounded version of Hilbert''s Tenth problem to the finite containment problem that these two problems are extremely hard, that, in fact, the complexity of each decision procedure exceeds any primitive recursive function infinitely often. The finite containment and quality problem are thus the first uncontrived, decidable problems with provably non-primitive recursive complexity.