### The PCTL satisfiability problem resolved!

The general/finite PCTL satisfiability problems are undecidable.

We show that for every fixed degree k\geq 3, the problem whether the termination/counter complexity of a given demonic VASS is O(n^{k}), \Omega(n^{k}), and \Theta(n^{k}) is coNP-complete, NP-complete, and DP-complete, respectively. We also classify the complexity of these problems for k\leq 2. This shows that the polynomial-time algorithm designed for strongly connected demonic VASS in previous works cannot be extended to the general case. Then, we prove that the same problems for VASS games are PSPACE-complete. Again, we classify the complexity also for k\leq 2.

Tractable subclasses of demonic VASS and VASS games are obtained by bounding certain structural parameters, which opens the way to applications in program analysis despite the presented lower complexity bounds.

Authors: Michal Ajdarów and Antonín Kučera