We show that for every fixed $k\geq 3$, the problem whether the termination/counter complexity of a given demonic VASS is $\mathcal{O}(n^k)$, $\Omega(n^{k})$, 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 $k\leq 2$. Interestingly, 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.

翻译：对于每个固定的 $k\geq 3 美元,我们显示,对于每个固定的 $k\ geq 3 美元, 特定恶魔VAS 的终止/ 对抗复杂程度是否为$\ mathcal{O} (n ⁇ k) $, $\\\ k} 和$\ Theta(n ⁇ k) $, 分别为 CNP 完成, NP- 完成, 和 DP- 完成。 我们还将这些问题的复杂性分类为 $k\leq 2 。 这显示, 先前作品中为紧密连接的恶魔 VAS 设计的多边- 时间算法不能扩大到一般情况。 然后, 我们证明 VAS 游戏的相同问题也是 PSPACE 完成的。 有意思的是, 魔鬼 VAS SS 和 VASS 游戏的可移动的子类通过约束某些结构参数来获得。 这为程序分析的应用打开了通道, 尽管显示的复杂度较低限制 。