We investigate the dimension-parametric complexity of the reachability problem in vector addition systems with states (VASS) and its extension with pushdown stack (pushdown VASS). Up to now, the problem is known to be $\mathcal{F}_k$-hard for VASS of dimension $3k+2$ (the complexity class $\mathcal{F}_k$ corresponds to the $k$th level of the fast-growing hierarchy), and no essentially better bound is known for pushdown VASS. We provide a new construction that improves the lower bound for VASS: $\mathcal{F}_k$-hardness in dimension $2k+3$. Furthermore, building on our new insights we show a new lower bound for pushdown VASS: $\mathcal{F}_k$-hardness in dimension $\frac k 2 + 4$. This dimension-parametric lower bound is strictly stronger than the upper bound for VASS, which suggests that the (still unknown) complexity of the reachability problem in pushdown VASS is higher than in plain VASS (where it is Ackermann-complete).

翻译：我们研究了带状态的向量加法系统（VASS）及其带下推栈的扩展（下推VASS）中可达性问题的维度参数复杂度。目前已知，对于维度为$3k+2$的VASS，该问题是$\mathcal{F}_k$困难的（复杂度类$\mathcal{F}_k$对应快速增长层次结构的第$k$层），而对于下推VASS尚无本质上更优的界。我们提出了一种新构造，改进了VASS的下界：在维度$2k+3$上达到$\mathcal{F}_k$困难性。此外，基于新见解，我们给出了下推VASS的新下界：在维度$\frac k 2 + 4$上达到$\mathcal{F}_k$困难性。这一维度参数下界严格强于VASS的上界，表明下推VASS中可达性问题的（仍未知的）复杂度高于普通VASS（后者为阿克曼完全问题）。