We study reachability problems for various nondeterministic polynomial maps in ${\mathbb{Z}}^n$. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is $\mathsf{\text{PSPACE}}$-hard for both two-dimensional affine maps and one-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form $\pm x+a_0$ is lower. In this case the reachability problem is $\mathsf{\text{PSPACE}}$ for any dimension and if the dimension is not fixed, then the problem is $\mathsf{\text{PSPACE}}$-complete. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps.

我们研究了各种非确定性多项式映射的可达性问题 ${\mathbb{Z}}^n$. 我们证明了非常简单的三维仿射图（具有自变量）的可达性问题是不可判定的，并且是$\mathsf{\text{太空}}$-对于二维仿射图和一维二次图都很难。然后我们证明了没有形式函数的地图的可达性问题的复杂性$\pm X+{\mathrm{一种}}_0$较低。在这种情况下，可达性问题是$\mathsf{\text{太空}}$ 对于任何维度，如果维度不固定，那么问题是 $\mathsf{\text{太空}}$-完全的。最后，我们通过将地图视为语言接受器来扩展模型，并证明二维仿射地图的普遍性问题是不可判定的。