We study the following reachability problem for piecewise affine maps: Given two vectors $\mathbf{s}, \mathbf{t} \in \mathbb{Q}^d$ and a piecewise affine map $f \colon \mathbb{Q}^d\rightarrow \mathbb{Q}^d$, does there exist $n\in \mathbb{N}$ such that $f^{n}(\mathbf{s}) = \mathbf{t}$? In this work, we focus on this reachability problem for a subclass of piecewise affine maps -- Bellman operators arising from Markov decision processes. We prove that the reachability problem for $\max$- and $\min$-Bellman operators is decidable in any dimension under either of the following conditions: (i) the target vector $\mathbf{t}$ is not the fixed point of the operator $f$; or (ii) the initial and target vectors $\mathbf{s}$ and $\mathbf{t}$ are comparable with respect to the componentwise order. Furthermore, we show that in the two-dimensional case, the reachability problem for Bellman operators is decidable for arbitrary $\mathbf{s}, \mathbf{t} \in \mathbb{Q}^2$. This stands in sharp contrast to the known undecidability of reachability for general piecewise affine maps in dimension $d = 2$.

翻译：本文研究以下分段仿射映射的可达性问题：给定两个向量 $\mathbf{s}, \mathbf{t} \in \mathbb{Q}^d$ 及一个分段仿射映射 $f \colon \mathbb{Q}^d\rightarrow \mathbb{Q}^d$，是否存在 $n\in \mathbb{N}$ 使得 $f^{n}(\mathbf{s}) = \mathbf{t}$？本工作聚焦于分段仿射映射的一个子类——源自马尔可夫决策过程的贝尔曼算子。我们证明，对于 $\max$- 与 $\min$-贝尔曼算子，在满足以下任一条件时，其可达性问题在任何维度下均可判定：(i) 目标向量 $\mathbf{t}$ 不是算子 $f$ 的不动点；或 (ii) 初始向量 $\mathbf{s}$ 与目标向量 $\mathbf{t}$ 在分量序意义下可比。进一步地，我们证明在二维情形下，对于任意 $\mathbf{s}, \mathbf{t} \in \mathbb{Q}^2$，贝尔曼算子的可达性问题是可判定的。这与已知的二维（$d = 2$）一般分段仿射映射可达性问题不可判定性形成鲜明对比。