We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices.

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关于仿射可达性问题

我们分析了维度 1 和 2 中的仿射可达性问题。我们表明，具有仿射更新的整数上的 1 寄存器机器的可达性问题是 PSPACE-hard 的，因此是 PSPACE-complete，加强了 Finkel 等人的结果。这需要多项式更新。基于二维整数矩阵的最新结果，我们证明了行列式为 +1 和 0 的二维整数矩阵的死亡率问题的 NP 完全性。受与没有控制状态的一维仿射可达性问题的紧密联系的启发，我们还研究有限生成的二维上三角整数矩阵半群中许多可达性问题的复杂性。