We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix $M \in \mathbb{Q}^{d \times d}$, an initial vector $x\in\mathbb{Q}^{d}$, a granularity $g\in \mathbb{Q}_+$ and a rounding operation $[\cdot]$ projecting a vector of $\mathbb{Q}^{d}$ onto another vector whose every entry is a multiple of $g$, we are interested in the behaviour of the orbit $\mathcal{O}={<}[x], [M[x]],[M[M[x]]],\dots{>}$, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability---whether a given target $y \in\mathbb{Q}^{d}$ belongs to $\mathcal{O}$---is PSPACE-complete for hyperbolic systems (when no eigenvalue of $M$ has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

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具有舍入的动力系统中的可达性

我们考虑具有离散线性更新但具有固定数字精度的动态系统中的可达性，即系统的值在每一步都被四舍五入。给定一个矩阵 $M \in \mathbb{Q}^{d \times d}$，一个初始向量 $x\in\mathbb{Q}^{d}$，一个粒度 $g\in \mathbb{Q} _+$ 和舍入运算 $[\cdot]$ 将 $\mathbb{Q}^{d}$ 的向量投影到另一个向量，其每个条目都是 $g$ 的倍数，我们对轨道 $\mathcal{O}={<}[x], [M[x]],[M[M[x]]],\dots{>}$，即线性动力系统的轨迹，其中状态在每一步后四舍五入。对于具有有界效果的任意舍入函数，我们证明了决定点对点可达性的复杂性---给定目标 $y \in\mathbb{Q}^{d}$ 是否属于 $\mathcal{O}$--- 是 PSPACE-complete对于双曲线系统（当 $M$ 的特征值没有模数为 1 时）。我们还建立了对几个自然类舍入函数的特征值没有任何限制的可判定性。