We consider the following Markov Reachability decision problems that view Markov Chains as Linear Dynamical Systems: given a finite, rational Markov Chain, source and target states, and a rational threshold, does the probability of reaching the target from the source at the step: (i) equal the threshold for some ? (ii) cross the threshold for some ? (iii) cross the threshold for infinitely many ? These problems are respectively known to be equivalent to the Skolem, Positivity, and Ultimate Positivity problems for Linear Recurrence Sequences (LRS), number-theoretic problems whose decidability has been open for decades. We present an elementary reduction from LRS Problems to Markov Reachability Problems that improves the state of the art as follows. (a) We map LRS to (irreducible and aperiodic) Markov Chains that are ubiquitous, not least by virtue of their spectral structure, and (b) our reduction maps LRS of order to Markov Chains of order : a substantial improvement over the previous reduction that mapped LRS of order to reducible and periodic Markov chains of order . This contribution is significant in view of the fact that the number-theoretic hardness of verifying Linear Dynamical Systems can often be mitigated by spectral assumptions and restrictions on order.

中文翻译：

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Skolem 和遍历马尔可夫链的正完备性

我们考虑以下马尔可夫可达性决策问题，将马尔可夫链视为线性动力系统：给定有限的、有理的马尔可夫链、源状态和目标状态以及有理阈值，在步骤中从源到达目标的概率是：（ i) 等于某些的阈值？(ii) 跨越某些门槛？(iii) 跨越无限多个的门槛？这些问题分别相当于线性递归序列 (LRS) 的 Skolem、正性和终极正性问题，这些数论问题的可判定性已经开放了数十年。我们提出了从 LRS 问题到马尔可夫可达性问题的基本简化，它改进了现有技术，如下所示。(a) 我们将 LRS 映射到无处不在的（不可约且非周期的）马尔可夫链，尤其是由于它们的谱结构，以及 (b) 我们的约简将有序的 LRS 映射到有序的马尔可夫链：比之前的约简有了实质性的改进将 LRS 的阶数映射到可简化且周期性的马尔可夫阶数链。鉴于验证线性动力系统的数论难度通常可以通过谱假设和阶数限制来减轻，这一贡献非常重要。