In this paper we develop the elements of the theory of algorithmic randomness in continuous-time Markov chains (CTMCs). Our main contribution is a rigorous, useful notion of what it means for an individual trajectory of a CTMC to be random. CTMCs have discrete state spaces and operate in continuous time. This, together with the fact that trajectories may or may not halt, presents challenges not encountered in more conventional developments of algorithmic randomness. Although we formulate algorithmic randomness in the general context of CTMCs, we are primarily interested in the computational} power of stochastic chemical reaction networks, which are special cases of CTMCs. This leads us to embrace situations in which the long-term behavior of a network depends essentially on its initial state and hence to eschew assumptions that are frequently made in Markov chain theory to avoid such dependencies. After defining the randomness of trajectories in terms of martingales (algorithmic betting strategies), we prove equivalent characterizations in terms of algorithmic measure theory and Kolmogorov complexity. As a preliminary application we prove that, in any stochastic chemical reaction network, every random trajectory with bounded molecular counts has the non-Zeno property that infinitely many reactions do not occur in any finite interval of time.

翻译：本文发展了连续时间马尔可夫链（CTMC）中算法随机性理论的基本要素。我们的主要贡献是提出了一个严格且实用的概念，用于界定连续时间马尔可夫链的单个轨迹何时可被视为随机的。连续时间马尔可夫链具有离散状态空间并在连续时间中运行。这一特性，加之轨迹可能停止也可能不停止，带来了在算法随机性更常规的发展中未曾遇到的挑战。尽管我们在连续时间马尔可夫链的一般框架下构建算法随机性，但我们主要关注随机化学反应网络的计算能力，而该网络是连续时间马尔可夫链的特例。这促使我们接受网络长期行为本质上依赖于其初始状态的情形，从而避免马尔可夫链理论中常用来规避此类依赖性的假设。在利用鞅（算法投注策略）定义轨迹的随机性之后，我们证明了其在算法测度论和柯尔莫哥洛夫复杂度方面的等价刻画。作为初步应用，我们证明在任何随机化学反应网络中，每个具有有限分子数量的随机轨迹都具有非芝诺性质，即在任何有限时间区间内不会发生无限多次反应。