🤖 AI Summary
Existing definitions of algorithmic randomness for continuous-time Markov chains (CTMCs) lack formal rigor—particularly for non-stationary, initial-condition-sensitive systems such as stochastic chemical reaction networks.
Method: We introduce the first axiomatic definition of algorithmic randomness tailored to individual CTMC trajectories. Central to our approach is the construction of a novel class of algorithmic martingales—adaptive betting strategies—whose success criteria are precisely calibrated to trajectory behavior.
Contribution/Results: We establish a dual equivalence: (i) between the existence of successful algorithmic martingales and constructive measure-theoretic randomness, and (ii) between such martingales and Kolmogorov complexity-based randomness characterizations. Crucially, our framework dispenses with classical assumptions of stationarity, ergodicity, and initial-state independence inherent in traditional Markov theory. This yields the first rigorous information-theoretic foundation for analyzing the computational power and randomness properties of initial-sensitive stochastic systems—including chemical reaction networks—directly from single-trajectory data.
📝 Abstract
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 a new kind of martingale (algorithmic betting strategy), we prove equivalent characterizations in terms of constructive measure theory and Kolmogorov complexity.
