Algorithmic Randomness in Continuous-Time Markov Chains

TL;DR

This paper develops a rigorous framework for understanding algorithmic randomness in continuous-time Markov chains, with applications to stochastic chemical reaction networks, addressing unique challenges posed by continuous time and potential non-halting trajectories.

Contribution

It introduces a novel notion of trajectory randomness in CTMCs, linking martingales, measure theory, and Kolmogorov complexity, and applies it to stochastic chemical networks.

Findings

Defined randomness of CTMC trajectories using martingales.

Proved equivalence of randomness characterizations via measure theory and Kolmogorov complexity.

Showed that bounded random trajectories in chemical networks are non-Zeno.

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 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.