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Algorithmic Randomness in Continuous-Time Markov Chains

Xiang Huang, Jack H. Lutz, Neil Lutz, Andrei N. Migunov

TL;DR

The paper develops a rigorous single-trajectory theory of algorithmic randomness for continuous-time Markov chains (CTMCs), with a central focus on stochastic chemical reaction networks. It builds a Schnorr-style martingale framework for both state sequences and sojourn times, formulates CTMC-specific measure theory, and proves analogs of Ville's, Schnorr's, and van Lambalgen's theorems for CTMC trajectories. A Kolmogorov-complexity viewpoint is provided, linking trajectory randomness to prefix-free descriptions and an information-content measure. As a key application, the authors show a non-Zeno property for random CTMC trajectories with bounded molecular counts in stochastic CRNs, ensuring that infinitely many reactions fail to occur in any finite time interval. Together, these results establish a robust, general framework for randomness of CTMC trajectories with implications for the computability and behavior of chemical reaction networks.

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.

Algorithmic Randomness in Continuous-Time Markov Chains

TL;DR

The paper develops a rigorous single-trajectory theory of algorithmic randomness for continuous-time Markov chains (CTMCs), with a central focus on stochastic chemical reaction networks. It builds a Schnorr-style martingale framework for both state sequences and sojourn times, formulates CTMC-specific measure theory, and proves analogs of Ville's, Schnorr's, and van Lambalgen's theorems for CTMC trajectories. A Kolmogorov-complexity viewpoint is provided, linking trajectory randomness to prefix-free descriptions and an information-content measure. As a key application, the authors show a non-Zeno property for random CTMC trajectories with bounded molecular counts in stochastic CRNs, ensuring that infinitely many reactions fail to occur in any finite time interval. Together, these results establish a robust, general framework for randomness of CTMC trajectories with implications for the computability and behavior of chemical reaction networks.

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.

Paper Structure

This paper contains 18 sections, 14 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Boolean transition systems
  3. Random state sequences
  4. Random sequences of sojourn times
  5. Measure and martingales for continuous-time Markov chains
  6. Continuous-time Markov chains
  7. Trajectories in CTMCs
  8. Trajectories in CTMCs
  9. Measure in CTMCs
  10. Martingales for CTMCs
  11. Analog of Ville's theorem for CTMC trajectories
  12. Random CTMC trajectories
  13. Algorithmic measure and randomness in CTMCs
  14. Analog of Schnorr's theorem for CTMCs
  15. Analog of van Lambalgen's theorem for CTMC trajectories
  16. ...and 3 more sections

Key Result

Theorem 2

Let $\mathscr{Q}$ be a probabilistic transition system, let $\sigma$ be an initialization of $\mathscr{Q}$, and let $\mu = \mu_{\mathscr{Q}, \sigma}$. For every set $E \subseteq \mathds{A}_{\mathscr{Q}}(\sigma)$, the following two conditions are equivalent.

Theorems & Definitions (24)

  • Theorem 2: Ville Ville39
  • Theorem 3: Schnorr DBLP:journals/mst/Schnorr71
  • Example 4
  • Theorem 6
  • Theorem 7
  • proof
  • Lemma 8: Kraft inequality for $C$-martingales
  • proof
  • Theorem 9
  • proof
  • ...and 14 more