Jack H. Lutz

Xiang Huang, Jack H. Lutz, Neil Lutz et al.

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, \emph{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.

Jack H. Lutz, Neil Lutz, Robyn R. Lutz et al.

Matter, especially DNA, is now programmed to carry out useful processes at the nanoscale. As these programs and processes become more complex and their envisioned safety-critical applications approach deployment, it is essential to develop methods for engineering trustworthiness into molecular programs. Some of this can be achieved by adapting existing software engineering methods, but molecular programming also presents new challenges that will require new methods. This paper presents a method for dealing with one such challenge, namely, the difficulty of ascertaining how robust a molecular program is to perturbations of the relative "clock speeds" of its various reactions. The method proposed here is game-theoretic. The robustness of a molecular program is quantified in terms of its ability to win (achieve its original objective) in games against other molecular programs that manipulate its relative clock speeds. This game-theoretic approach is general enough to quantify the security of a molecular program against malicious manipulations of its relative clock speeds. However, this preliminary report focuses on games against nature, games in which the molecular program's opponent perturbs clock speeds randomly (indifferently) according to the probabilities inherent in chemical kinetics.