On the Runtime of Chemical Reaction Networks Beyond Idealized Conditions

This paper studies the (discrete) \emph{chemical reaction network (CRN)} computational model that emerged in the last two decades as an abstraction for molecular programming. The correctness of CRN protocols is typically established under one of two possible schedulers that determine how the execution advances: (1) a \emph{stochastic scheduler} that obeys the (continuous time) Markov process dictated by the standard model of stochastic chemical kinetics; or (2) an \emph{adversarial scheduler} whose only commitment is to maintain a certain fairness condition. The latter scheduler is justified by the fact that the former one crucially assumes ``idealized conditions'' that more often than not, do not hold in real wet-lab experiments. However, when it comes to analyzing the \emph{runtime} of CRN protocols, the existing literature focuses strictly on the stochastic scheduler, thus raising the research question that drives this work: Is there a meaningful way to quantify the runtime of CRNs without the idealized conditions assumption? The main conceptual contribution of the current paper is to answer this question in the affirmative, formulating a new runtime measure for CRN protocols that does not rely on idealized conditions. This runtime measure is based on an adapted (weaker) fairness condition as well as a novel scheme that enables partitioning the execution into short \emph{rounds} and charging the runtime for each round individually (inspired by definitions for the runtime of asynchronous distributed algorithms). Following that, we turn to investigate various fundamental computational tasks and establish (often tight) bounds on the runtime of the corresponding CRN protocols operating under the adversarial scheduler. This includes an almost complete chart of the runtime complexity landscape of predicate decidability tasks.