Robust Real-time Computing with Chemical Reaction Networks

TL;DR

This paper advances the understanding of real-time computation with chemical reaction networks by establishing the exact class of algebraic numbers they can compute and introducing a robust, perturbation-tolerant model with finite memory.

Contribution

It proves that algebraic numbers exactly characterize Lyapunov CRN-computable reals and develops a robust, finite-memory CRN model resilient to perturbations.

Findings

Algebraic numbers equal the class of Lyapunov CRN-computable reals.

Robust CRNs can approximate values in finite time and tolerate small perturbations.

Finite-memory CRNs are equivalent to Turing machines under real-time simulation.

Abstract

Recent research into analog computing has introduced new notions of computing real numbers. Huang, Klinge, Lathrop, Li, and Lutz defined a notion of computing real numbers in real-time with chemical reaction networks (CRNs), introducing the classes $\mathbb{R}_\text{LCRN}$ (the class of all Lyapunov CRN-computable real numbers) and $\mathbb{R}_\text{RTCRN}$ (the class of all real-time CRN-computable numbers). In their paper, they show the inclusion of the real algebraic numbers $ALG \subseteq \mathbb{R}_\text{LCRN} \subseteq \mathbb{R}_\text{RTCRN}$ and that $ALG \subsetneqq \mathbb{R}_\text{RTCRN}$ but leave open where the inclusion is proper. In this paper, we resolve this open problem and show $ALG= \mathbb{R}_\text{LCRN} \subsetneqq \mathbb{R}_\text{RTCRN}$. However, their definition of real-time computation is fragile in the sense that it is sensitive to perturbations in initial conditions. To resolve this flaw, we further require a CRN to withstand these perturbations. In doing so, we arrive at a discrete model of memory. This approach has several benefits. First, a bounded CRN may compute values approximately in finite time. Second, a CRN can tolerate small perturbations of its species' concentrations. Third, taking a measurement of a CRN's state only requires precision proportional to the exactness of these approximations. Lastly, if a CRN requires only finite memory, this model and Turing machines are equivalent under real-time simulations.