Robust Real-time Computing with Chemical Reaction Networks

• Published in: arXiv.org
• Main Authors: Fletcher, Willem, Klinge, Titus H, Lathrop, James I, Nye, Dawn A, Rayman, Matthew
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 07.09.2021
• Subjects: Chemical reactions; Computer memory; Initial conditions; Logic; Perturbation; Real numbers; Real time; Turing machines
• ISSN: 2331-8422

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.