Real-time computing and robust memory with deterministic chemical reaction networks

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 \(\text { ALG} \subseteq \mathbb {R}_\text {LCRN}\subseteq \mathbb {R}_\text {RTCRN}\) and that \(\text { ALG} \subsetneqq \mathbb {R}_\text {RTCRN}\) but leave open whether the inclusion is proper. In this paper, we resolve this open problem and show that \({ ALG} = \mathbb {R}_\text {LCRN}\) and, as a consequence, \(\mathbb {R}_\text {LCRN}\subsetneqq \mathbb {R}_\text {RTCRN}\). However, the definition of real-time computation by Huang et al. 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.