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

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.