  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.

Emerging Technologies