Real-time computability of real numbers by chemical reaction networks

Robust Real-time Computing with Chemical Reaction Networks

“…The definition of real-time computable by a CRN used in this paper is given by [12,13]. We repeat the definition here for convenience.…”

“…As a consequence, a bounded CRN may only compute the algebraic numbers reliably in the sense that they exist inside of a potential well. Since Huang et al proved that ALG Ř R RTCRN and ALG Ď R LCRN Ď R RTCRN [13], it suffices to show that ALG " R LCRN to resolve that R LCRN Ř R RTCRN . We prove this result in two parts.…”

“…Thus, zpX i q " |α| is algebraic, and therefore α P ALG. As the converse is known [13], we have ALG " R LCRN . 4…”

“…To compute α P R "in the limit," a designated variable xptq must satisfy lim tÑ8 xptq " α. Computing real numbers in this way has also been investigated in population protocols [3] and chemical reaction networks (CRNs) [12]. Huang et al defined a number α P R to be real-time computable by chemical reaction networks, written α P R RTCRN , if there exists a CRN with integral rate constants and a designated species X such that, if all species concentrations are initialized to zero, then xptq converges to α exponentially quickly [13]. This means that after n seconds, the concentration of X is within 2 ´n of α, so the CRN gains one bit of accuracy every second.…”