Real-Time Computability of Real Numbers by Chemical Reaction Networks

TL;DR

This paper investigates the capability of deterministic chemical reaction networks to compute real numbers in real time, demonstrating that algebraic and some transcendental numbers are computable in this model, with implications for classical computational conjectures.

Contribution

It introduces a model of real-time computation by chemical reaction networks and proves that algebraic and some transcendental numbers are computable within this framework.

Findings

Algebraic numbers are real-time computable by chemical reaction networks.

Some transcendental numbers are also real-time computable.

Implications for the Hartmanis-Stearns conjecture are discussed.

Abstract

We explore the class of real numbers that are computed in real time by deterministic chemical reaction networks that are integral in the sense that all their reaction rate constants are positive integers. We say that such a reaction network computes a real number $\alpha$ in real time if it has a designated species $X$ such that, when all species concentrations are set to zero at time $t = 0$, the concentration $x(t)$ of $X$ is within $2^{-t}$ of $|{\alpha}|$ at all times $t \ge 1$, and the concentrations of all other species are bounded. We show that every algebraic number and some transcendental numbers are real time computable by chemical reaction networks in this sense. We discuss possible implications of this for the 1965 Hartmanis-Stearns conjecture, which says that no irrational algebraic number is real time computable by a Turing machine.