Computing with reaction networks at input-independent speed: exponential and logarithmic functions

TL;DR

This paper develops chemical reaction network modules that compute exponential and logarithmic functions at fixed, input-independent speed, enabling efficient and accurate transcendental function computation without relying on power series.

Contribution

It introduces reaction network modules for exponential and logarithmic functions that operate at fixed speed and can be combined with existing arithmetic modules for complex computations.

Findings

Modules achieve arbitrary accuracy with sufficient time.

Reaction networks operate at input-independent speed.

Modules can be combined for complex transcendental computations.

Abstract

The concept of input-independent computational time for chemistry-based analog computers was introduced in Anderson-Joshi (2025), where it was shown that arithmetic operations can be computed in a fixed time independent of the input values. Here, by inputs we mean the numerical values encoded by the initial concentrations of designated input species, with the underlying reaction network and rate constants held fixed. Combining these operations via power series to approximate transcendental functions is possible in principle, but the number of chemical species required grows with the number of terms retained, and achieving sufficient accuracy may demand many terms -- a burden that is especially severe for slowly converging series such as the power series for the logarithm. In this paper, we begin the program of directly computing transcendental functions by chemical reaction networks by focusing on the exponential and logarithmic functions, two widely used transcendental functions, and constructing reaction network modules that compute them without relying on truncated power series. We show that the resulting modules are mass-action systems, and prove that they achieve arbitrary accuracy given sufficient time while operating at input-independent speed. Moreover, these modules can be freely combined with each other and with the arithmetic modules of Anderson-Joshi (2025): any such composite computation also runs at input-independent speed, with the total computation time bounded independently of the input values \emph{and} of the number of elementary steps in the computation. Logarithm and exponential functions serve as foundational cases, and the constructions developed here are intended to serve as templates for the direct computation of more general transcendental functions by chemical reaction networks.