Computing Threshold Circuits with Bimolecular Void Reactions in Step Chemical Reaction Networks

TL;DR

This paper demonstrates how step chemical reaction networks with only molecular deletion rules can simulate threshold formulas and circuits efficiently, establishing optimal bounds on resource requirements for such simulations.

Contribution

It introduces a novel method for simulating threshold circuits using step CRNs with void rules, proving optimal resource bounds for this approach.

Findings

Step CRNs with void rules of size (2,0) can simulate threshold formulas.

These systems can simulate threshold circuits with exponential volume.

A matching exponential lower bound on volume is established for circuit simulation.

Abstract

Step Chemical Reaction Networks (step CRNs) are an augmentation of the Chemical Reaction Network (CRN) model where additional species may be introduced to the system in a sequence of ``steps.'' We study step CRN systems using a weak subset of reaction rules, \emph{void} rules, in which molecular species can only be deleted. We demonstrate that step CRNs with only void rules of size (2,0) can simulate threshold formulas (TFs) under linear resources. These limited systems can also simulate threshold \emph{circuits} (TCs) by modifying the volume of the system to be exponential. We then prove a matching exponential lower bound on the required volume for simulating threshold circuits in a step CRN with (2,0)-size rules under a restricted \emph{gate-wise} simulation, thus showing our construction is optimal for simulating circuits in this way.