Computing Threshold Circuits with Void Reactions in Step Chemical Reaction Networks

TL;DR

This work introduces Step CRNs, a staged extension of traditional CRNs, and shows that even a restricted class of void rules becomes computationally powerful when augmented with steps. The authors construct threshold circuits using only true void $(3,0)$ rules with precise circuit‑level indexing, achieving TC computation in $O(D\log F_{out})$ steps and favorable space bounds; they also demonstrate that using $(2,0)$ and $(2,1)$ rules (with or without catalysts) preserves the ability to compute TC with competitive resource costs, including $O(G)$ species and $O(W)$ volume. A lower bound for constant‑size rules ($\,\Omega(\log k)$ steps for $k$‑CNOT) and a coNP‑hardness result for strict function verification establish fundamental limits of the model. The results together argue that steps are essential for the computational power of void rules and identify promising directions for robustness and broader staged CRN frameworks with potential practical molecular implementations.

Abstract

We introduce a new model of \emph{step} Chemical Reaction Networks (step CRNs), motivated by the step-wise addition of materials in standard lab procedures. Step CRNs have ordered reactants that transform into products via reaction rules over a series of steps. We study an important subset of weak reaction rules, \emph{void} rules, in which chemical species may only be deleted but never changed. We demonstrate the capabilities of these simple limited systems to simulate threshold circuits and compute functions using various configurations of rule sizes and step constructions, and prove that without steps, void rules are incapable of these computations, which further motivates the step model. Additionally, we prove the coNP-completeness of verifying if a given step CRN computes a function, holding even for $O(1)$ step systems.

Figures (3)

• Figure 1: An example step CRN system. The test tubes show the species added at each step and the system with those elements added. The CRN species and void rule-set are shown on the left.
• Figure 2: Example AND gate and steps to compute using $(3, 0)$ rules. Note the gate indexing of the wires ($i:1$ and $i:2$) and the input indexing for the next level ($i:1$ since there is only one gate). The process of computing the gate is shown on the right in steps. The new species added at each step are above and the remaining ones are in the system. The lines show the rules that would be executed during each step. To see the added species and rules in detail, see Table \ref{['tab:(3,0)_AND_OR']}.
• Figure 3: (a) Example indexing pattern of wires for the step CRN method using $O(W^2)$ species. (b) Example indexing pattern of wires for the step CRN method using $O(G)$ species. (c) Example circuit (with indexing) for Table \ref{['tab:(3,0)_example']}. (d) Example circuit (with indexing) for Table \ref{['tab:(2,0)_example']}.

Theorems & Definitions (24)

• Definition 2.1: Discrete Chemical Reaction Network
• Definition 2.2: Void and Autogenesis rules
• Definition 2.3
• Definition 2.4: size-$(i,j)$ rules
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof