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

Rachel Anderson, Bin Fu, Aiden Massie, Gourab Mukhopadhyay, Adrian Salinas, Robert Schweller, Evan Tomai, Tim Wylie

TL;DR

This work analyzes Step Chemical Reaction Networks restricted to bimolecular void rules of size $(2,0)$ and shows they can compute Threshold Formulas under linear resources and Threshold Circuits with exponential volume via gate-wise simulation. It introduces a detailed TF/TC construction that uses a depth-wise, bit-encoded representation and $(2,0)$ rule interactions to realize Boolean computations, achieving $O(G)$ species, $O(D)$ steps, and $O(G)$ volume for TFs and $O(G F_{out}^D)$ volume for TCs. A matching exponential lower bound, $2^{\, ext{Omega}(D)}$, is proven for gate-wise simulation under these rules, demonstrating the necessity of exponential volume in this restricted setting and highlighting a power boundary relative to $(3,0)$ void rules. Overall, the paper delineates the capabilities and limits of $(2,0)$ step CRNs for logic computation, offering insights into CRN-based computation and the cost of simulating circuits within constrained reaction models.

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.

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

TL;DR

This work analyzes Step Chemical Reaction Networks restricted to bimolecular void rules of size and shows they can compute Threshold Formulas under linear resources and Threshold Circuits with exponential volume via gate-wise simulation. It introduces a detailed TF/TC construction that uses a depth-wise, bit-encoded representation and rule interactions to realize Boolean computations, achieving species, steps, and volume for TFs and volume for TCs. A matching exponential lower bound, , is proven for gate-wise simulation under these rules, demonstrating the necessity of exponential volume in this restricted setting and highlighting a power boundary relative to void rules. Overall, the paper delineates the capabilities and limits of step CRNs for logic computation, offering insights into CRN-based computation and the cost of simulating circuits within constrained reaction models.

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.
Paper Structure (16 sections, 5 theorems, 6 equations, 4 figures, 4 tables)

This paper contains 16 sections, 5 theorems, 6 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chemical Reaction Networks
  4. Void Rules
  5. Step Chemical Reaction Networks
  6. Computing Functions in Step CRNs
  7. Boolean and Threshold Circuits
  8. Computation of Threshold Formulas with (2, 0) Rules
  9. Bit Representation
  10. Logic Gate Computation
  11. Formula Computation Example
  12. Threshold Formula Computation
  13. Computation of TCs with Exponential Volume
  14. Threshold Circuit Computation
  15. Exponential Volume Lower Bound for Gate-Wise Simulation
  16. ...and 1 more sections

Key Result

Theorem 1

Threshold formulas (TF) can be computed with multiple-input relaxation by a step CRN with only $(2,0)$ rules with upper bounds of $O(G)$ species, $O(D)$ steps, and $O(G)$ volume.

Figures (4)

  • 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: (a) The input bits of a threshold formula and their representation as species. (b) An indexed threshold formula with the input species shown in Figure \ref{['fig:bitrep']}.
  • Figure 3: (a) A threshold formula consisting of a single three-input AND gate. (b) Reaction rules and added species for the step CRN that compute the formula in \ref{['fig:andgate']}. (c) The step CRN computing the formula in Figure \ref{['fig:andgate']}. The black lines connecting species represents a reaction applied to them. (d) The step CRN computing the circuit in Figure \ref{['fig:andgate']}, but with three true inputs.
  • Figure 4: (a) A NOT gate with a fan-out of three. (b) Computing the NOT gate in Figure \ref{['fig:unboundedfanout']} in (2, 0) rules.

Theorems & Definitions (15)

  • Definition 1: Discrete Chemical Reaction Network
  • Definition 2: Void and Autogenesis rules
  • Definition 3
  • Definition 4: size-$(i,j)$ rules
  • Theorem 1
  • proof
  • Theorem 2
  • Definition 5
  • Lemma 1
  • proof
  • ...and 5 more