The computational power of discrete chemical reaction networks with bounded executions

TL;DR

This paper explores the computational capabilities of chemical reaction networks with limited reaction steps, revealing how initial conditions and voting rules influence their ability to compute complex functions.

Contribution

It characterizes execution bounded CRNs using a nonnegative linear potential function and analyzes how initial leader presence and voting constraints affect computational power.

Findings

CRNs with an initial leader can compute all semilinear predicates in O(n log n) time.

Without an initial leader, CRNs are limited to computing only eventually constant predicates.

A characterization of execution bounded CRNs via a decreasing potential function is provided.

Abstract

Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as $A + B \to C$, representing that if a molecule of $A$ and $B$ collide, they disappear and a molecule of $C$ is produced. CRNs can compute Boolean-valued predicates $\phi:\mathbb{N}^d \to \{0,1\}$ and integer-valued functions $f:\mathbb{N}^d \to \mathbb{N}$; for instance $X_1 + X_2 \to Y$ computes the function $\min(x_1,x_2)$. We study the computational power of execution bounded CRNs, in which only a finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as $A \rightleftharpoons B$). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in $O(n \log n)$ parallel time by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is "noncollapsing" (does not shrink from initially large to final $O(1)$ size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is to characterize execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction, a result that may be of independent interest.