Composable computation in discrete chemical reaction networks

TL;DR

This paper characterizes which integer-valued functions can be stably computed by output-oblivious chemical reaction networks, showing they must be the minimum of finitely many nondecreasing quilt-affine functions for large inputs.

Contribution

It provides a precise characterization of functions computable by output-oblivious CRNs with an initial leader, linking computability to quilt-affine functions.

Findings

Functions computable are the minimum of finitely many quilt-affine functions.

Output-oblivious CRNs can be composed by renaming output species.

Characterization applies to large input values.

Abstract

We study the composability of discrete chemical reaction networks (CRNs) that stably compute (i.e., with probability 0 of error) integer-valued functions $f:\mathbb{N}^d\to\mathbb{N}$. We consider output-oblivious CRNs in which the output species is never a reactant (input) to any reaction. The class of output-oblivious CRNs is fundamental, appearing in earlier studies of CRN computation, because it is precisely the class of CRNs that can be composed by simply renaming the output of the upstream CRN to match the input of the downstream CRN. Our main theorem precisely characterizes the functions $f$ stably computable by output-oblivious CRNs with an initial leader. The key necessary condition is that for sufficiently large inputs, $f$ is the minimum of a finite number of nondecreasing quilt-affine functions. (An affine function is linear with a constant offset; a quilt-affine function is linear with a periodic offset).