Leaderless deterministic chemical reaction networks

TL;DR

This paper proves that leaderless deterministic chemical reaction networks can compute all semilinear functions efficiently, removing the need for initial 'leader' molecules, and provides bounds on their expected computation time.

Contribution

It demonstrates that all semilinear functions are deterministically computable by leaderless CRNs, extending previous results that required leaders, with an analysis of their expected runtime.

Findings

Leaderless CRNs can compute all semilinear functions.

Expected computation time is O(n), where n is total input molecules.

Time is slower than leader-based CRNs but faster than naive constructions.

Abstract

This paper answers an open question of Chen, Doty, and Soloveichik [1], who showed that a function f:N^k --> N^l is deterministically computable by a stochastic chemical reaction network (CRN) if and only if the graph of f is a semilinear subset of N^{k+l}. That construction crucially used "leaders": the ability to start in an initial configuration with constant but non-zero counts of species other than the k species X_1,...,X_k representing the input to the function f. The authors asked whether deterministic CRNs without a leader retain the same power. We answer this question affirmatively, showing that every semilinear function is deterministically computable by a CRN whose initial configuration contains only the input species X_1,...,X_k, and zero counts of every other species. We show that this CRN completes in expected time O(n), where n is the total number of input molecules. This time bound is slower than the O(log^5 n) achieved in [1], but faster than the O(n log n) achieved by the direct construction of [1] (Theorem 4.1 in the latest online version of [1]), since the fast construction of that paper (Theorem 4.4) relied heavily on the use of a fast, error-prone CRN that computes arbitrary computable functions, and which crucially uses a leader.