Leaderless deterministic chemical reaction networks
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 nonzero 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, errorprone CRN that computes arbitrary computable functions, and which crucially uses a leader.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 arXiv:
 arXiv:1304.4519
 Bibcode:
 2013arXiv1304.4519D
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Data Structures and Algorithms;
 Quantitative Biology  Molecular Networks
 EPrint:
 arXiv admin note: substantial text overlap with arXiv:1204.4176