Time Complexity of Computation and Construction in the Chemical Reaction Network-Controlled Tile Assembly Model

Abstract

In isolation, chemical reaction networks and tile-based self-assembly are well-studied models of chemical computation. Previously, we introduced the chemical reaction network-controlled tile assembly model (CRN-TAM), in which a stochastic chemical reaction network can act as a non-local control and signalling system for tile-based assembly, and showed that the CRN-TAM can perform several tasks related to the simulation of Turing machines and construction of algorithmic shapes with lower space or program complexity than in either of its parent models. Here, we introduce a kinetic variant of the CRN-TAM and investigate the time complexity of computation and construction. We analyze the time complexity of decision problems in the CRN-TAM, and show that decidable languages can be decided as efficiently by CRN-TAM programs as by Turing machines. We also give a lower bound for the space-time complexity of CRN-TAM computation that rules out efficient parallel stack machines. We provide efficient parallel implementations of non-deterministic computations, showing among other things that CRN-TAM programs can decide languages in \(\mathsf {NTIME}(f(n)) \cap \mathsf {coNTIME}(f(n))\) in \(\mathcal {O}(f(n) + n + \log c)\) time with \(1 - \exp (-c)\) probability, using volume exponential in n. Lastly, we provide basic mechanisms for parallel computations that share information and illustrate the limits of parallel computation in the CRN-TAM.