A Limit to the Power of Multiple Nucleation in Self-assembly

Abstract

Majumder, Reif and Sahu presented in [7] a model of reversible, error-permitting tile self-assembly, and showed that restricted classes of tile assembly systems achieved equilibrium in (expected) polynomial time. One open question they asked was how the model would change if it permitted multiple nucleation, i.e.,independent groups of tiles growing before attaching to the original seed assembly. This paper provides a partial answer, by proving that no tile assembly model can use multiple nucleation to achieve speedup from polynomial time to constant time without sacrificing computational power: if a tile assembly system $\mathcal{T}$ uses multiple nucleation to tile a surface in constant time (independent of the size of the surface), then $\mathcal{T}$ is unable to solve computational problems that have low complexity in the (single-seeded) Winfree-Rothemund Tile Assembly Model. The proof technique defines a new model of distributed computing that simulates tile assembly, so a tile assembly model can be described as a distributed computing model.