Computability and Complexity in Self-assembly

Abstract

This paper explores the impact of geometry on computability and complexity in Winfree’s model of nanoscale self-assembly. We work in the two-dimensional tile assembly model, i.e., in the discrete Euclidean plane ℤ×ℤ. Our first main theorem says that there is a roughly quadratic function f such that a set A⊆ℤ+ is computably enumerable if and only if the set X A ={(f(n),0)∣n∈A}—a simple representation of A as a set of points on the x-axis—self-assembles in Winfree’s sense. In contrast, our second main theorem says that there are decidable sets D⊆ℤ×ℤ that do not self-assemble in Winfree’s sense. Our first main theorem is established by an explicit translation of an arbitrary Turing machine M to a modular tile assembly system $\mathcal{T}_{M}$, together with a proof that $\mathcal{T}_{M}$ carries out concurrent simulations of M on all positive integer inputs.