Computability and Complexity in Self-assembly

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 x Z. Our first main theorem says that there is a roughly quadratic function f such that a set A subset of Z(+) is computably enumerable if and only if the set X-A = {(f (n), 0) vertical bar n is an element of 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 x 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 T-M, together with a proof that T-M carries out concurrent simulations of M on all positive integer inputs.