Intrinsic universality and the computational power of self-assembly

Abstract

Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.