Intrinsic universality and the computational power of self-assembly

TL;DR

This paper surveys recent advances in tile self-assembly, highlighting the existence of universal tile sets, the limitations of certain models, and the potential for a single tile to simulate all systems, revealing the field's computational depth.

Contribution

It introduces the concept of a universal tile set that captures the geometry and dynamics of any system, and explores the hierarchy of simulation power across different models.

Findings

Existence of a universal tile set that simulates any tile assembly system.

Noncooperative (temperature 1) model is weaker than the full model.

A single polygonal tile can simulate any tile assembly system.

Abstract

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with 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 can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.