Intrinsic universality in tile self-assembly requires cooperation

TL;DR

This paper proves that noncooperative tile self-assembly models are not intrinsically universal, highlighting the importance of cooperative binding for complex, universal behavior in nanoscale self-assembly systems.

Contribution

It establishes a fundamental limitation of noncooperative tile assembly models and contrasts their computational power with cooperative models, showing the necessity of cooperation for universality.

Findings

Noncooperative models are not intrinsically universal.

Cooperative binding enables universal self-assembly.

Three-dimensional noncooperative systems can simulate two-dimensional ones.

Abstract

We prove a negative result on the power of a model of algorithmic self-assembly for which it has been notoriously difficult to find general techniques and results. Specifically, we prove that Winfree's abstract Tile Assembly Model, when restricted to use noncooperative tile binding, is not intrinsically universal. This stands in stark contrast to the recent result that, via cooperative binding, the abstract Tile Assembly Model is indeed intrinsically universal. Noncooperative self-assembly, also known as "temperature 1", is where tiles bind to each other if they match on one or more sides, whereas cooperative binding requires binding on multiple sides. Our result shows that the change from single- to multi-sided binding qualitatively improves the kinds of dynamics and behavior that these models of nanoscale self-assembly are capable of. Our lower bound on simulation power holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. On the positive side, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. Our negative result can be interpreted to mean that Turing universal algorithmic behavior in self-assembly does not imply the ability to simulate arbitrary algorithmic self-assembly processes.