Deterministic 2-Dimensional Temperature-1 Tile Assembly Systems Cannot Compute

TL;DR

This paper proves that deterministic 2D temperature-1 tile assembly systems are incapable of universal computation by characterizing their maximal assemblies as para-periodic and finite, thus limiting their computational power.

Contribution

It introduces the concept of para-periodic assemblies, characterizes maximal assemblies in confluent 1-TAS, and proves their computational limitations.

Findings

Maximal assemblies in confluent 1-TAS are para-periodic.

Such systems have at most one maximal assembly, which is either a grid or a union of combs.

Finite descriptions (quipu) can be algorithmically generated for these assemblies.

Abstract

We consider non cooperative binding in so called `temperature 1', in deterministic (here called {\it confluent}) tile self-assembly systems (1-TAS) and prove the standing conjecture that such systems do not have universal computational power. We call a TAS whose maximal assemblies contain at least one ultimately periodic assembly path {\it para-periodic}. We observe that a confluent 1-TAS has at most one maximal producible assembly, $\alpha_{max}$, that can be considered a union of path assemblies, and we show that such a system is always para-periodic. This result is obtained through a superposition and a combination of two paths that produce a new path with desired properties, a technique that we call \emph{co-grow} of two paths. Moreover we provide a characterization of an $\alpha_{max}$ of a confluent 1-TAS as one of two possible cases, so called, a grid or a disjoint union of combs. To a given $\alpha_{max}$ we can associate a finite labeled graph, called \emph{quipu}, such that the union of all labels of paths in the quipu equals $\alpha_{max}$, therefore giving a finite description for $\alpha_{max}$. This finite description implies that $\alpha_{max}$ is a union of semi-affine subsets of $\mathbb{Z}^2$ and since such a finite description can be algorithmicly generated from any 1-TAS, 1-TAS cannot have universal computational power.

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