On the directed tile assembly systems at temperature 1

TL;DR

This paper proves that directed self-assembly at temperature 1 cannot perform complex computations like a Turing machine, by developing a 2D pumping lemma and classifying possible computational behaviors.

Contribution

It harmonizes previous work to definitively solve the directed temperature 1 conjecture and provides an optimal description of bi-periodic structures in tile assembly systems.

Findings

Temperature 1 systems cannot simulate Turing machines.

A 2D pumping lemma is established for classification.

Bi-periodic structures are characterized precisely.

Abstract

We show here that a model called directed self-assembly at temperature 1 is unable to do complex computations like the ones of a Turing machine. Since this model can be seen as a generalization of finite automata to 2D languages, a logical approach is to proceed in two steps. The first one is to develop a 2D pumping lemma and the second one is to use this pumping lemma to classify the different types of possible computation. Previously, Meunier at al have proven a pumping lemma and Doty et al, assuming the existence of a pumping lemma, have classified the different types of terminal assembly. Thus the combination of these two papers solves the directed temperature 1 conjecture ... but in an imperfect way. Indeed, since the work of Doty et al is anterior to the pumping lemma of Meunier et al, the authors assumed a different and stronger pumping lemma. Nevertheless, all the demonstrations made in Doty et al still hold with the pumping lemma of Meunier et al. In this paper, we harmonize the notations between these two articles in order to clearly solve the directed temperature 1 conjecture. We are also able to give an optimal description of the bi-periodic structures which may appear in some tile assembly system.