当前位置: X-MOL 学术arXiv.cs.FL › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On the directed tile assembly systems at temperature 1
arXiv - CS - Formal Languages and Automata Theory Pub Date : 2020-11-18 , DOI: arxiv-2011.09675
Pierre-\'Etienne Meunier and Damien Regnault

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.

中文翻译:

在温度为 1 的定向瓷砖组装系统上

我们在这里展示了在温度 1 下称为定向自组装的模型无法像图灵机那样进行复杂的计算。由于这个模型可以被看作是有限自动机对二维语言的推广,所以一个合乎逻辑的方法是分两步进行。第一个是开发一个 2D 泵引理,第二个是使用这个泵引理对不同类型的可能计算进行分类。此前,Meunier 等人已经证明了一个泵引理,Doty 等人假设存在一个泵引理,对不同类型的端子组件进行了分类。因此,这两篇论文的结合解决了定向温度 1 猜想……但以一种不完美的方式。事实上,由于 Doty 等人的工作先于 Meunier 等人的抽水引理,作者假设了一个不同的和更强的泵引理。尽管如此,Doty 等人所做的所有演示仍然适用于 Meunier 等人的抽水引理。在本文中,我们协调了这两篇文章之间的符号,以清楚地解决定向温度 1 猜想。我们还能够给出可能出现在某些瓷砖组装系统中的双周期结构的最佳描述。
更新日期:2020-11-20
down
wechat
bug