The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation

Pierre-Etienne Meunier, D. Woods
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引用次数: 29

Abstract

The field of algorithmic self-assembly is concerned with the computational and expressive power of nanoscale self-assembling molecular systems. In the well-studied cooperative, or temperature 2, abstract tile assembly model it is known that there is a tile set to simulate any Turing machine and an intrinsically universal tile set that simulates the shapes and dynamics of any instance of the model, up to spatial rescaling. It has been an open question as to whether the seemingly simpler noncooperative, or temperature 1, model is capable of such behaviour. Here we show that this is not the case by showing that there is no tile set in the noncooperative model that is intrinsically universal, nor one capable of time-bounded Turing machine simulation within a bounded region of the plane. Although the noncooperative model intuitively seems to lack the complexity and power of the cooperative model it has been exceedingly hard to prove this. One reason is that there have been few tools to analyse the structure of complicated paths in the plane. This paper provides a number of such tools. A second reason is that almost every obvious and small generalisation to the model (e.g. allowing error, 3D, non-square tiles, signals/wires on tiles, tiles that repel each other, parallel synchronous growth) endows it with great computational, and sometimes simulation, power. Our main results show that all of these generalisations provably increase computational and/or simulation power. Our results hold for both deterministic and nondeterministic noncooperative systems. Our first main result stands in stark contrast with the fact that for both the cooperative tile assembly model, and for 3D noncooperative tile assembly, there are respective intrinsically universal tilesets. Our second main result gives a new technique (reduction to simulation) for proving negative results about computation in tile assembly.
非合作拼装模型本身不具有通用性,也不能进行有界图灵机仿真
算法自组装领域关注的是纳米级自组装分子系统的计算和表达能力。在经过充分研究的协作(或温度2)抽象瓦片组装模型中,已知有一个瓦片集可以模拟任何图灵机,还有一个本质上通用的瓦片集可以模拟模型的任何实例的形状和动态,直至空间重新缩放。表面上看起来更简单的非合作模式(即温度1)是否具有这种行为,一直是一个悬而未决的问题。在这里,我们通过展示在非合作模型中没有本质上通用的贴图集,也没有能够在平面的有界区域内进行有时间限制的图灵机模拟来证明情况并非如此。尽管从直觉上看,非合作模型似乎缺乏合作模型的复杂性和威力,但证明这一点却极其困难。一个原因是,几乎没有工具来分析平面上复杂路径的结构。本文提供了一些这样的工具。第二个原因是,几乎每一个明显的和小的一般化模型(例如,允许误差,3D,非正方形瓷砖,瓷砖上的信号/电线,相互排斥的瓷砖,并行同步增长)都赋予它强大的计算能力,有时是模拟能力。我们的主要结果表明,所有这些推广都可以证明提高计算和/或模拟能力。我们的结果适用于确定性和非确定性非合作系统。我们的第一个主要结果与以下事实形成鲜明对比:对于合作瓷砖组装模型和3D非合作瓷砖组装,都存在各自的内在通用瓷砖集。我们的第二个主要结果给出了一种新的技术(还原到模拟)来证明关于瓷砖组装计算的负面结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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