双手瓷砖组装模型本质上并不通用
The two-handed tile assembly model is not intrinsically universal
摘要：
双手瓦片组装模型（Two-Handed Tile Assembly Model，2 HAM）是一种瓦片组装模型，其中成对的大型组件可以绑定或自组装在一起。为了粘合，两个组件必须具有匹配的胶水，这些胶水可以同时彼此接触，并且以至少为温度$τ$的强度粘在一起，其中$τ$是某个固定的正整数。我们问2 HAM是否本质上是通用的，换句话说，我们问：是否有一个通用的2 HAM瓦片集$U$，可以用来模拟模型的任何实例？我们的主要结果是对这个问题的否定回答。我们发现，对于所有的$τ' < τ$，每个温度- $τ'$ 2 HAM瓷砖系统不模拟至少一个温度- $τ$ 2 HAM瓷砖系统。这个不可能的结果证明了2 HAM本质上不是通用的，这与本质上通用的更简单的（仅单个瓦片添加）抽象瓦片组装模型形成鲜明对比（“瓦片组装模型本质上是通用的”，FOCS 2012）。然而，在积极的一面，我们证明，对于每一个固定的温度$τ\geq 2$，温度- $τ$ 2 HAM瓦片系统确实是固有的通用：换句话说，对于每个$τ$有一个单一的通用2 HAM瓦片集$U$，当适当地初始化，是能够模拟任何温度- $τ$ 2 HAM瓦片系统的行为。作为这些结果的推论，我们发现一个无限的2 HAM系统的无限层次的无限集，严格增加每个层次内的模拟能力。最后，我们表明，对于每个$τ$，有一个温度- $τ$ 2 HAM系统，同时模拟所有的温度- $τ$ 2 HAM系统。
英文摘要：
The well-studied Two-Handed Tile Assembly Model (2HAM) is a model of tile assembly in which pairs of large assemblies can bind, or self-assemble, together. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature $τ$, where $τ$ is some fixed positive integer. We ask whether the 2HAM is intrinsically universal, in other words we ask: is there a single universal 2HAM tile set $U$ which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all $τ' < τ$, each temperature-$τ'$ 2HAM tile system does not simulate at least one temperature-$τ$ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler (single-tile addition only) abstract Tile Assembly Model which is intrinsically universal ("The tile assembly model is intrinsically universal", FOCS 2012). However, on the positive side, we prove that, for every fixed temperature $τ\geq 2$, temperature-$τ$ 2HAM tile systems are indeed intrinsically universal: in other words, for each $τ$ there is a single universal 2HAM tile set $U$ that, when appropriately initialized, is capable of simulating the behavior of any temperature-$τ$ 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each $τ$, there is a temperature-$τ$ 2HAM system that simultaneously simulates all temperature-$τ$ 2HAM systems.
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