非决定论在自组装中的力量
The Power of Nondeterminism in Self-Assembly
摘要：
我们研究了Winfree的抽象瓦片组装模型（aTAM）中的非确定性的作用，该模型被认为是从DNA构建的人工分子自组装系统的模型。特别实际重要的是找到最小化资源的瓦片系统，例如不同瓦片类型的数量，每个瓦片类型对应于必须在aTAM的实际分子实现中定制合成的一组DNA链。我们试图确定在何种程度上使用瓷砖系统中的非确定性影响所需的资源，这样的分子形状建设算法。我们首先展示了一个“分子可计算性理论”的结果：存在一个无限的形状S，它是由一个瓦片系统唯一组装的，但不是由任何确定性瓦片系统组装的。然后，我们在有限的“分子复杂性理论”情况下展示了类似的现象：存在一个有限的形状S，它由具有c个瓦片类型的瓦片系统唯一组装，但每个唯一组装S的确定性瓦片系统都具有超过c个瓦片类型。事实上，我们扩展的技术，得到一个更强的（经典的复杂性理论）的结果，显示的问题，找到最小数量的瓷砖类型，唯一地组装一个给定的有限形状是Sigma-P-2-完成。相比之下，Adleman、Cheng、Goel、Huang、Kempe、Moisset de Espanés和Rothemund（自组装中的组合优化问题，STOC 2002）证明了找到唯一组装形状的最小数量的确定性瓦片类型的问题是NP完全的。结论是，非确定性赋予额外的权力，从一个小的瓷砖系统组装的形状，但除非多项式层次崩溃，它是计算更难以利用这种权力，找到最小的瓷砖系统的大小，相比，找到最小的确定性瓷砖系统的大小。
英文摘要：
We investigate the role of nondeterminism in Winfree's abstract Tile Assembly Model (aTAM), which was conceived to model artificial molecular self-assembling systems constructed from DNA. Of particular practical importance is to find tile systems that minimize resources such as the number of distinct tile types, each of which corresponds to a set of DNA strands that must be custom-synthesized in actual molecular implementations of the aTAM. We seek to identify to what extent the use of nondeterminism in tile systems affects the resources required by such molecular shape-building algorithms. We first show a "molecular computability theoretic" result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We then show an analogous phenomenon in the finitary "molecular complexity theoretic" case: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is Sigma-P-2-complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape was shown to be NP-complete by Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). The conclusion is that nondeterminism confers extra power to assemble a shape from a small tile system, but unless the polynomial hierarchy collapses, it is computationally more difficult to exploit this power by finding the size of the smallest tile system, compared to finding the size of the smallest deterministic tile system.
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