本文研究了任意有限形状的放大模型的自组装。我们在Aggarwal，Cheng，Goldwasser，Kao和Schweller（自组装广义模型的复杂性，SODA 2004）介绍的多温度模型中工作。多温度模型是Winfree的抽象瓦片组装模型的自然推广，其中瓦片系统的温度允许随着自组装的进行而上下移动。我们首先展示了两个恒定大小的瓷砖集，其中任意形状的放大版本自组装。我们的第一个瓦片集的属性，每个缩放的形状自组装通过一个渐进的“柯尔莫哥洛夫最佳”温度序列，但缩放因子的大小正在组装的形状增长。相比之下，我们的第二个瓦片集通过温度序列组装每个缩放的形状，该温度序列的长度与形状中的点的数量成比例，但缩放因子是与组装的形状无关的常数。然后，我们表明，在多温度模型中，没有恒定大小的瓦片集可以唯一地组装任意（非缩放的，连接的）形状，即，缩放对于自组装是必要的。这回答了Kao和Schweller（Reducing Tile Complexity for Self-Assembly Through Temperature Programming，SODA 2006）的一个未决问题，他们询问是否存在这样的瓦片组。
英文摘要：
This paper concerns the self-assembly of scaled-up versions of arbitrary finite shapes. We work in the multiple temperature model that was introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller (Complexities for Generalized Models of Self-Assembly, SODA 2004). The multiple temperature model is a natural generalization of Winfree's abstract tile assembly model, where the temperature of a tile system is allowed to be shifted up and down as self-assembly proceeds. We first exhibit two constant-size tile sets in which scaled-up versions of arbitrary shapes self-assemble. Our first tile set has the property that each scaled shape self-assembles via an asymptotically "Kolmogorov-optimum" temperature sequence but the scaling factor grows with the size of the shape being assembled. In contrast, our second tile set assembles each scaled shape via a temperature sequence whose length is proportional to the number of points in the shape but the scaling factor is a constant independent of the shape being assembled. We then show that there is no constant-size tile set that can uniquely assemble an arbitrary (non-scaled, connected) shape in the multiple temperature model, i.e., the scaling is necessary for self-assembly. This answers an open question of Kao and Schweller (Reducing Tile Complexity for Self-Assembly Through Temperature Programming, SODA 2006), who asked whether such a tile set existed.
We prove the unique assembly and unique shape verification problems, benchmark measures of self-assembly model power, are $\mathrm{coNP}^{\mathrm{NP}}$-hard and contained in $\mathrm{PSPACE}$ (and in $\mathrmΠ^\mathrm{P}_{2s}$ for staged systems with $s$ stages). En route, we prove that unique shape verification problem in the 2HAM is $\mathrm{coNP}^{\mathrm{NP}}$-complete.
工作在Winfree的抽象瓷砖组装模型，我们表明，一个恒定大小的瓷砖组装系统可以通过编程相对瓷砖浓度，以建立一个n × n平方与高概率，任何足够大的n。这回答了Kao和Schweller（Randomized Self-Assembly for Approximate Decomposition，ICALP 2008）的一个开放性问题，他们展示了如何使用瓷砖浓度编程构建一个近似n x n的正方形，并询问近似是否可以以高概率精确。我们展示了如何修改这种技术来回答Kao和Schweller的另一个问题，通过展示一个恒定大小的瓷砖组装系统可以通过瓷砖浓度来编程，以组装任意有限 * 缩放形状 *，这些形状是通过将每个点替换为c x c块点来修改的，对于某个整数c。此外，我们表现出一个平稳的权衡之间的指定位n通过瓦片浓度与指定它们通过硬编码的瓦片类型，这允许瓦片浓度编程被用于指定一小部分的位的“输入”到一个瓦片组装系统，在浓度的约束下，只能指定到一个有限的精度。最后，考虑到瓷砖浓度规划模型的一些不切实际的方面，我们展示了如何修改施工，只使用浓度是任意接近均匀。
We prove a result which strongly hints at the computational weakness of a model of tile assembly that has so far resisted many attempts of formal analysis or positive constructions. Specifically, we prove that, in Winfree's abstract Tile Assembly Model, when restricted to use only noncooperative bindings, any long enough path starting from the seed that can grow in all terminal assemblies is pumpable, meaning that this path can be extended into an infinite, ultimately periodic path. This result can be seen as a geometric generalization of the pumping lemma of finite state automata, and closes …
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,…
We consider non-cooperative binding, so-called 'temperature 1', in deterministic or directed (called here confluent) tile self-assembly systems in two dimensions and show a necessary and sufficient condition for such system to have an ultimately periodic assembly path. We prove that an infinite maximal assembly has an ultimately periodic assembly path if and only if it contains an infinite assembly path that does not intersect a periodic path in the Z2 grid. Moreover we show that every infinite assembly must satisfy this condition, and therefore, contains an ultimately periodic path. This resu…
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…
We study the role that parallelism plays in time complexity of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the "hierarchical" aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the "seeded" aTAM, tiles attach one at a time to a growing assembly. Adleman, Cheng, Goel, and Huang ("Running Time and Program Size for Self-Assembled Squares", STOC 2001) showed how to assemble an n x n square in O(n) time in the seeded aTAM using O(log n / log log n) unique tile types, where both of these paramet…
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. …
图案自组装瓦片集综合（Pattern self-assembled tile set synthesis，PATS）是一个以最小化可唯一自组装给定矩形图案的直线瓦片组装系统为目标的组合优化问题，是NP难问题.当PATS由常数c参数化时，它变得实际上有意义，使得任何给定的图案都保证最多包含c种颜色（c-PATS）。我们首先研究简单的模式和性质的最小RTAS他们。在此基础上，设计了一个59色的图案，将3SAT约简为59色的图案，并证明了59-PATS是NP-难的。
We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we call it the Maze-Walking Tile Assembly Model. The model has a number of implementation prospects, one being DNA strands that attach to a DNA origami substrate. Intuitively, the model suggests a separation of signal routing and computation: the input structure (maze) supplies a routing diagram, and the programmer's tile set provides the computational ability. We…
We consider non cooperative binding in so called `temperature 1', in deterministic (here called {\it confluent}) tile self-assembly systems (1-TAS) and prove the standing conjecture that such systems do not have universal computational power. We call a TAS whose maximal assemblies contain at least one ultimately periodic assembly path {\it para-periodic}. We observe that a confluent 1-TAS has at most one maximal producible assembly, $α_{max}$, that can be considered a union of path assemblies, and we show that such a system is always para-periodic. This result is obtained through a superpositi…
We prove a Pumping Lemma for the noncooperative abstract Tile Assembly Model, a model central to the theory of algorithmic self-assembly since the beginning of the field. This theory suggests, and our result proves, that small differences in the nature of adhesive bindings between abstract square molecules gives rise to vastly different expressive capabilities. In the cooperative abstract Tile Assembly Model, square tiles attach to each other using multi-sided cooperation of one, two or more sides. This precise control of tile binding is directly exploited for algorithmic tasks including growt…
We show the first asymptotically efficient constructions in the so-called "noncooperative planar tile assembly" model. Algorithmic self-assembly is the study of the local, distributed, asynchronous algorithms ran by molecules to self-organise, in particular during crystal growth. The general cooperative model, also called "temperature 2", uses synchronisation to simulate Turing machines, build shapes using the smallest possible amount of tile types, and other algorithmic tasks. However, in the non-cooperative ("temperature 1") model, the growth process is entirely asynchronous, and mostly reli…
One of the most fundamental and well-studied problems in Tile Self-Assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity of this problem in the 2-Handed Assembly Model (2HAM) at a constant temperature is a long-standing open problem since the model was introduced. Previously, only membership in the class coNP was known and that the problem is in P if the temperature is one ($τ=1$). The problem is known to be hard for many generalizations of the model, such as allowing one step…
