统计自相似分形的自组装
Self-Assembly of a Statistically Self-Similar Fractal
摘要：
我们证明了存在一个瓷砖组装系统，自组装的统计自相似Sierpinski三角的Winfree-Rothemund瓷砖组装模型。这似乎是第一篇考虑随机分形自组装的论文，而不是确定性分形或有限的有界形状。我们的技术贡献包括一种方法来记住，并使用，无限长的前缀的一个无限的编码序列在每个阶段的分形结构;一个瓦片组装机制嵌套递归;和定义的“几乎无处不在的本地确定性”，以描述一个tiltaxis的组件是本地确定的，条件是zeta维零集（无限多）的“输入”瓦片。这最后一个类似于图灵机的随机计算的定义，在图灵机中，算法相对于提供建议但本身不计算的硬币翻转的预言序列是确定性的。关键词：瓦片自组装，统计自相似谢尔宾斯基三角形。
英文摘要：
We demonstrate existence of a tile assembly system that self-assembles the statistically self-similar Sierpinski Triangle in the Winfree-Rothemund Tile Assembly Model. This appears to be the first paper that considers self-assembly of a random fractal, instead of a deterministic fractal or a finite, bounded shape. Our technical contributions include a way to remember, and use, unboundedly-long prefixes of an infinite coding sequence at each stage of fractal construction; a tile assembly mechanism for nested recursion; and a definition of "almost-everywhere local determinism," to describe a tileset whose assembly is locally determined, conditional upon a zeta-dimension zero set of (infinitely many) "input" tiles. This last is similar to the definition of randomized computation for Turing machines, in which an algorithm is deterministic relative to an oracle sequence of coin flips that provides advice but does not itself compute. Keywords: tile self-assembly, statistically self-similar Sierpinski Triangle.
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