定点瓦片集及其应用
Fixed-point tile sets and their applications
摘要：
R.伯杰同时证明了多米诺骨牌问题的不可判定性。事实证明，非周期瓦片集出现在许多主题中，从逻辑（Entscheidungsproblem）到物理（quasicrystals）。我们提出了一个新的非周期瓦片集，是基于Kleene的不动点结构，而不是几何参数的建设。这种构造类似于J·冯·诺依曼的自复制自动机; P. Gacs在纠错计算的上下文中也使用了类似的想法。这种构造是相当灵活的，因此它可以在许多方面使用：我们展示了如何使用它来实现替换规则，构造强非周期瓦片集（任何瓦片都远离任何周期瓦片），给出多米诺问题的不可判定性的新证明和相关结果，有效地刻画有限型二维移位的闭一维子移位项（改进了M. Hochman），构造仅具有复杂平铺的图块集合，以及构造不具有周期性（或接近周期性）平铺的“鲁棒的”非周期性图块集合，即使我们允许一些（足够稀疏的）平铺错误。对于后者，我们开发了一个层次分类的点随机集到不同等级的岛屿。最后，我们结合和修改我们的工具来证明我们的主要结果：存在一个瓦片集，使所有的平铺具有高的Kolmogorov复杂度，即使（足够稀疏）平铺错误是允许的。
英文摘要：
An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann self-reproducing automata; similar ideas were also used by P. Gacs in the context of error-correcting computations. This construction it rather flexible, so it can be used in many ways: we show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, characterize effectively closed 1D subshift it terms of 2D shifts of finite type (improvement of a result by M. Hochman), to construct a tile set which has only complex tilings, and to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.
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