多米诺问题对鲁棒瓷砖集是可判定的
The domino problem is decidable for robust tilesets
AI总结：
本文证明鲁棒Wang瓷砖集的多米诺问题可判定，确立多个著名瓷砖集的鲁棒性，并提供证明瓷砖集可铺满平面的可靠方法。
摘要：
铺砖理论中最基本的问题之一是多米诺问题：给定一组瓷砖和铺砖规则，判定是否存在一种方式，使用瓷砖的副本并遵循其规则来铺满平面。该问题在一般情况下已知是不可判定的，甚至对于Wang瓷砖集也是如此。Wang瓷砖是单位正方形瓷砖，其边缘带有颜色，当它们在公共边缘上共享相同颜色时可以组装，这一点由Berger在20世纪60年代证明。本文聚焦于Wang瓷砖集。我们证明，对于鲁棒瓷砖集，即要么无法铺满平面，要么如果可以铺满，则满足某种可证明的特定不变量的瓷砖集，多米诺问题可判定。我们确立了文献中考虑的多个著名瓷砖集是鲁棒的。我们给出论据表明，除非瓷砖集由非鲁棒图灵机产生，否则所有瓷砖集都是鲁棒的：如果一台图灵机不停机且这种不停机是不可证明的，则称该图灵机为非鲁棒的。作为我们工作的一个附带效果，我们提供了一种可靠且相对完整的方法来证明瓷砖集可以铺满平面。我们的分析还为文献中各种瓷砖集证明之间观察到的相似性，以及使用计算机方法系统研究瓷砖集时实验观察到的现象提供了解释。
英文摘要：
One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be undecidable in general and even for sets of Wang tiles, which are unit square tiles wearing colours on their edges which can be assembled provided they share the same colour on their common edge, as proven by Berger in the 1960s. In this paper, we focus on Wang tilesets. We prove that the domino problem is decidable for robust tilesets, i.e. tilesets that either cannot tile the plane or can but, if so, satisfy some particular invariant provably. We establish that several famous tilesets considered in the literature are robust. We give arguments that this is true for all tilesets unless they are produced from non-robust Turing machines: a Turing machine is said to be non-robust if it does not halt and furthermore does so non-provably. As a side effect of our work, we provide a sound and relatively complete method for proving that a tileset can tile the plane. Our analysis also provides explanations for the observed similarities between proofs in the literature for various tilesets, as well as of phenomena that have been observed experimentally in the systematic study of tilesets using computer methods.
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