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The undecidability of the domino problem. (English) Zbl 07287528
Akiyama, Shigeki (ed.) et al., Substitution and tiling dynamics: introduction to self-inducing structures. Lecture notes from the research school on tiling dynamical systems, CIRM Jean-Morlet Chair, Marseille, France, Fall 2017. Cham: Springer (ISBN 978-3-030-57665-3/pbk; 978-3-030-57666-0/ebook). Lecture Notes in Mathematics 2273, 293-357 (2020).
Summary: One of the most fundamental problems in tiling theory is to decide, given a surface, a set of tiles and a tiling rule, whether there exists a way to tile the surface using the set of tiles and following the rules. As proven by R. Berger [The undecidability of the domino problem. Boston: Harvard University (PhD. thesis) (1964); see also Zbl 0199.30802] in the 1960s, this problem is undecidable in general. When formulated in terms of tilings of the discrete plane $$\mathbb{Z}^2$$ by unit tiles with colored constraints, this is called the Domino Problem and was introduced by H. Wang [“Proving theorems by pattern recognition II”, Bell Syst. Tech. J. 40, No. 1, 1–41 (1961; doi:10.1002/j.1538-7305.1961.tb03975.x)] in an effort to solve satisfaction problems for $$\forall \exists \forall$$ formulas by translating the problem into a geometric problem. There exist a few different proofs of this result. The most well-known proof is probably the proof by R. M. Robinson [Invent. Math. 12, 177–209 (1971; Zbl 0197.46801)] which is a variation on the proof of Berger. A relatively new proof by J. Kari [Lect. Notes Comput. Sci. 4664, 72–79 (2007; Zbl 1211.03062)] has some nice ramifications for tilings of surfaces and groups. In terms of ingredients, one can divide the proofs in 4 categories. The remaining two categories are given by the proof of S. O. Aanderaa and H. R. Lewis [J. Symb. Log. 39, 519–548 (1974; Zbl 0301.02042)] and the fixed point method of B. Durand et al. [J. Comput. Syst. Sci. 78, No. 3, 731–764 (2012; Zbl 1244.05049)]. In this course, we will give a brief description of the problem and to the meaning of the word “undecidable”, and then give the four different proofs. As we will explain, the undecidability of the Domino Problem has as a consequence the existence of an aperiodic tileset. All four sections will be organized in such a way that the interested reader can first extract from the proof the aperiodic tileset into consideration, before we go into more details to actually prove the undecidability of the problem.
For the entire collection see [Zbl 1454.37001].
##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 37B52 Tiling dynamics 52C23 Quasicrystals and aperiodic tilings in discrete geometry
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