Fixed Point and Aperiodic Tilings

arXiv:0802.2432v5 [cs.CC] 27 Jan 2010 Fixed Point and Aperiodic Tilings Bruno Durand1, Andrei Romashchenko2,3, Alexander Shen1,3 1 LIF, CNRS & Univ. de Provence, Marseille 2 LIP, ENS de Lyon & CNRS 3 Institute for Information Transmission Problems, Moscow Abstract. An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that ape- riodic tile sets appear in many topics ranging from logic (the Entschei- dungsproblem) 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. G´acs in the context of error-correcting computations. The flexibility of this construction allows us 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. This property was not known for any of the existing aperiodic tile sets. 1 Introduction In this paper4, tiles are unit squares with colored sides. Tiles are considered as prototypes: we may place translated copies of the same tile into different cells of 4 The first version of this preprint was published in arxiv and hal on 18 Feb 2008. Later this paper was published in proceedings of the DLT conference: [DLT08] B.Durand, A.Romashchenko, A.Shen. Fixed Point and Aperiodic Tilings. Proc. 12th international conference on Developments in Language Theory. Kyoto, Japan, September 2008, pp. 537–548. A short journal version of this work was presented in: [EATCS] B.Durand, A.Romashchenko, A.Shen. Fixed point theorem and ape- riodic tilings. Bulletin of the EATCS (The Logic in Computer Science Column by Yuri Gurevich). no 97 (2009) pp. 126–136. Also this article became a part of a long paper on a fixed-point technique in tilings: [DRS09] B.Durand, A.Romashchenko, A.Shen. Fixed-point tile sets and their applications. 2009, hal:00424024 and arXiv:0910.2415 (50 pages). Since the present paper is only a preliminary preprint, we encourage the reader to refer directly to [EATCS] or [DRS09] (this footnote is added on Jan 13, 2010). a cell paper (rotations are not allowed). Tiles in the neighbor cells should match (common side should have the same color in both). Formally speaking, we consider a finite set C of colors. A tile is a quadruple of colors (left, right, top and bottom ones), i.e., an element of C4. A tile set is a subset τ ⊂C4. A tiling of the plane with tiles from τ (τ-tiling) is a mapping U : Z2 →τ that respects the color matching condition. A tiling U is periodic if it has a period, i.e., a non-zero vector T ∈Z2 such that U(x + T ) = U(x) for all x ∈Z2. Otherwise the tiling is aperiodic. The following classical result was proved by Berger in a paper [2] where he used this construction as a main tool to prove Berger’s theorem: the domino problem (to find out whether a given tile set has tilings or not) is undecidable. Theorem 1. There exists a tile set τ such that τ-tilings exist and all of them are aperiodic. [2] The first tile set of Berger was rather complicated. Later many other con- structions were suggested. Some of them are simplified versions of the Berger’s construction ([17], see also the expositions in [1,5,13]). Some others are based on polygonal tilings (including famous Penrose and Ammann tilings, see [10]). An ingenious construction suggested in [11] is based on the multiplication in a kind of positional number system and gives a small aperiodic set of 14 tiles (in [3] an improved version with 13 tiles is presented). Another nice construction with a short and simple proof (based explicitly on ideas of self-similarity) was recently proposed by N. Ollinger [16]. In this paper we present yet another construction of aperiodic tile set. It does not provide a small tile set; however, we find it interesting because: • The existence of an aperiodic tile set becomes a simple application of a classical construction used in Kleene’s fixed point (recursion) theorem, in von Neumann’s self-reproducing automata [15] and, more recently, in G´acs’ reliable cellular automata [7,8]; we do not use any geometric tricks. The construction of an aperiodic tile set is not only an interesting result but an important tool (recall that it was invented to prove that domino problem is undecidable); our construction makes this tool easier to use (see Theorem 3). • The construction is rather general, so it is flexible enough to achieve some additional properties of the tile set. Our main result is Theorem 6: there exists a “robust” aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. It is not clear whether this can be achieved for previously known aperiodic tile sets; however, the math- ematical model for a processes like quasicrystals’ growth or DNA-computati

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