Abstract
Tile sets and tilings of the plane appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). The idea is to enforce some global properties (of the entire tiling) by means of local rules (for neighbor tiles). A fundamental question: Can local rules enforce a complex (highly irregular) structure of a tiling?
The minimal (and weak) notion of irregularity is aperiodicity. R. Berger constructed a tile set such that every tiling is aperiodic. Though Berger’s tilings are not periodic, they are very regular in an intuitive sense.
In [3] a stronger result was proven: There exists a tile set such that all n×n squares in all tilings have Kolmogorov complexity Ω(n), i.e., contain Ω(n) bits of information. Such a tiling cannot be periodic or even computable.
In the present paper we apply the fixed-point argument from [5] to give a new construction of a tile set that enforces high Kolmogorov complexity tilings (thus providing an alternative proof of the results of [3]). The new construction is quite flexible, and we use it to prove a much stronger result: there exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed.
Supported by NAFIT ANR-08-EMER-008-01 and RFBR 09-01-00709-a grants.
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References
Berger, R.: The Undecidability of the Domino Problem. Mem. Amer. Math. Soc. 66 (1966)
Culik, K.: An Aperiodic Set of 13 Wang Tiles. Discrete Math. 160, 245–251 (1996)
Durand, B., Levin, L., Shen, A.: Complex Tilings. J. Symbolic Logic 73(2), 593–613 (2008); see also Proc. 33rd Ann. ACM Symp. Theory Computing, pp. 732–739 (2001), www.arxiv.org/cs.CC/0107008
Durand, B., Levin, L., Shen, A.: Local Rules and Global Order, or Aperiodic Tilings. Math. Intelligencer 27(1), 64–68 (2004)
Durand, B., Romashchenko, A., Shen, A.: Fixed point and aperiodic tilings. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 276–288. Springer, Heidelberg (2008), http://arxiv.org/abs/0802.2432
Gács, P.: Reliable Cellular Automata with Self-Organization. In: Proc. 38th Ann. Symp. Found. Comput. Sci., pp. 90–97 (1997)
Gács, P.: Reliable Cellular Automata with Self-Organization. J. Stat. Phys. 103(1/2), 45–267 (2001)
Kari, J.: A Small Aperiodic Set of Wang tiles. Discrete Math. 160, 259–264 (1996)
Rogers, H.: The Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge (1987)
Lafitte, G., Weiss, M.: Computability of Tilings. In: IFIP TCS 2008, pp. 187–201 (2008)
Levin, L.: Aperiodic Tilings: Breaking Translational Symmetry. Computer J. 48(6), 642–645 (2005); http://www.arxiv.org/cs.DM/0409024
von Neumann, J.: Theory of Self-reproducing Automata. In: Burks, A. (ed.), University of Illinois Press (1966)
Robinson, R.: Undecidability and Nonperiodicity for Tilings of the Plane. Inventiones Mathematicae 12, 177–209 (1971)
Bienvenu, L., Romashchenko, A., Shen, A.: Sparse sets. Journées Automates Cellulaires 2008 (Uzès), 18–28, MCCME Publishers (2008); http://hal.archives-ouvertes.fr/docs/00/27/40/10/PDF/18-28.pdf
Rumyantsev, A.Y., Ushakov, M.A.: Forbidden substrings, kolmogorov complexity and almost periodic sequences. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 396–407. Springer, Heidelberg (2006)
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Durand, B., Romashchenko, A., Shen, A. (2009). High Complexity Tilings with Sparse Errors. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_34
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DOI: https://doi.org/10.1007/978-3-642-02927-1_34
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