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Complex tilings

Published online by Cambridge University Press:  12 March 2014

Bruno Durand,
Leonid A. Levin and
Alexander Shen
Bruno Durand
Affiliation:
Laboratoire D'Informatique Fondamentale De Marseille, 39, Rue Joliot-Curie, 13453 Marseille, France, E-mail: Bruno.Durand@lif.univ-mrs.fr
Leonid A. Levin
Affiliation:
Boston University, Computer Science Department, 111 Cummington St., Boston. MA 02215, USA, E-mail: Lnd@bu.edu
Alexander Shen
Affiliation:
Laboratoire D'Informatique Fondamentale De Marseille, and, Institute of Problems of Information Transmission, Moscow., Russia, E-mail: alexander.shen@lif.univ-mrs.fr, E-mail: shen@mccme.ru
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Abstract

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings—that we also develop in terms of Turing degrees of unsolvability.

Keywords

TilingsKolmogorov complexityrecursion theory

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 2 , June 2008, pp. 593 - 613
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[]Allauzen, Cyril and Durand, Bruno, Appendix A: “Tiling problems”, In [Börger, , Grädel, , and Gurevich, , 1996], pp. 407–420.Google Scholar
[1966]Berger, Robert, The undecidability of the domino problem, Memoirs of the American Mathematical Society, vol. 66, 1966.Google Scholar
[1996]Börger, Egon, Grädel, Erich, and Gurevich, Yuri (editors), The classical decision problem. Springer-Verlag, 1996.Google Scholar
[2000]Cervelle, Julien and Durand, Bruno, Tilings: Recursivity and regularity, STACS'00, Lecture Notes in Computer Science, vol. 1770, Springer Verlag, 2000.Google Scholar
[1999]Durand, Bruno, Tilings and quasiperiodicity, Theoretical Computer Science, vol. 221 (1999), pp. 61–75.CrossRefGoogle Scholar
[2001]Durand, Bruno, Levin, Leonid, and Shen, Alexander, Complex tilings, STOC, 2001, Extended version: http://ww.arxiv.org/cs.CC/0107008, pp. 732–739.CrossRefGoogle Scholar
[2004]Durand, Bruno, Levin, Leonid, and Shen, Alexander, Local rules and global order, or aperiodic tilings. Mathematical Intelligencer, vol. 27 (2004), no. 1, pp. 64–68.CrossRefGoogle Scholar
[2001]Gacs, Peter, Reliable cellular automata with self-organization. Journal of Statistical Physics, vol. 103 (2001), no. 1/2, pp. 45–267.CrossRefGoogle Scholar
[1991]Gurevich, Yuri, Average case completeness, Journal of Computer and System Sciences, vol. 42 (1991), pp. 346–398.CrossRefGoogle Scholar
[1972]Gurevich, Yuri and Koriakov, Igor, A remark on Berger's paper on the domino problem, Siberian Journal of Mathematics, vol. 13 (1972), pp. 459–463, in Russian.CrossRefGoogle Scholar
[1974]Hanf, William, Nonrecursive tilings of the plane. I, this Journal, vol. 39 (1974), no. 2, pp. 283–285.Google Scholar
[1991]Ingersent, Kevin, Matching rules for quasicrystalline tilings, Quasicrystals. The state of the art, World Scientific, 1991, pp. 185–212.CrossRefGoogle Scholar
[1986]Levin, Leonid, Average case complete problems, SIAM Journal on Computing, vol. 15 (1986), no. 1, pp. 285–286.CrossRefGoogle Scholar
[2005]Levin, Leonid, Aperiodic tilings: Breaking translational symmetry, The Computer Journal, vol. 48 (2005), no. 6, pp. 642–645, on-line: http://www.arxiv.org/cs.DM/0409024.CrossRefGoogle Scholar
[1997]Li, Ming and Vitányi, Paul, An introduction to Kolmogorov complexity and its applications, second ed., Springer-Verlag, 1997.CrossRefGoogle Scholar
[2000]Muchnik, Andrej, personal communication, 2000.Google Scholar
[1974]Myers, Dale, Nonrecursive tilings of the plane, ii, this Journal, vol. 39 (1974), no. 2, pp. 286–294.Google Scholar
[1989]Odifreddi, Piergiorgio, Classical recursion theory, North-Holland, 1989.Google Scholar
[1971]Robinson, Raphael, Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, vol. 12 (1971), pp. 177–209.CrossRefGoogle Scholar
[1961]Wang, Hao, Proving theorems by pattern recognition II, Bell System Technical Journal, vol. 40 (1961), pp. 1–41.CrossRefGoogle Scholar
[1962]Wang, Hao. Dominoes and the ∀∃∀-case of the decision problem, Proceedings of the Symposium on Mathematical Theory of Automata, Brooklyn Polytechnic Institute, New York, 1962, pp. 23–55.Google Scholar