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An Almost Totally Universal Tile Set

  • Grégory Lafitte
  • Michael Weiss
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)

Abstract

Wang tiles are unit size squares with colored edges. In this paper, we approach one aspect of the study of tilings computability: the quest for a universal tile set. Using a complex construction, based on Robinson’s classical construction and its different modifications, we build a tile set μ (pronounced ayin) which almost always simulates any tile set. By way of Banach-Mazur games on tilings topological spaces, we prove that the set of μ-tilings which do not satisfy the universality condition is meager in the set of μ-tilings.

Keywords

Topological Space Turing Machine West Side Winning Strategy Colored Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AD96]
    Allauzen, C., Durand, B.: Appendix A: Tiling problems. The classical decision problem, 407–420 (1996)Google Scholar
  2. [BDJ08]
    Ballier, A., Durand, B., Jeandel, E.: Structural aspects of tilings. In: Proceeding of the Symposium on Theoretical Aspects of Computer Science, pp. 61–72 (2008)Google Scholar
  3. [Ber66]
    Berger, R.: The undecidability of the domino problem. Mem. Amer. Math Soc. 66, 1–72 (1966)Google Scholar
  4. [DLS01]
    Durand, B., Levin, L.A., Shen, A.: Complex tilings. In: STOC, pp. 732–739 (2001)Google Scholar
  5. [DLS04]
    Durand, B., Levin, L.A., Shen, A.: Local rules and global order. Mathematical Intelligencer 27(1), 64–68 (2004)CrossRefMathSciNetGoogle Scholar
  6. [DRS08]
    Durand, B., Romashchenko, A.E., Shen, A.: Fixed point and aperiodic tilings. In: Developments in Language Theory, pp. 276–288 (2008)Google Scholar
  7. [Han74]
    Hanf, W.P.: Nonrecursive tilings of the plane. I. J. Symb. Log 39(2), 283–285 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [LW07]
    Lafitte, G., Weiss, M.: Universal tilings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 367–380. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. [LW08a]
    Lafitte, G., Weiss, M.: Simulations between tilings. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) Logic and Theory of Algorithms, 4th Conference on Computability in Europe, CiE 2008, Athens, Greece, June 2008, University of Athens (2008)Google Scholar
  10. [LW08b]
    Lafitte, G., Weiss, M.: Computability of tilings. In: International Federation for Information Processing, Fifth IFIP International Conference on Theoretical Computer Science, vol. 273, pp. 187–201 (2008)Google Scholar
  11. [LW08c]
    Lafitte, G., Weiss, M.: A topological study of tilings. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 375–387. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. [Mye74]
    Myers, D.: Nonrecursive tilings of the plane. II. J. Symb. Log 39(2), 286–294 (1974)zbMATHCrossRefGoogle Scholar
  13. [Oxt57]
    Oxtoby, J.C.: Tilings: recursivity and regularity. Contribution to the theory of games III(39), 159–163 (1957)Google Scholar
  14. [Rob71]
    Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inv. Math. 12, 117–209 (1971)CrossRefGoogle Scholar
  15. [Wan61]
    Wang, H.: Proving theorems by pattern recognition II. Bell Systems Journal 40, 1–41 (1961)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Grégory Lafitte
    • 1
  • Michael Weiss
    • 2
  1. 1.Laboratoire d’Informatique Fondamentale de Marseille (LIF)CNRS – Aix-Marseille UniversitéMarseille Cedex 13France
  2. 2.Bicocca Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di MilanoMilanoItaly

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