Cellular Automata and Hyperbolic Spaces

Part of the Emergence, Complexity and Computation book series (ECC, volume 2)

Abstract

In this paper we look at the possibility to implement cellular automata in hyperbolic spaces and at a few consequences it may have, both on theory and on more practical problems.

Keywords

cellular automata hyperbolic geometry tilings 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ben-Jacob, E.: Social behavior of bacteria: from physics to complex organization. European Physical Journal B 65(3), 315–322 (2008)CrossRefGoogle Scholar
  2. 2.
    Berger, R.: The undecidability of the domino problem. Memoirs of the American Mathematical Society 66, 1–72 (1966)Google Scholar
  3. 3.
    Bonola, R.: Non-Euclidean Geometry. Dover (2007)Google Scholar
  4. 4.
    Chelghoum, K., Margenstern, M., Martin, B., Pecci, I.: Palette hyperbolique: un outil pour interagir avec des ensembles de données. In: IHM 2004, Namur (2004)Google Scholar
  5. 5.
    Cook, M.: Universality in Elementary Cellular Automata. Complex Systems 15(1), 1–40 (2004)MathSciNetMATHGoogle Scholar
  6. 6.
    Coxeter, H.S.M.: Non-Euclidean Geometry. Mathematical Association of America (1998)Google Scholar
  7. 7.
    Herrmann, F., Margenstern, M.: A universal cellular automaton in the hyperbolic plane. Theoretical Computer Science 296, 327–364 (2003)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Iwamoto, C., Margenstern, M., Morita, K., Worsch, T.: Polynomial Time Cellular Automata in the Hyperbolic Plane Accept Exactly the PSPACE Languages. In: SCI 2002 (2002)Google Scholar
  9. 9.
    Margenstern, M.: New Tools for Cellular Automata of the Hyperbolic Plane. Journal of Universal Computer Science 6(12), 1226–1252 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Margenstern, M.: A universal cellular automaton with five states in the 3D hyperbolic space. Journal of Cellular Automata 1(4), 315–351 (2006)MathSciNetGoogle Scholar
  11. 11.
    Margenstern, M.: Cellular Automata in Hyperbolic Spaces. Theory, vol. 1, 422 p. Old City Publishing, Philadelphia (2007)Google Scholar
  12. 12.
    Margenstern, M.: The Domino Problem of the Hyperbolic Plane Is Undecidable. Theoretical Computer Science 407, 29–84 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Margenstern, M.: Cellular Automata in Hyperbolic Spaces. Implementation and computations, vol. 2, 360 p. Old City Publishing, Philadelphia (2008)Google Scholar
  14. 14.
    Margenstern, M.: An upper bound on the number of states for a strongly universal hyperbolic cellular automaton on the pentagrid. In: JAC 2010, Turku, Finland, December 15-17 (2010) (accepted)Google Scholar
  15. 15.
    Margenstern, M.: A universal cellular automaton on the heptagrid of the hyperbolic plane with four states. Theoretical Computer Science 412, 33–56 (2011)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Margenstern, M.: Bacteria, Turing machines and hyperbolic cellular automata. In: Zenil, H. (ed.) A Computable Universe: Understanding and Exploring Nature as Computation, ch. 12. World Scientific (in press, 2012)Google Scholar
  17. 17.
    Margenstern, M.: A protocol for a message system for the tiles of the heptagrid, in the hyperbolic plane. International Journal of Satellite Communications Policy and Management (in press)Google Scholar
  18. 18.
    Margenstern, M.: Universal cellular automata with two states in the hyperbolic plane. Journal of Cellular Automata (in press)Google Scholar
  19. 19.
    Margenstern, M., Martin, B., Umeo, H., Yamano, S., Nishioka, K.: A Proposal for a Japanese Keyboard on Cellular Phones. In: Umeo, H., Morishita, S., Nishinari, K., Komatsuzaki, T., Bandini, S. (eds.) ACRI 2008. LNCS, vol. 5191, pp. 299–306. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Margenstern, M., Morita, K.: NP problems are tractable in the space of cellular automata in the hyperbolic plane. Theoretical Computer Science 259, 99–128 (2001)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Margenstern, M., Song, Y.: A universal cellular automaton on the ternary heptagrid. Electronic Notes in Theoretical Computer Science 223, 167–185 (2008)CrossRefGoogle Scholar
  22. 22.
    Margenstern, M., Song, Y.: A new universal cellular automaton on the pentagrid. Parallel Processing Letters 19(2), 227–246 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs (1967)MATHGoogle Scholar
  24. 24.
    Morgenstein, D., Kreinovich, V.: Which Algorithms are Feasible and Which are not Depends on the Geometry of Space-Time. Geocombinatorics 4(3), 80–97 (1995)MATHGoogle Scholar
  25. 25.
    Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12, 177–209 (1971)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Stewart, I.: A Subway Named Turing, Mathematical Recreations. Scientific American, 90–92 (1994)Google Scholar
  27. 27.
    Taimina, D.: Crocheting Adventures with Hyperbolic Planes, 148 p. A K Peters, Ltd., Wellesley (2009)Google Scholar
  28. 28.
    Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Université de LorraineNancyFrance

Personalised recommendations