Filling Curves Constructed in Cellular Automata with Aperiodic Tiling

  • Gaétan RichardEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10248)


In many constructions on cellular automata, information is transmitted with signals propagating through a defined background. In this paper, we investigate the possibility of using aperiodic tiling inside zones delimited by signals. More precisely, we study curves delineated by CA-constructible functions and prove that most of them can be filled with the NW-deterministic tile set defined by Kari [1]. The achieved results also hint a new possible way to study deterministic tile sets.


  1. 1.
    Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21(3), 571–586 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334(1–3), 3–33 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ganguly, N., Sikdar, B.K., Deutsch, A., Canright, G., Chaudhuri, P.P.: A survey on cellular automata. Technical report, Centre for High Performance Computing, Dresden University of Technology (2003)Google Scholar
  4. 4.
    Richard, G.: On the synchronisation problem over cellular automata. In: 34th International Symposium on Theoretical Aspects of Computer Science (2017, to appear)Google Scholar
  5. 5.
    Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. J. ACM 12(3), 388–394 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theor. Comput. Sci. 217(1), 53–80 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Iwamoto, C., Hatsuyama, T., Morita, K., Imai, K.: Constructible functions in cellular automata and their applications to hierarchy results. Theor. Comput. Sci. 270(1–2), 797–809 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Berger, R.: The undecidability of the domino problem. Ph.D. thesis, Harvard University (1964)Google Scholar
  9. 9.
    Robinson, R.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12, 177–209 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Penrose, R.: The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10(2), 266–271 (1974)Google Scholar
  11. 11.
    Culik, K., Kari, J.: An aperiodic set of Wang cubes. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 137–146. Springer, Heidelberg (1996). doi: 10.1007/3-540-60922-9_12 CrossRefGoogle Scholar
  12. 12.
    Ben-Abraham, S.I., Gähler, F.: Covering cluster description of octagonal MnSiAl quasicrystals. Phys. Rev. B 60, 860–864 (1999)CrossRefGoogle Scholar
  13. 13.
    Fernique, T., Ollinger, N.: Combinatorial substitutions and sofic tilings. In: Kari, J. (ed.) Proceedings of the Second Symposium on Cellular Automata “Journeacute;es Automates Cellulaires”, JAC 2010, Turku, 15–17 December 2010, pp. 100–110. Turku Center for Computer Science (2010)Google Scholar
  14. 14.
    Patitz, M.J.: An introduction to tile-based self-assembly and a survey of recent results. Nat. Comput. 13(2), 195–224 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2017

Authors and Affiliations

  1. 1.Normandie Univ, UNICAEN, ENSICAEN, CNRS, GREYCCaenFrance

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