On Universality of Radius 1/2 Number-Conserving Cellular Automata

  • Katsunobu Imai
  • Artiom Alhazov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6079)


A number-conserving cellular automaton (NCCA) is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modeling of the physical conservation laws of mass or energy. In this paper we show a construction method of radius 1/2 NCCAs. The local transition function is expressed via a single unary function which can be regarded as ‘flows’ of numbers. In spite of the strong constraint, we constructed radius 1/2 NCCAs that simulate any radius 1/2 cellular automata or any radius 1 NCCA. We also consider the state complexity of these non-splitting simulations (4n 2 + 2n + 1 and 8n 2 + 12n − 16, respectively). These results also imply existence of an intrinsically universal radius 1/2 NCCA.


Cellular automata number-conservation intrinsic universality one-way automata state complexity 


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  1. 1.
    Boccara, N., Fukś, H.: Number-conserving cellular automaton rules. Fundamenta Informaticae 52, 1–13 (2003)Google Scholar
  2. 2.
    Durand, B., Formenti, E., Grange, A., Róka, Z.: Number conserving cellular automata: new results on decidability and dynamics. In: Morvan, M., Rémila, É. (eds.) Proceedings of Discrete Models for Complex Systems, DMCS 2003. Discrete Mathematics and Theoretical Computer Science, vol. AB, pp. 129–140 (2003)Google Scholar
  3. 3.
    Durand, B., Formenti, E., Róka, Z.: Number conserving cellular automata I: decidability. Theoretical Computer Science 299(1-3), 523–535 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fukś, H., Sullivan, K.: Enumeration of number-conserving cellular automata rules with two inputs. Journal of Cellular Automata 2(2), 141–148 (2007)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hattori, T., Takesue, S.: Additive conserved quantities in discrete-time lattice dynamical systems. Pysica 49D, 295–322 (1991)MathSciNetGoogle Scholar
  6. 6.
    Imai, K., Fujita, K., Iwamoto, C., Morita, K.: Embedding a logically universal model and a self-reproducing model into number-conserving cellular automata. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002. LNCS, vol. 2509, pp. 164–175. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Imai, K., Ikazaki, A., Iwamoto, C., Morita, K.: A logically universal number-conserving cellular automaton with a unary table-lookup function. Trans. IEICE E87-D(3), 694–699 (2004)Google Scholar
  8. 8.
    Kasai, Y.: Number-conserving cellular automata with universality under errors, Master’s thesis, Hiroshima University (2003)Google Scholar
  9. 9.
    Moreira, A.: Universality and decidability of number-conserving cellular automata. Theoretical Computer Science 292, 711–721 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Moreira, A., Boccara, N., Goles, E.: On conservative and monotone one-dimensional cellular automata and their particle representation. Theoretical Computer Science 325, 285–316 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Morita, K.: Computation-universality of one-dimensional one-way reversible cellular automata. Information Processing Letters 42, 325–329 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nagel, K., Schreckenberg, M.: A cellular automaton for freeway traffic. Journal of Physics I(2), 2221–2229 (1992)Google Scholar
  13. 13.
    Ollinger, N., Richard, G.: A Particular Universal Cellular Automaton. In: Proc. International Workshop on The Complexity of Simple Programs (CSP 2008), pp. 205–214 (2008)Google Scholar
  14. 14.
    Scharanko, A., Oliveira, P.: Derivation of one-dimensional, reversible, number-conserving cellular automata rules. In: Proc. 15th International Workshop on Cellular Automata and Discrete Complex Systems (Automata 2009), pp. 335–345 (2009)Google Scholar
  15. 15.
    Tanimoto, N., Imai, K.: A Characterization of von Neumann Neighbor number-conserving cellular automata. Journal of Cellular Automata 4(1), 39–54 (2009)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Katsunobu Imai
    • 1
  • Artiom Alhazov
    • 1
    • 2
  1. 1.Department of Information Engineering, Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Institute of Mathematics and Computer ScienceAcademy of Sciences of MoldovaMoldova

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