Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields \(\pmb{ {\mathbb{Z}}}_{p}\)

Article

Abstract

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field ℤ2 (del Rey and Rodriguez Sánchez in Appl. Math. Comput., 2011, doi:10.1016/j.amc.2011.03.033). In this paper, we study one-dimensional linear cellular automata with periodic boundary conditions over any finite field ℤp. For any given p≥2, we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. More specifically, for any given values (from any fixed field ℤp) of the coefficients of the local rules, we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells. As an example, we give the full criteria for the reversibility of the one-dimensional linear cellular automata over the fields ℤ2 and ℤ3.

Keywords

Cellular automata Periodic boundary condition Reversibility Matrix representations 

References

  1. 1.
    Akın, H.: The topological entropy of invertible cellular automata. J. Comput. Appl. Math. 213(2), 501–508 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Akın, H., Siap, I.: On cellular automata over Galois rings. Inf. Process. Lett. 103(1), 24–27 (2007) MATHCrossRefGoogle Scholar
  3. 3.
    Czeizler, E.: On the size of inverse neighborhoods for one-dimensional reversible cellular automata. Theor. Comput. Sci. 325, 273–284 (2004) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Hernández Encinas, L., del Rey, A.M.: Inverse rules of ECA with rule number 150. Appl. Math. Comput. 189, 1782–1786 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Manzini, G., Margara, L.: Invertible linear cellular automata over: algorithmic and dynamical aspects. J. Comput. Syst. Sci. 56, 60–67 (1998) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    del Rey, A.M., Rodriguez Sánchez, G.: Reversibility of linear cellular automata, Appl. Math. Comput. (2011). doi:10.1016/j.amc.2011.03.033
  7. 7.
    Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219–258 (1984) MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Morita, K.: Reversible cellular automata. J. Inf. Process. Soc. Jpn. 35, 315–321 (1994) Google Scholar
  9. 9.
    Morita, K.: Reversible computing and cellular automata—a survey. Theor. Comput. Sci. 395, 101–131 (2008) MATHCrossRefGoogle Scholar
  10. 10.
    Wolfram Research, Inc.: Mathematica Edition: Version 7.0. Wolfram Research, Inc., Champaign (2008) Google Scholar
  11. 11.
    Nobe, A., Yura, F.: On reversibility of cellular automata with periodic boundary conditions. J. Phys. A, Math. Gen. 37, 5789–5804 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Mora, J.C.S.T.: Matrix methods and local properties of reversible one-dimensional cellular automata. J. Phys. A, Math. Gen. 35, 5563–5573 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Mora, J.C.S.T., Vergara, S.V.C., Martýnez, G.J., McIntosh, H.V.: Procedures for calculating reversible one-dimensional cellular automata. Physica D 202, 134–141 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Seck, J.C., Juarez, G., McIntosh, H.V.: The inverse behaviour of a reversible one-dimensional cellular automaton obtained by a single Welch diagram. J. Cell. Autom. 1, 25–39 (2006) MathSciNetMATHGoogle Scholar
  15. 15.
    von Neumann, J.: Theory and organization of complicated automata. In: Burks, A.W. (ed.) The Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1996) Google Scholar
  16. 16.
    Wolfram, S.: A New Kind of Science. Wolfram Media Inc., Champaign (2002) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationZirve UniversityGaziantepTurkey
  2. 2.Department of MathematicsYıldız Technical UniversityIstanbulTurkey

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