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 ℤ p . 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 ℤ p ) 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.
Similar content being viewed by others
References
Akın, H.: The topological entropy of invertible cellular automata. J. Comput. Appl. Math. 213(2), 501–508 (2008)
Akın, H., Siap, I.: On cellular automata over Galois rings. Inf. Process. Lett. 103(1), 24–27 (2007)
Czeizler, E.: On the size of inverse neighborhoods for one-dimensional reversible cellular automata. Theor. Comput. Sci. 325, 273–284 (2004)
Hernández Encinas, L., del Rey, A.M.: Inverse rules of ECA with rule number 150. Appl. Math. Comput. 189, 1782–1786 (2007)
Manzini, G., Margara, L.: Invertible linear cellular automata over: algorithmic and dynamical aspects. J. Comput. Syst. Sci. 56, 60–67 (1998)
del Rey, A.M., Rodriguez Sánchez, G.: Reversibility of linear cellular automata, Appl. Math. Comput. (2011). doi:10.1016/j.amc.2011.03.033
Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219–258 (1984)
Morita, K.: Reversible cellular automata. J. Inf. Process. Soc. Jpn. 35, 315–321 (1994)
Morita, K.: Reversible computing and cellular automata—a survey. Theor. Comput. Sci. 395, 101–131 (2008)
Wolfram Research, Inc.: Mathematica Edition: Version 7.0. Wolfram Research, Inc., Champaign (2008)
Nobe, A., Yura, F.: On reversibility of cellular automata with periodic boundary conditions. J. Phys. A, Math. Gen. 37, 5789–5804 (2004)
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)
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)
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)
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)
Wolfram, S.: A New Kind of Science. Wolfram Media Inc., Champaign (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cinkir, Z., Akin, H. & Siap, I. Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields \(\pmb{ {\mathbb{Z}}}_{p}\) . J Stat Phys 143, 807–823 (2011). https://doi.org/10.1007/s10955-011-0202-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-011-0202-2