Skip to main content
SpringerLink
Log in

Reversibility Problem of Multidimensional Finite Cellular Automata

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility problem of multidimensional linear cellular automata under boundary conditions. This work proposes a criterion for testing the reversibility of a multidimensional linear cellular automaton under null boundary condition and an algorithm for the computation of its reverse, if it exists. The investigation of the dynamical behavior of a multidimensional linear cellular automaton under null boundary condition is equivalent to elucidating the properties of the block Toeplitz matrix. The proposed criterion significantly reduces the computational cost whenever the number of cells or the dimension is large; the discussion can also apply to cellular automata under periodic boundary conditions with a minor modification.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. Roughly speaking, the computational cost of characterizing the eigenvalues/eigenvectores/determinant of an \(n \times n\) matrix is \(O(n^3)\), and the computational cost of our approach is \(O(n^{\frac{3}{d}})\), where d is the dimension of the considered system.

References

  1. Abdoa, A.A., Lianb, S., Ismailc, I.A., Amina, M., Diaba, H.: A cryptosystem based on elementary cellular automata. Commun. Nonlinear Sci. Numer. Simul. 18, 136–147 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  2. Amoroso, S., Patt, Y.N.: Decision procedures for surjectivity and injectivity of parallelmaps for tessellation structures. J. Comput. Syst. Sci. 6, 448–464 (1972)

    Article  MATH  Google Scholar 

  3. Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cappellari, L., Milani, S., Cruz-Reyes, C., Calvagno, G.: Resolution scalable image coding with reversible cellular automata. IEEE Trans. Image Process. 20, 1461–1468 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  5. Chang, C.-H., Chang, H.: On the Bernoulli automorphism of reversible linear cellular automata. Inform. Sci. 345, 217–225 (2016)

    Article  MathSciNet  Google Scholar 

  6. Chang, C.-H., Su, J.-Y.: Reversibility of linear cellular automata on cayley trees with periodic boundary condition (2016). arXiv:1603.01679

  7. Cinkir, Z., Akın, H., Siap, I.: Reversibility of 1D cellular automata with periodic boundary over finite fields \(\mathbb{Z}_p\). J. Stat. Phys. 143, 807–823 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. del Rey, A.M., Sánchez, G.R.: Reversibility of linear cellular automata. Appl. Math. Comput. 217, 8360–8366 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Dennunzio, A., Formenti, E., Weiss, M.: Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues. Theor. Comput. Sci. 516, 40–59 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Encinas, L.H., del Rey, A.M.: Inverse rules of ECA with rule number 150. Appl. Math. Comput. 189, 1782–1786 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Gray, R.M.: Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information. Now Publishers Inc, Hanover (2006)

    MATH  Google Scholar 

  12. Gutierrez-Gutierrez, J., Crespo, P.M.: Block Toeplitz Matrices: Asymptotic Results and Applications. Foundations and Trends in Communications and Information. Now Publishers Inc, Hanover (2012)

    MATH  Google Scholar 

  13. Hishiki, T., Torikai, H.: A novel rotate-and-fire digital spiking neuron and its neuron-like bifurcations and responses. IEEE Trans. Neural Netw. 22, 752–767 (2011)

    Article  Google Scholar 

  14. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  15. Ito, M., Osato, N., Nasu, M.: Linear cellular automata over \(\mathbb{Z}_m\). J. Comput. Syst. Sci. 27, 125–140 (1983)

    Article  MATH  Google Scholar 

  16. Kari, J.: Reversibility of 2D cellular automata is undecidable. Phys. D 45, 386–395 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kari, J.: Reversibility and surjectivity problems of cellular automata. J. Comput. Syst. Sci. 48, 149–182 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kari, J.: Theory of cellular automata: a survey. Theor. Comput. Sci. 334, 3–33 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kazmi, N., Hossain, M.A., Phillips, R.M.: A hybrid cellular automaton model of solid tumor growth and bioreductive drug transport. IEEE ACM Trans. Comput. Biol. Bioinfo. 9, 1595–1606 (2012)

    Article  Google Scholar 

  20. Kippenberger, S., Bernd, A., Thaçi, D., Kaufmann, R., Meissner, M.: Modeling pattern formation in skin diseases by a cellular automaton. J. Invest. Dermatol. 133, 567–571 (2013)

    Article  Google Scholar 

  21. Köroğlu, M.E., Siap, I., Akın, H.: The reversibility problem for a family of two-dimensional cellular automata. Turk. J. Math. 40, 665–678 (2016)

    Article  MathSciNet  Google Scholar 

  22. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  23. Manzini, G., Margara, L.: Invertible linear cellular automata over \(\mathbb{Z}_m\): algorithmic and dynamical aspects. J. Comput. Syst. Sci. 56, 60–97 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morita, K.: Reversible Cellular Automata. Handbook of Natural Computing. Springer-Verlag, Berlin (2012)

    MATH  Google Scholar 

  25. Morita, K., Harao, M.: Computation universality of 1 dimensional reversible (injective) cellular automata. IEICE Trans. E72, 758–762 (1989)

    Google Scholar 

  26. Mullen, G.I., Mummert, C.: Finite Fields and Applications. American Mathematical Society, Boston (2007)

    Book  MATH  Google Scholar 

  27. Nasu, M.: The dynamics of expansive invertible onesided cellular automata. Trans. Am. Math. Soc. 354, 4067–4084 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nobe, A., Yura, F.: On reversibility of cellular automata with periodic boundary conditions. J. Phys. A Math. General 37, 5789–5804 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Ortega, J.M.: Matrix Theory. Plenum Press, New York (1987)

    Book  MATH  Google Scholar 

  30. Roman, S.: Field Theory, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  31. Sahin, U., Ugus, S., Akın, H., Siap, I.: Three-state von Neumann cellular automata and pattern generation. Appl. Math. Model. 39, 2003–2024 (2015)

    Article  MathSciNet  Google Scholar 

  32. Seck-Tuoh-Mora, J.C., Martínez, G.J., Alonso-Sanz, R., Hernández-Romero, N.: Invertible behavior in elementary cellular automata with memory. Inform. Sci. 199, 125–132 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Toffoli, T.: Computation and construction universality of reversible cellular automata. J. Comput. Syst. Sci. 15, 213–231 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. Viriyasitavat, W., Bai, F., Tonguz, O.K.: Dynamics of network connectivity in urban vehicular networks. IEEE J. Sel. Area Commun. 29, 515–533 (2011)

    Article  Google Scholar 

  35. Yamagishi, M.: Elliptic curves over finite fields and reversibility of additive cellular automata on square grids. Finite Fields Appl. 19, 105–119 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang, B., Wang, C., Xiang, A.: Reversibility of general 1d linear cellular automata over the binary field \(\mathbb{Z}_2\) under null boundary conditions. Inform. Sci. 324, 23–31 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to express our deep gratitude for the anonymous referees’ valuable and constructive comments, which have significantly improved the quality and readability of this paper. This work is partially supported by the Ministry of Science and Technology, ROC (Contract No MOST 105-2115-M-390 -001 -MY2).

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National University of Kaohsiung, 81148, Kaohsiung, Taiwan, ROC

    Chih-Hung Chang & Jing-Yi Su

  2. Gokkusagi Mahallesi, 1164. Cadde, No: 9/4, Cankaya-Ankara, Turkey

    Hasan Akın

  3. Department of Mathematics, Adıyaman University, Adıyaman, Turkey

    Ferhat Şah

Authors
  1. Chih-Hung Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing-Yi Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hasan Akın
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ferhat Şah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chih-Hung Chang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, CH., Su, JY., Akın, H. et al. Reversibility Problem of Multidimensional Finite Cellular Automata. J Stat Phys 168, 208–231 (2017). https://doi.org/10.1007/s10955-017-1799-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-017-1799-6

Keywords

Search

Navigation