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On computing the entropy of cellular automata

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Automata, Languages and Programming (ICALP 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1443))

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Abstract

We show how to compute the entropy of two important classes of cellular automata namely, linear and positively expansive cellular automata. In particular, we prove a closed formula for the topological entropy of D-dimensional (D ≥ 1) linear cellular automata over the ring Z m (m ≥ 2) and we provide an algorithm for computing the topological entropy of positively expansive cellular automata.

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References

  1. S. Amoroso and Y. N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. Journal of Computer and System Sciences, 6:448–464, 1972.

    MATH  MathSciNet  Google Scholar 

  2. F. Blanchard, P. Kurka, and A. Maass. Topological and measure-theoretic properties of one-dimensional cellular automata. Physica D, 103:86–99, 1997.

    Article  MathSciNet  Google Scholar 

  3. G. Cattaneo, E. Formenti, G. Manzini, and L. Margara. Ergodicity and regularity for linear cellular automata over Zm. Theoretical Computer Science. To appear.

    Google Scholar 

  4. M. D'amico, G. Manzini, and L. Margara. On computing the entropy of cellular automata. Technical Report B4-98-04, Istituto di Matematica Computazionale, CNR, Pisa, Italy, 1998.

    Google Scholar 

  5. G. A. Hedlund. Endomorphisms and automorphisms of the shift dynamical system. Mathematical Systems Theory, 3:320–375, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. P. Hurd, J. Kari, and K. Culik. The topological entropy of cellular automata is uncomputable. Ergodic Theory and Dynamical Systems, 12:255–265, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Ito, N. Osato, and M. Nasu. Linear cellular automata over Z m . Journal of Computer and System Sciences, 27:125–140, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Kari. The nilpotency problem of one-dimensional cellular automata. SIAM Journal on Computing, 21(3):571–586, June 1992.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Kari. Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences, 48(1):149–182, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Manzini and L. Margara. Invertible linear cellular automata over Z m : Algorithmic and dynamical aspects. Journal of Computer and System Sciences. To appear.

    Google Scholar 

  11. G. Manzini and L. Margara. A complete and efficiently computable topological classification of D-dimensional linear cellular automata over Z m . In 24th International Colloquium on Automata Languages and Programming (ICALP '97). LNCS n. 1256, Springer Verlag, 1997.

    Google Scholar 

  12. G. Manzini and L. Margara. Attractors of D-dimensional linear cellular automata. In 15th Annual Symposium on Theoretical Aspects of Computer Science (STACS '98). Springer Verlag, 1998.

    Google Scholar 

  13. M. Nasu. Textile systems for endomorphisms and automorphisms of the shift. Memoirs of the Amer. Math. Soc, 114(546), 1995.

    Google Scholar 

  14. M. A. Shereshevsky. Lyapunov exponents for one-dimensional cellular automata. Journal of Nonlinear Science, 2(1):1, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. A. Shereshevsky. Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag. Mathem. N.S., 4:203–210, 1993.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127, Bologna, Italy

    Michele D'amico

  2. Dipartimento di Scienze e Tecnologie Avanzate, Università di Torino, Via Cavour 84, 15100, Alessandria, Italy

    Giovanni Manzini

  3. Istituto di Matematica Computazionale, Via S. Maria, 46, 56126, Pisa, Italy

    Giovanni Manzini

  4. Dipartimento di Scienze dell'Informazione, Università di Bologna, Mura Anteo Zamboni 7, 40127, Bologna, Italy

    Luciano Margara

Authors
  1. Michele D'amico
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  2. Giovanni Manzini
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  3. Luciano Margara
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Editor information

Kim G. Larsen Sven Skyum Glynn Winskel

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© 1998 Springer-Verlag Berlin Heidelberg

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D'amico, M., Manzini, G., Margara, L. (1998). On computing the entropy of cellular automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055076

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  • DOI: https://doi.org/10.1007/BFb0055076

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64781-2

  • Online ISBN: 978-3-540-68681-1

  • eBook Packages: Springer Book Archive

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