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A shift-invariant metric on S zz inducing a non-trivial topology

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Mathematical Foundations of Computer Science 1997 (MFCS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1295))

Abstract

In this paper we discuss the meaning of sensitivity and its implications in CA behavior. A new shift-invariant metric is given. The metric topology induced by this metric is perfect but not compact. Moreover we prove that the new space is “suitable” for the study of the dynamical behavior of CA. In this context sensitivity assumes a stronger meaning than before (usually S ZZ is given the product topology). Now cellular automata are sensitive if they are not only capable of “transporting” the information but if they are also able to create new information. We also provide an experimental evidence of the fact that (in the new topology) sensitivity is linked to the fractal dimension of the space-time pattern generated by cellular automata evolutions.

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References

  1. Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney's definition of chaos. American Mathematical Montly 99 (1992) 332–334

    Google Scholar 

  2. Cattaneo, G., Formenti, E., Manzini, G., Margara, L.: On ergodic linear cellular automata over Zm. Proceeding of STAGS '97.

    Google Scholar 

  3. Cattaneo, G., Formenti, E., Margara, L., Mazoyer, J.: A shift-invariant metric on S ZZ. Ecole normal Superiéure de Lyon, Laboratoire de l'Informatique du Paralélisme rep. ?? (1997)

    Google Scholar 

  4. Devaney, R. L.: Introduction to chaotic dynamical systems. Addison-Wesley, second edition (1989)

    Google Scholar 

  5. v. Haeseler, H., Peitgen, H.-O., Skordev, G.: Cellular automata, matrix substitutions and fractals. Annales Mathematical Artificial Intelligence 8 (1993) 345–362

    Article  Google Scholar 

  6. v. Haeseler, H., Peitgen, H.-O., Skordev, G.: Global analysis of self-similarity features of cellular automata: some selected examples. Physica D 86 (1995) 64–80

    Article  Google Scholar 

  7. Hedlund, G. A.: Endomorphism and automorphism of the shift dynamical system. Mathematical System Theory, 3 (1969) 320–375

    Article  Google Scholar 

  8. Kelley, J. L.: General topology, volume 27 of Graduate texts in Mathematics. Springer-Verlag (1975)

    Google Scholar 

  9. Knudsen, C.: Chaos without nonperiodicity. American Mathematical Montly 101 (1994) 563–565

    Google Scholar 

  10. Margara, L.: Cellular automata and non periodic orbits. Complex Systems (to appear)

    Google Scholar 

  11. Wolfram, S.: Theory and Applications of Cellular Automata. World Scientific, Singapore (1986)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze dell'Informazione, Laboratorio di Fondamenti Fisici, 39 Via Comelico, 20135, Milano, Italy

    G. Cattaneo

  2. Ecole Normale Superiéure de Lyon, Laboratoire de l'Informatique du Parallélisme, 46 Allée d'Italie, 69364, Lyon, France

    E. Formenti & J. Mazoyer

  3. Dipartimento di Scienze dell'Informazione, Universitá di Bologna, Italy

    L. Margara

Authors
  1. G. Cattaneo
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  2. E. Formenti
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  3. L. Margara
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  4. J. Mazoyer
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Igor Prívara Peter Ružička

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© 1997 Springer-Verlag Berlin Heidelberg

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Cattaneo, G., Formenti, E., Margara, L., Mazoyer, J. (1997). A shift-invariant metric on S zz inducing a non-trivial topology. In: Prívara, I., Ružička, P. (eds) Mathematical Foundations of Computer Science 1997. MFCS 1997. Lecture Notes in Computer Science, vol 1295. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0029961

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  • DOI: https://doi.org/10.1007/BFb0029961

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63437-9

  • Online ISBN: 978-3-540-69547-9

  • eBook Packages: Springer Book Archive

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