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The Most General Conservation Law for a Cellular Automaton

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Computer Science – Theory and Applications (CSR 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5010))

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Abstract

We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a most general (group-valued or semigroup-valued) conservation law, encapsulating all conservation laws of that range. For one-dimensional CA the semigroup corresponding to such a most general conservation law has an effectively constructible finite presentation, while for higher-dimensional ones no such effective construction exists.

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References

  1. Biryukov, A.P.: Some algorithmic problems for finitely defined commutative semigroups. Siberian Mathematical Journal 8, 384–391 (1967)

    MATH  Google Scholar 

  2. Blondel, V.D., Cassaigne, J., Nichitiu, C.: On the presence of periodic configurations in Turing machines and in counter machines. Theoretical Computer Science 289, 573–590 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boccara, N., Fukś, H.: Number-conserving cellular automaton rules. Fundamenta Informaticae 52, 1–13 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Conway, J.H., Lagarias, J.C.: Tiling with polyominoes and combinatorial group theory. Journal of Combinatorial Theory A 53, 183–208 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Durand, B., Formenti, E., Róka, Z.: Number conserving cellular automata I: decidability. Theoretical Computer Science 299, 523–535 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Formenti, E., Grange, A.: Number conserving cellular automata II: dynamics. Theoretical Computer Science 304, 269–290 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Grillet, P.A.: Semigroups: An Introduction to the Structure Theory. Dekker, New York (1995)

    Google Scholar 

  8. Hattori, T., Takesue, S.: Additive conserved quantities in discrete-time lattice dynamical systems. Physica D 49, 295–322 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Kari, J.: Theory of cellular automata: A survey. Theoretical Computer Science 334, 3–33 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  12. Moore, E.F.: Machine models of self-reproduction. In: Proceedings of Symposia in Applied Mathematics, pp. 17–33. AMS (1962)

    Google Scholar 

  13. Myhill, J.: The converse of Moore’s Garden-of-Eden theorem. Proceedings of the American Mathematical Society 14, 685–686 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pivato, M.: Conservation laws in cellular automata. Nonlinearity 15, 1781–1793 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Robison, A.D.: Fast computation of additive cellular automata. Complex Systems 1, 211–216 (1987)

    MathSciNet  Google Scholar 

  16. Thurston, W.P.: Conway’s tiling groups. American Mathematical Monthly 97, 757–773 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Départment d’Informatique, Parc Valrose, Université de Nice-Sophia Antipolis, 06108, Nice Cedex 2, France

    Enrico Formenti

  2. Department of Mathematics, University of Turku, FI-20014, Turku, Finland

    Jarkko Kari & Siamak Taati

  3. Turku Centre for Computer Science, FI-20520, Turku, Finland

    Siamak Taati

Authors
  1. Enrico Formenti
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  2. Jarkko Kari
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  3. Siamak Taati
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Editor information

Edward A. Hirsch Alexander A. Razborov Alexei Semenov Anatol Slissenko

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© 2008 Springer-Verlag Berlin Heidelberg

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Formenti, E., Kari, J., Taati, S. (2008). The Most General Conservation Law for a Cellular Automaton . In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_21

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  • DOI: https://doi.org/10.1007/978-3-540-79709-8_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79708-1

  • Online ISBN: 978-3-540-79709-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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