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P-completeness of Cellular Automaton Rule 110

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Automata, Languages and Programming (ICALP 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4051))

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Abstract

We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is P-complete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polynomial time, this is an exponential improvement on their previously known simulation time overhead.

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References

  1. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55, 601–644 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cook, M.: Universality in elementary cellular automata. Complex Systems 15, 1–40 (2004)

    MATH  MathSciNet  Google Scholar 

  3. Cocke, J., Minsky, M.: Universality of tag systems with P = 2. Journal of the ACM 11, 15–20 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  4. Rogozhin, Y.: Small universal Turing machines. TCS 168, 215–240 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baiocchi, C.: Three small universal Turing machines. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 1–10. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  6. Kudlek, M., Rogozhin, Y.: A universal Turing machine with 3 states and 9 symbols. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 311–318. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory, Symp. in Pure Math., vol. 5, pp. 229–238. AMS (1962)

    Google Scholar 

  8. Neary, T., Woods, D.: A small fast universal Turing machine. Technical Report NUIM-CS-TR-2005-12, Dept. of Computer Science, NUI Maynooth (2005)

    Google Scholar 

  9. Neary, T., Woods, D.: Small fast universal Turing machines. TCS (to appear)

    Google Scholar 

  10. Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to parallel computation: P-completeness theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  11. Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4, 299–318 (1990)

    MATH  MathSciNet  Google Scholar 

  12. Ollinger, N.: The quest for small universal cellular automata. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 318–329. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  13. Moore, C.: Majority-vote cellular automata, Ising dynamics and P-completeness. Journal of Statistical Physics 88, 795–805 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Moore, C.: Quasi-linear cellular automata. Physica D 103, 100–132 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Moore, C.: Predicting non-linear cellular automata quickly by decomposing them into linear ones. Physica D 111, 27–41 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Aaronson, S.: Book review: A new kind of science. Quantum Information and Computation 2, 410–423 (2002)

    Google Scholar 

  17. Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. TASS, Department of Computer Science, National University of Ireland Maynooth, Ireland

    Turlough Neary

  2. Department of Mathematics and Boole Centre for Research in Informatics, University College Cork, Ireland

    Damien Woods

Authors
  1. Turlough Neary
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  2. Damien Woods
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università Ca’ Foscari, Venice

    Michele Bugliesi

  2. Interdisciplinary Institute for BroadBand Technology (IBBT), Belgium

    Bart Preneel

  3. ECS, University of Southampton, UK

    Vladimiro Sassone

  4. FB Informatik, Univ. Dortmund, LS2, 44221, Dortmund, Germany

    Ingo Wegener

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© 2006 Springer-Verlag Berlin Heidelberg

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Neary, T., Woods, D. (2006). P-completeness of Cellular Automaton Rule 110. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_13

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  • DOI: https://doi.org/10.1007/11786986_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35904-3

  • Online ISBN: 978-3-540-35905-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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