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Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata

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Algorithmic Bioprocesses

Part of the book series: Natural Computing Series ((NCS))

Abstract

Using the fact that the tiling problem of Wang tiles is undecidable even if the given tile set is deterministic by two opposite corners, it is shown that the question whether there exists a trajectory which belongs to the given open and closed set is undecidable for one-dimensional reversible cellular automata. This result holds even if the cellular automaton is mixing. Furthermore, it is shown that left expansivity of a reversible cellular automaton is an undecidable property. Also, the tile set construction gives yet another proof for the universality of one-dimensional reversible cellular automata.

J. Kari’s research supported by the Academy of Finland grant 211967.

V. Lukkarila’s research supported by the Finnish Cultural Foundation and the Fund of Vilho, Yrjö and Kalle Väisälä.

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References

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

    MATH  MathSciNet  Google Scholar 

  2. Bennett CH (1973) Logical reversibility of computation. IBM J Res Dev 6:525–532

    Google Scholar 

  3. Berger R (1966) The undecidability of the domino problem. Mem Am Math Soc 66:1–72

    Google Scholar 

  4. Blanchard F, Maass A (1997) Dynamical properties of expansive one-sided cellular automata. Israel J Math 99:149–174

    Article  MATH  MathSciNet  Google Scholar 

  5. Dubacq J-C (1995) How to simulate any Turing machine by reversible one-dimensional cellular automaton. Int J Found Comput Sci 6(4):395–402

    Article  MATH  Google Scholar 

  6. Durand B, Formenti E, Varouchas G (2003) On undecidability of equicontinuity classification for cellular automata. Discrete Math Theor Comput Sci, pp 117–128

    Google Scholar 

  7. Finelli M, Manzini G, Margara L (1998) Lyapunov exponents versus expansivity and sensitivity in cellular automata. J Complex 14:210–233

    Article  MATH  MathSciNet  Google Scholar 

  8. Hooper P (1966) The undecidability of the Turing machine immortality problem. J Symbol Log 31(2):219–234

    Article  MATH  MathSciNet  Google Scholar 

  9. Hopcroft JE, Ullman JD (1979) Introduction to automata theory, languages and computation. Addison–Wesley, Readings

    MATH  Google Scholar 

  10. Kari J (1992) The nilpotency problem of one-dimensional cellular automata. SIAM J Comput 21:571–586

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Kari J, Ollinger N (2008) Periodicity and immortality in reversible computing. In: MFCS 2008. Lecture notes in computer science, vol 5162. Springer, Berlin, pp 419–430

    Google Scholar 

  14. Kurka P (2001) Topological dynamics of cellular automata. In: Markus B, Rosenthal J (eds) Codes, systems and graphical models. IMA volumes in mathematics and its applications, vol 123. Springer, Berlin, pp 447–498

    Google Scholar 

  15. Lukkarila V (2008) Sensitivity and topological mixing are undecidable for reversible one-dimensional cellular automata. J Cell Autom (submitted)

    Google Scholar 

  16. Lukkarila V (2009) The 4-way deterministic tiling problem is undecidable. Theor Comput Sci 410:1516–1533

    Article  MATH  MathSciNet  Google Scholar 

  17. Manna Z (1974) Mathematical theory of computation. McGraw–Hill, New York

    MATH  Google Scholar 

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

    Google Scholar 

  19. Nasu M (1995) Textile systems for endomorphisms and automorphisms of the shift. Mem Am Math Soc, vol 546. AMS, Providence

    Google Scholar 

  20. Robinson RM (1971) Undecidability and nonperiodicity for tilings of the plane. Invent Math 12:177–209

    Article  MathSciNet  Google Scholar 

  21. Shereshevsky MA (1993) Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms. Indag Math N S 4:203–210

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Jarkko Kari & Ville Lukkarila

  2. Turku Centre for Computer Science, 20520, Turku, Finland

    Jarkko Kari & Ville Lukkarila

Authors
  1. Jarkko Kari
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  2. Ville Lukkarila
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Corresponding author

Correspondence to Jarkko Kari .

Editor information

Editors and Affiliations

  1. Dept. Computer Science, University of British Columbia, Main Mall 201-2366, Vancouver, V6T 1Z4, Canada

    Anne Condon

  2. Dept. Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    David Harel

  3. Leiden Inst. Advanced Computer Science, Leiden University, Niels Bohrweg 1, Leiden, 2333 CA, Netherlands

    Joost N. Kok

  4. Turku Centre for Computer Science, Lemminkaisenkatu 14 A, Turku, 20520, Finland

    Arto Salomaa

  5. Computer Science, Computation,, California Inst. of Technology, Pasadena, 91125, U.S.A.

    Erik Winfree

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Kari, J., Lukkarila, V. (2009). Some Undecidable Dynamical Properties for One-Dimensional Reversible Cellular Automata. In: Condon, A., Harel, D., Kok, J., Salomaa, A., Winfree, E. (eds) Algorithmic Bioprocesses. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88869-7_32

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  • DOI: https://doi.org/10.1007/978-3-540-88869-7_32

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88868-0

  • Online ISBN: 978-3-540-88869-7

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