Skip to main content
SpringerLink
Log in

Universality and the Halting Problem for Cellular Automata in Hyperbolic Spaces: The Side of the Halting Problem

  • Conference paper
Unconventional Computation and Natural Computation (UCNC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7445))

Abstract

In this paper, we remind results on universality for cellular automata in hyperbolic spaces, mainly results about weak universality, and we deal with the halting problem in the same settings. This latter problem is very close to that of strong universality. The paper focuses on the halting problem and it can be seen as a preliminary approach to strong universality about cellular automata in hyperbolic spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    MathSciNet  MATH  Google Scholar 

  2. Herrmann, F., Margenstern, M.: A universal cellular automaton in the hyperbolic plane. Theoretical Computer Science 296, 327–364 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lindgren, K., Nordahl, M.G.: Universal Computation in Simple One-Dimensional Cellular Automata. Complex Systems 4, 299–318 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Margenstern, M.: Frontier between decidability and undecidability a survey. Theoretical Computer Science 231(2), 217–251 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Margenstern, M.: Two railway circuits: a universal circuit and an NP-difficult one. Computer Science Journal of Moldova 9, 1–35 (2001)

    Google Scholar 

  6. Margenstern, M.: A universal cellular automaton with five states in the 3D hyperbolic space. Journal of Cellular Automata 1(4), 315–351 (2006)

    MathSciNet  Google Scholar 

  7. Margenstern, M.: Cellular Automata in Hyperbolic Spaces, Volume 1, Theory, 422 p. OCP, Philadelphia (2007)

    Google Scholar 

  8. Margenstern, M.: Cellular Automata in Hyperbolic Spaces, Volume 2, Implementation and computations, 360 p. OCP, Philadelphia (2008)

    Google Scholar 

  9. Margenstern, M.: Surprising Areas in the Quest for Small Universal Devices. Electronic Notes in Theoretical Computer Science 225, 201–220 (2009)

    Article  Google Scholar 

  10. Margenstern, M.: Turing machines with two letters and two states. Complex Systems (2010) (accepted)

    Google Scholar 

  11. Margenstern, M.: A weakly universal cellular automaton in the hyperbolic 3D space with three states. arXiv:1002.4290[cs.FL], 54 p. (2010)

    Google Scholar 

  12. Margenstern, M.: A universal cellular automaton on the heptagrid of the hyperbolic plane with four states. Theoretical Computer Science (2010) (to appear)

    Google Scholar 

  13. Margenstern, M.: About the embedding of one dimensional cellular automata into hyperbolic cellular automata. arXiv:1004.1830[cs.FL], 19 p. (2010)

    Google Scholar 

  14. Margenstern, M.: An upper bound on the number of states for a strongly universal hyperbolic cellular automaton on the pentagrid. In: JAC 2010, Turku, Finland. TUCS Proceedings (2010)

    Google Scholar 

  15. Margenstern, M.: A weakly universal cellular automaton in the hyperbolic 3D space with three states. In: Discrete Mathematics and Theoretical Computer Science Proceedings, Automata 2010, pp. 91–110 (2010)

    Google Scholar 

  16. Margenstern, M.: An Algorithmic Approach to Tilings of Hyperbolic Spaces: 10 Years Later. In: Gheorghe, M., Hinze, T., Păun, G., Rozenberg, G., Salomaa, A. (eds.) CMC 2010. LNCS, vol. 6501, pp. 37–52. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  17. Margenstern, M.: Towards the Frontier between Decidability and Undecidability for Hyperbolic Cellular Automata. In: Kučera, A., Potapov, I. (eds.) RP 2010. LNCS, vol. 6227, pp. 120–132. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  18. Margenstern, M.: An Upper Bound on the Number of States for a Strongly Universal Hyperbolic Cellular Automaton on the Pentagrid. Turku Center for Computer Science (2010), ISBN 978-952-12-2503-1; Proceedings of JAC 2010, pp. 168–179 (2010)

    Google Scholar 

  19. Margenstern, M.: A New Weakly Universal Cellular Automaton in the 3D Hyperbolic Space with Two States. In: Delzanno, G., Potapov, I. (eds.) RP 2011. LNCS, vol. 6945, pp. 205–217. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  20. Margenstern, M., Skordev, G.: Tools for devising cellular automata in the hyperbolic 3D space. Fundamenta Informaticae 58(2), 369–398 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Margenstern, M., Song, Y.: A universal cellular automaton on the ternary heptagrid. Electronic Notes in Theoretical Computer Science 223, 167–185 (2008)

    Article  Google Scholar 

  22. Margenstern, M., Song, Y.: A new universal cellular automaton on the pentagrid. Parallel Processing Letters 19(2), 227–246 (2009)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  24. Neary, T., Woods, D.: Four Small Universal Turing Machines. Fundamenta Informaticae 91(1), 105–126 (2009)

    MathSciNet  Google Scholar 

  25. Rogozhin, Y.: Small Universal Turing Machines. Theoretical Computer Science 168(2), 215–240 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stewart, I.: A Subway Named Turing. Mathematical Recreations in Scientific American, pp. 90–92 (1994)

    Google Scholar 

  27. Turing, A.M.: On computable real numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Ser. 2 42, 230–265 (1936)

    Article  Google Scholar 

  28. Wolfram, S.: A new kind of science. Wolfram Media, Inc. (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Lorraine LITA EA 3097, UFR MIM, Campus du Saulcy, 57045, Metz Cédex 1, France

    Maurice Margenstern

Authors
  1. Maurice Margenstern
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire d’Informatique, Fondamentale d’Orléan et Départment d’informatique, UFR Sciences, Université d’Orléans, 45067, Orléans Cedex 2, France

    Jérôme Durand-Lose

  2. Department of Mathematics and Statistics, University of South Florida, 33620, Tampa, FL, USA

    Nataša Jonoska

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Margenstern, M. (2012). Universality and the Halting Problem for Cellular Automata in Hyperbolic Spaces: The Side of the Halting Problem. In: Durand-Lose, J., Jonoska, N. (eds) Unconventional Computation and Natural Computation. UCNC 2012. Lecture Notes in Computer Science, vol 7445. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32894-7_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-32894-7_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32893-0

  • Online ISBN: 978-3-642-32894-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation