Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Mathematical Foundations of Computer Science

MFCS 1993: Mathematical Foundations of Computer Science 1993 pp 433–441Cite as

Global properties of 2D cellular automata: some complexity results

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 711))

Abstract

In this paper, we prove the co-NP-completeness of the following decision problem: “given a 2-dimensional cellular automaton A (even with Von Neumann neighborhood), is A injective when restricted to finite configurations not greater than its length?” In order to prove this result, we introduce two decision problems concerning respectively Turing Machines and tilings that we prove NP-complete. Then, we transform problems concerning tilings into problems concerning cellular automata.

This work was partially supported by the Esprit Basic Research Action “Algebraic and Semantical Methods In Computer Science” and by the PRC “Mathématique et Informatique”.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Amoroso and Y.N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. J. Comp. Syst. Sci., 6:448–464, 1972.

    Google Scholar 

  2. R. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 66, 1966.

    Google Scholar 

  3. J. Kari. Reversibility and surjectivity problems of cellular automata. to appear in Journal of Computer and System Sciences.

    Google Scholar 

  4. J. Kari. Reversability of 2d cellular automata is undecidable. Physica, D 45:379–385, 1990.

    Google Scholar 

  5. E.F. Moore. Machine models of self-reproduction. Proc. Symp. Apl. Math., 14:13–33, 1962.

    Google Scholar 

  6. J. Myhill. The converse to Moore's garden-of-eden theorem. Proc. Am. Math. Soc., 14:685–686, 1963.

    Google Scholar 

  7. D. Richardson. Tesselations with local transformations. Journal of Computer and System Sciences, 6:373–388, 1972.

    Google Scholar 

  8. R.M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12:177–209, 1971.

    Google Scholar 

  9. K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5:19–30, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de l'Informatique du Parallélisme Unité de Recherche Associée 1398 du CNRS, Ecole Normale Supérieure de Lyon, 46, Allée d'Italie, 69364, Lyon Cedex 07, France

    Bruno Durand

Authors
  1. Bruno Durand
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Andrzej M. Borzyszkowski Stefan Sokołowski

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Durand, B. (1993). Global properties of 2D cellular automata: some complexity results. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_35

Download citation

  • DOI: https://doi.org/10.1007/3-540-57182-5_35

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57182-7

  • Online ISBN: 978-3-540-47927-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation