Global properties of 2D cellular automata: some complexity results

  • Bruno Durand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 711)


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.


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  1. 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. 2.
    R. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 66, 1966.Google Scholar
  3. 3.
    J. Kari. Reversibility and surjectivity problems of cellular automata. to appear in Journal of Computer and System Sciences.Google Scholar
  4. 4.
    J. Kari. Reversability of 2d cellular automata is undecidable. Physica, D 45:379–385, 1990.Google Scholar
  5. 5.
    E.F. Moore. Machine models of self-reproduction. Proc. Symp. Apl. Math., 14:13–33, 1962.Google Scholar
  6. 6.
    J. Myhill. The converse to Moore's garden-of-eden theorem. Proc. Am. Math. Soc., 14:685–686, 1963.Google Scholar
  7. 7.
    D. Richardson. Tesselations with local transformations. Journal of Computer and System Sciences, 6:373–388, 1972.Google Scholar
  8. 8.
    R.M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12:177–209, 1971.Google Scholar
  9. 9.
    K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5:19–30, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Bruno Durand
    • 1
  1. 1.Laboratoire de l'Informatique du Parallélisme Unité de Recherche Associée 1398 du CNRSEcole Normale Supérieure de LyonLyon Cedex 07France

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