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”.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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.
R. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 66, 1966.
J. Kari. Reversibility and surjectivity problems of cellular automata. to appear in Journal of Computer and System Sciences.
J. Kari. Reversability of 2d cellular automata is undecidable. Physica, D 45:379–385, 1990.
E.F. Moore. Machine models of self-reproduction. Proc. Symp. Apl. Math., 14:13–33, 1962.
J. Myhill. The converse to Moore's garden-of-eden theorem. Proc. Am. Math. Soc., 14:685–686, 1963.
D. Richardson. Tesselations with local transformations. Journal of Computer and System Sciences, 6:373–388, 1972.
R.M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12:177–209, 1971.
K. Sutner. De Bruijn graphs and linear cellular automata. Complex Systems, 5:19–30, 1991.
Author information
Authors and Affiliations
Editor information
Rights 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