Abstract
In this paper, we prove the co-RNP-completeness (RNP=Random NP) of the following decision problem: “Given a 2-dimensional cellular automaton A, is A reversible when restricted to finite configurations extending a given row?” In order to prove this result, we introduce a polynomial reduction from problems concerning finite tilings into problems concerning cellular automata. Then we add to tile sets and cellular automata two probability functions and we prove that these problems are not only co-NP-complete, but co-RNP-complete too.
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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.
S. Ben-David, B. Chor, O. Goldreich, and M. Luby. On the theory of average case complexity. In STOC, pages 204–216, 1989.
R. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, 66, 1966.
A. Blass and Y. Gurevich. Randomizing reductions of search problems. SIAM Journal on Computing, 22(5):949–975, 1993.
A. Blass and Y. Gurevich. Matrix transformation is complete for the average case. SIAM Journal on computing, (to appear).
B. Durand. Global properties of 2D cellular automata: some complexity results. In MFCS'93, Lecture Notes in Computer Science. Springer Verlag, September 1993.
B. Durand. Undecidability of the surjectivity problem for 2D cellular automata: A simplified proof. In FCT'93, Lecture Notes in Computer Science. Springer Verlag, August 1993.
B. Durand. Automates cellulaires: réversibilité et complexité. PhD thesis, Ecole Normale Supérieure de Lyon, 1994.
B. Durand. Inversion of 2d cellular automata: some complexity results. Theoretical Computer Science, (to appear).
B. Durand. The surjectivity problem for 2D cellular automata. Journal of Computer and Systems Science, (to appear).
Y. Gurevich. Complete and incomplete randomized NP problems. In FOCS, 1987.
Y. Gurevich. Average case completeness. Journal of Computer and System Sciences, 42:346–398, 1991.
J. Kari. Reversability of 2D cellular automata is undecidable. Physica, D 45:379–385, 1990.
J. Kari. Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences, 48:149–182, 1994.
L. Levin. Average case complete problems. SIAM J. Comput, 15(1):285–286, February 1986.
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.
T. Toffoli and N. Margolus. Invertible cellular automata: a review. Physica, D 45:229–253, 1990.
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© 1995 Springer-Verlag Berlin Heidelberg
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Durand, B. (1995). A random NP-complete problem for inversion of 2D cellular automata. In: Mayr, E.W., Puech, C. (eds) STACS 95. STACS 1995. Lecture Notes in Computer Science, vol 900. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59042-0_65
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DOI: https://doi.org/10.1007/3-540-59042-0_65
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