Complex tilings
Bruno Durand, Leonid A. Levin, and Alexander Shen
Source: J. Symbolic Logic Volume 73, Issue 2
(2008), 593-613.
Abstract
We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with 𝒪(n) Kolmogorov complexity of its (n×n)-squares. We construct tile sets for which this bound is tight: all (n×n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings—that we also develop in terms of Turing degrees of unsolvability.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208359062
Digital Object Identifier: doi:10.2178/jsl/1208359062
Mathematical Reviews number (MathSciNet): MR2414467
Zentralblatt MATH identifier: 1141.03021
References
Cyril Allauzen and Bruno Durand, Appendix A: ``Tiling problems'', In [bgg?], pp. 407--420.
Robert Berger, The undecidability of the domino problem, Memoirs of the American Mathematical Society, vol. 66, 1966.
Mathematical Reviews (MathSciNet): MR216954
Egon Börger, Erich Grädel, and Yuri Gurevich (editors), The classical decision problem, Springer-Verlag, 1996.
Mathematical Reviews (MathSciNet): MR1482227
Julien Cervelle and Bruno Durand, Tilings: Recursivity and regularity, STACS'00, Lecture Notes in Computer Science, vol. 1770, Springer Verlag, 2000.
Mathematical Reviews (MathSciNet): MR1781757
Zentralblatt MATH: 0953.03055
Digital Object Identifier: doi:10.1007/3-540-46541-3_41
Bruno Durand, Tilings and quasiperiodicity, Theoretical Computer Science, vol. 221 (1999), pp. 61--75.
Mathematical Reviews (MathSciNet): MR1700820
Digital Object Identifier: doi:10.1016/S0304-3975(99)00027-4
Zentralblatt MATH: 1062.05502
Bruno Durand, Leonid Levin, and Alexander Shen, Complex tilings, STOC, 2001, Extended version: family http://www.arxiv.org/cs.CC/0107008, pp. 732--739.
Mathematical Reviews (MathSciNet): MR2120376
Digital Object Identifier: doi:10.1145/380752.380880
--------, Local rules and global order, or aperiodic tilings, Mathematical Intelligencer, vol. 27 (2004), no. 1, pp. 64--68.
Mathematical Reviews (MathSciNet): MR2145827
Peter Gacs, Reliable cellular automata with self-organization, Journal of Statistical Physics, vol. 103 (2001), no. 1/2, pp. 45--267.
Mathematical Reviews (MathSciNet): MR1828729
Digital Object Identifier: doi:10.1023/A:1004823720305
Zentralblatt MATH: 0973.68158
Yuri Gurevich, Average case completeness, Journal of Computer and System Sciences, vol. 42 (1991), pp. 346--398.
Mathematical Reviews (MathSciNet): MR1110429
Digital Object Identifier: doi:10.1016/0022-0000(91)90007-R
Zentralblatt MATH: 0825.68420
Yuri Gurevich and Igor Koriakov, A remark on Berger's paper on the domino problem, Siberian Journal of Mathematics, vol. 13 (1972), pp. 459--463, in Russian.
Mathematical Reviews (MathSciNet): MR295909
William Hanf, Nonrecursive tilings of the plane. I, Journal of Symbolic Logic, vol. 39 (1974), no. 2, pp. 283--285.
Mathematical Reviews (MathSciNet): MR363855
Digital Object Identifier: doi:10.2307/2272640
Project Euclid: euclid.jsl/1183739044
Zentralblatt MATH: 0299.02054
Kevin Ingersent, Matching rules for quasicrystalline tilings, Quasicrystals. The state of the art, World Scientific, 1991, pp. 185--212.
Mathematical Reviews (MathSciNet): MR1749510
Leonid Levin, Average case complete problems, SIAM Journal on Computing, vol. 15 (1986), no. 1, pp. 285--286.
Mathematical Reviews (MathSciNet): MR822205
Digital Object Identifier: doi:10.1137/0215020
Zentralblatt MATH: 0589.68032
--------, Aperiodic tilings: Breaking translational symmetry, The Computer Journal, vol. 48 (2005), no. 6, pp. 642--645, on-line: family http://www.arxiv.org/cs.DM/0409024.
Ming Li and Paul Vitányi, An introduction to Kolmogorov complexity and its applications, second ed., Springer-Verlag, 1997.
Mathematical Reviews (MathSciNet): MR1438307
Zentralblatt MATH: 0866.68051
Andrej Muchnik personal communication, 2000.,
Dale Myers, Nonrecursive tilings of the plane. ii, Journal of Symbolic Logic, vol. 39 (1974), no. 2, pp. 286--294.
Mathematical Reviews (MathSciNet): MR363856
Digital Object Identifier: doi:10.2307/2272641
Project Euclid: euclid.jsl/1183739045
Zentralblatt MATH: 0299.02055
Piergiorgio Odifreddi, Classical recursion theory, North-Holland, 1989.
Mathematical Reviews (MathSciNet): MR982269
Zentralblatt MATH: 0661.03029
Raphael Robinson, Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, vol. 12 (1971), pp. 177--209.
Mathematical Reviews (MathSciNet): MR297572
Digital Object Identifier: doi:10.1007/BF01418780
Zentralblatt MATH: 0197.46801
Hao Wang, Proving theorems by pattern recognition II, Bell System Technical Journal, vol. 40 (1961), pp. 1--41.
--------, Dominoes and the $\forall\exists\forall$-case of the decision problem, Proceedings of the Symposium on Mathematical Theory of Automata, Brooklyn Polytechnic Institute, New York, 1962, pp. 23--55.
