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Tilings: simulation and universality

  • GREGORY LAFITTE (a1) and MICHAEL WEISS (a2)

Abstract

Wang tiles are unit-size squares with coloured edges. In this paper, we approach one aspect of the study of tiling computability: the quest for a universal tile set. Using a complex construction, based on Robinson's classical construction and its different modifications, we build a tile set (pronounced ayin) that almost always simulates any tile set. By way of Banach–Mazur games on tilings topological spaces, we prove that the set of -tilings that do not satisfy the universality condition is meagre in the set of -tilings.

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Allauzen, C. and Durand, B. (1997) Appendix A: Tiling problems. In: Börger, E., Grädel, E. and Gurevich, Y. (eds.) The classical decision problem, Springer-Verlag 407420.
Berger, R. (1966) The undecidability of the domino problem. Mem. Amer. Math Soc. 66 172.
Cervelle, J. and Durand, B. (2004) Tilings: recursivity and regularity. Theoretical Computer Science 310 (1-3)469477.
Delorme, M., Mazoyer, J., Ollinger, N. and Theyssier, G. (2010) Bulking I: an abstract theory of bulking. Available at http://hal.archives-ouvertes.fr/hal-00451732/fr/.
Durand, B. (1999) Tilings and quasiperiodicity. Theoretical Computer Science 221 (1-2)6175.
Durand, B., Levin, L. A. and Shen, A. (2001) Complex tilings. In: Proceedings of the thirty-third annual ACM Symposium on Theory of Computation (STOC), ACM 732739.
Durand, B., Levin, L. A. and Shen, A. (2004) Local rules and global order. Mathematical Intelligencer 27 (1)6468.
Durand, B., Romashchenko, A. E. and Shen, A. (2008) Fixed point and aperiodic tilings. In: Ito, M. and Toyama, M. (eds.) Proceedings of the conference Developments in Language Theory. Springer-Verlag Lecture Notes in Computer Science 5257 276–288.
Hanf, W. P. (1974) Nonrecursive tilings of the plane. I. Journal of Symbolic Logic 39 (2)283285.
Lafitte, G. and Weiss, M. (2007) Universal tilings. In: Thomas, W. and Weil, P. (eds.) Twenty-Fourth International Symposium on Theoretical Aspects of Computer Science (STACS'2007). Springer-Verlag Lecture Notes in Computer Science 4393 367–380.
Lafitte, G. and Weiss, M. (2008a) Computability of tilings. In: Ausiello, G., Karhumäki, J., Mauri, G. and Ong, C.-H. L. (eds.) Proceedings of the Fifth IFIP International Conference on Theoretical Computer Science (IFIP-TCS'2008). IFIP Conference Proceedings, Springer-Verlag 273 187–201.
Lafitte, G. and Weiss, M. (2008b) Simulations between tilings. In: Beckmann, A., Dimitracopoulos, C. and Löwe, B. (eds.) Logic and Theory of Algorithms, Fourth Conference on Computability in Europe (CiE'2008), University of Athens 264–273.
Lafitte, G. and Weiss, M. (2008c) A topological study of tilings. In: Agrawal, M., Du, D., Duan, Z. and Li, A. (eds.), Fifth Annual Conference on Theory and Applications of Models of Computation (TAMC'2008). Springer-Verlag Lecture Notes in Computer Science 4978 375–387.
Lafitte, G. and Weiss, M. (2009) An almost totally universal tile set. In: Chen, J. and Cooper, S. B. (eds.) Sixth Annual Conference on Theory and Applications of Models of Computation (TAMC'2009). Springer-Verlag Lecture Notes in Computer Science 5532 271–280.
Myers, D. (1974) Nonrecursive tilings of the plane. II. Journal of Symbolic Logic 39 (2)286294.
Oxtoby, J. (1957) The Banach–Mazur game and Banach category theorem. In: Contribution to the theory of games, Volume III. Annals of Mathematical Studies, Princeton University Press 39159163.
Robinson, R. M. (1971) Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12 (3)177209.
Wang, H. (1961) Proving theorems by pattern recognition II. Bell System Tech. Journal 40 (1)141.

Tilings: simulation and universality

  • GREGORY LAFITTE (a1) and MICHAEL WEISS (a2)

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