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Turing degrees of multidimensional SFTs. (English) Zbl 1417.03241
Summary: In this paper, we are interested in computability aspects of subshifts and in particular Turing degrees of two-dimensional subshifts of finite type (SFTs) (i.e., tilings). To be more precise, we prove that, given any $$\Pi_1^0$$ class $$P$$ of $$\{0,1\}^\mathbb N$$, there is an SFT $$X$$ such that $$P\times\mathbb Z^2$$ is recursively homeomorphic to $$X\setminus U$$, where $$U$$ is a computable set of points. As a consequence, if $$P$$ contains a computable member, $$P$$ and $$X$$ have the exact same set of Turing degrees. On the other hand, we prove that, if $$X$$ contains only non-computable members, some of its members always have different but comparable degrees. This gives a fairly complete study of Turing degrees of SFTs.

##### MSC:
 03D28 Other Turing degree structures 37B10 Symbolic dynamics 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry)
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##### References:
 [1] A. Ballier, B. Durand, E. Jeandel, Structural aspects of tilings, in: 25th International Symposium on Theoretical Aspects of Computer Science, STACS, 2008. · Zbl 1258.05023 [2] Ballier, A.; Jeandel, E., Computing (or not) quasi-periodicity functions of tilings, (Symposium on Cellular Automata (JAC), (2010)), 54-64 [3] R. Berger, The undecidability of the domino problem, Ph.D. Thesis, Harvard University, 1964. · Zbl 0199.30802 [4] Berger, R., (The Undecidability of the Domino Problem, Memoirs of the American Mathematical Society, vol. 66, (1966)) · Zbl 0199.30802 [5] Cenzer, D.; Dashti, A.; King, J. L.F., Computable symbolic dynamics, Mathematical Logic Quarterly, 54, 5, 460-469, (2008) · Zbl 1170.03029 [6] D. Cenzer, A. Dashti, F. Toska, S. Wyman, Computability of countable subshifts in one dimension. Theory of Computing Systems, 1-2010.1007/s00224-011-9358-z. 2012. http://dx.doi.org/10.1007/s00224-011-9358-z. · Zbl 1285.03055 [7] Cenzer, D.; Remmel, J., $$\Pi_1^0$$ classes in mathematics, (Handbook of Recursive Mathematics - Volume 2: Recursive Algebra, Analysis and Combinatorics, Studies in Logic and the Foundations of Mathematics, vol. 139, (1998), Elsevier), 623-821, Ch. 13 · Zbl 0941.03044 [8] D. Cenzer, J. Remmel, Effectively closed sets, ASL Lecture Notes in Logic, 2011 (in preparation). · Zbl 1140.03026 [9] Culik, K., An aperiodic set of 13 Wang tiles, Discrete Mathematics, 160, 245-251, (1996) · Zbl 0865.05033 [10] A. Dashti, Effective symbolic dynamics, Ph.D. Thesis, University of Florida, 2008. · Zbl 1262.37008 [11] Durand, B., Tilings and quasiperiodicity, Theoretical Computer Science, 221, 1-2, 61-75, (1999) · Zbl 1062.05502 [12] Durand, B.; Levin, L. A.; Shen, A., Complex tilings, Journal of Symbolic Logic, 73, 2, 593-613, (2008) · Zbl 1141.03021 [13] Hanf, W., Non recursive tilings of the plane I, Journal of Symbolic Logic, 39, 2, 283-285, (1974) · Zbl 0299.02054 [14] Jockusch, C. G.; Soare, R. I., Degrees of members of $$\Pi_1^0$$ classes, Pacific Journal of Mathematics, 40, 605-616, (1972) · Zbl 0209.02201 [15] Jockusch, C. G.; Soare, R. I., $$\Pi_1^0$$ classes and degrees of theories, Transactions of the American Mathematical Society, 173, 33-56, (1972) · Zbl 0262.02041 [16] Kari, J., A small aperiodic set of Wang tiles, Discrete Mathematics, 160, 259-264, (1996) · Zbl 0861.05017 [17] Kechris, A. S., Classical descriptive set theory, (Graduate Texts in Mathematics, vol. 156, (1995), Springer-Verlag New York) · Zbl 0819.04002 [18] Lind, D., Multi-dimensional symbolic dynamics, Proceedings of Symposia in Applied Mathematics, 60, 81-120, (2004) [19] Lind, D.; Marcus, B., An introduction to symbolic dynamics and coding, (1995), Cambridge University Press New York, NY, USA · Zbl 1106.37301 [20] Myers, D., Non recursive tilings of the plane II, Journal of Symbolic Logic, 39, 2, 286-294, (1974) · Zbl 0299.02055 [21] Robinson, R. M., Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, 12, (1971) · Zbl 0197.46801 [22] Simpson, S. G., Mass problems associated with effectively closed sets, Tohoku Mathematical Journal. Second Series, 63, 4, 489-517, (2011-2012) · Zbl 1246.03064 [23] Simpson, S. G., Medvedev degrees of 2-dimensional subshifts of finite type, Ergodic Theory and Dynamical Systems, (2011) [24] Wang, H., Proving theorems by pattern recognition II, Bell Systems Technical Journal, 40, 1-41, (1961) [25] Wang, H., Dominoes and the $$\forall \exists \forall$$ case of the decision problem, Mathematical Theory of Automata, 23-55, (1963)
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