Abstract
In this paper, we prove that given any \(\it \Pi^0_1\) subset P of {0,1}ℕ there is a tileset τ with a countable set of configurations C such that P is recursively homeomorphic to C ∖ U where U is a computable set of configurations. As a consequence, if P is countable, this tileset has the exact same set of Turing degrees.
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References
Ballier, A., Durand, B., Jeandel, E.: Structural aspects of tilings. In: 25th International Symposium on Theoretical Aspects of Computer Science (STACS) (2008)
Berger, R.: The Undecidability of the Domino Problem. PhD thesis, Harvard University (1964)
Berger, R.: The Undecidability of the Domino Problem. Memoirs of the American Mathematical Society, vol. 66. The American Mathematical Society, Providence (1966)
Cenzer, D., Dashti, A., King, J.L.F.: Computable symbolic dynamics. Mathematical Logic Quarterly 54(5), 460–469 (2008)
Cenzer, D., Dashti, A., Toska, F., Wyman, S.: Computability of Countable Subshifts. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 88–97. Springer, Heidelberg (2010)
Cenzer, D., Remmel, J.B.: \(\Pi_1^0\) classes in mathematics. In: Handbook of Recursive Mathematics - Volume 2: Recursive Algebra, Analysis and Combinatorics, ch. 13. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 623–821. Elsevier, Amsterdam (1998)
Cenzer, D., Remmel, J.: Effectively Closed Sets. ASL Lecture Notes in Logic (2011) (in preparation)
Dashti, A.: Effective Symbolic Dynamics. PhD thesis, University of Florida (2008)
Durand, B., Levin, L.A., Shen, A.: Complex tilings. Journal of Symbolic Logic 73(2), 593–613 (2008)
Hanf, W.: Non Recursive Tilings of the Plane I. Journal of Symbolic Logic 39(2), 283–285 (1974)
Kechris, A.S.: Classical descriptive set theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)
Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, New York (1995)
Myers, D.: Non Recursive Tilings of the Plane II. Journal of Symbolic Logic 39(2), 286–294 (1974)
Robinson, R.M.: Undecidability and Nonperiodicity for Tilings of the Plane. Inventiones Math. 12 (1971)
Simpson, S.: Mass Problems Associated with Effectively Closed Sets (in preparation)
Simpson, S.G.: Medvedev Degrees of 2-Dimensional Subshifts of Finite Type. Ergodic Theory and Dynamical Systems (2011)
Wang, H.: Proving theorems by Pattern Recognition II. Bell Systems Technical Journal 40, 1–41 (1961)
Wang, H.: Dominoes and the ∀ ∃ ∀ case of the decision problem. Mathematical Theory of Automata, 23–55 (1963)
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Jeandel, E., Vanier, P. (2011). \(\it \Pi^0_1\) Sets and Tilings. In: Ogihara, M., Tarui, J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20877-5_24
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DOI: https://doi.org/10.1007/978-3-642-20877-5_24
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