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Hardness of conjugacy, embedding and factorization of multidimensional subshifts. (English) Zbl 1328.68074
Summary: Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts in dimensions \(d > 1\) for subshifts of finite type and sofic shifts and in dimensions \(d \geq 1\) for effective shifts. In particular, we prove that the conjugacy, factorization and embedding problems are \(\Sigma_3^0\)-complete for sofic and effective subshifts and that they are \(\Sigma_1^0\)-complete for SFTs, except for factorization which is also \(\Sigma_3^0\)-complete.

MSC:
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
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[1] Aubrun, N.; Sablik, M., Simulation of effective subshifts by two-dimensional subshifts of finite type, Acta Appl. Math., (2013) · Zbl 1283.37014
[2] Berger, R., The undecidability of the domino problem, (1964), Harvard University, Ph.D. thesis
[3] Boyer, L.; Theyssier, G., On factor universality in symbolic spaces, (MFCS, (2010)), 209-220 · Zbl 1287.68115
[4] Boyle, M., Lower entropy factors of sofic systems, Ergod. Theory Dyn. Syst., 3, 541-551, (1983) · Zbl 0529.28014
[5] Boyle, M., Open problems in symbolic dynamics, Contemp. Math., 469, 69-118, (2008) · Zbl 1158.37007
[6] Hedlund, G. A., Endomorphisms and automorphisms of the shift dynamical system, Theory Comput. Syst., 3, 4, 320-375, (1969) · Zbl 0182.56901
[7] Hochman, M., A note on universality in multidimensional symbolic dynamics, Discrete Contin. Dyn. Syst., Ser. S, 2, 2, (2009) · Zbl 1175.37019
[8] Hochman, M., On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math., 176, 1, (Apr. 2009)
[9] Jeandel, E.; Vanier, P., \(\operatorname{\Pi}_1^0\) sets and tilings, (Theory and Applications of Models of Computation, TAMC, Lecture Notes in Computer Science, vol. 6648, (2011)), 230-239 · Zbl 1333.03109
[10] Jeandel, E.; Vanier, P., Hardness of conjugacy, embedding and factorization of multidimensional subshifts of finite type, (Portier, N.; Wilke, T., STACS, LIPIcs, vol. 20, (2013), Schloss Dagstuhl - Leibniz-Zentrum für Informatik), 490-501 · Zbl 1354.68098
[11] Kozen, D., Theory of computation, (2006), Springer New York
[12] Krieger, W., On the subsystems of topological Markov chains, Ergod. Theory Dyn. Syst., 2, 02, 195-202, (1982) · Zbl 0508.54032
[13] Lind, D.; Marcus, B., An introduction to symbolic dynamics and coding, (1995), Cambridge University Press New York, NY, USA · Zbl 1106.37301
[14] Lind, D. A., Multi-dimensional symbolic dynamics, (Williams, S. G., Symbolic Dynamics and Its Applications, Proceedings of Symposia in Applied Mathematics, vol. 60, (2004), American Mathematical Society), 61-79 · Zbl 1070.37009
[15] Robinson, R. M., Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12, (1971) · Zbl 0197.46801
[16] Rogers, H., Theory of recursive functions and effective computability, (1987), MIT Press Cambridge, MA, USA · Zbl 0183.01401
[17] Wang, H., Proving theorems by pattern recognition II, Bell Syst. Tech. J., 40, 1-41, (1961)
[18] Williams, R. F., Classification of subshifts of finite type, Ann. Math., 98, 120-153, (1973) · Zbl 0282.58008
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