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Characterizations of periods of multi-dimensional shifts. (English) Zbl 1336.37015
Following recent works that characterize dynamical invariants of multi-dimensional shifts of finite type (SFT) in computability-theoretical terms, this article gives dynamical characterizations with the original help of complexity theory.
Namely, if numbers are all represented unarily:
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the sets of strong periods of SFT are exactly those in $$\mathsf{NP}_1$$ (unary languages decidable in polynomial time by a nondeterministic Turing machine);
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the sets of horizontal periods of SFT are exactly those in $$\mathsf{NSPACE}_1(n)$$ (unary languages decidable in linear space by a nondeterministic Turing machine);
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the sets of periodicity vectors of 2D SFT are exactly the sets of pairs of integers in $$\mathsf{NSPACE}_1(n)$$;
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the sets of functions counting the points that have a given strong period, in SFT, are exactly those in $$\#\mathsf{P}$$ (counting accepting paths in a nondeterministic polynomial-time computation).
In contrast, it is eventually shown that the sets of periodicity vectors for effective or sofic subshifts are exactly the sets in $$\Pi^0_1$$ (co-recursively enumerable).
The constructions involve a smart combination classical marking widgets, a Turing machine embedding, and an aperiodic deterministic tile set.

##### MSC:
 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 37E15 Combinatorial dynamics (types of periodic orbits) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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##### References:
 [1] Berstel, Publications du LaCIM 20 (1995) [2] Borchert, J. Autom. Lang. Comb. 13 pp 161– (2008) [3] DOI: 10.4007/annals.2010.171.2011 · Zbl 1192.37022 · doi:10.4007/annals.2010.171.2011 [4] Berger, The Undecidability of the Domino Problem (1966) · Zbl 0199.30802 [5] van Emde Boas, Handbook of Theoretical Computer Science vol A: Algorithms and Complexity pp 1– (1990) · Zbl 0900.68265 [6] DOI: 10.1007/978-3-642-15025-8_12 · Zbl 1287.37012 · doi:10.1007/978-3-642-15025-8_12 [7] DOI: 10.1007/978-3-642-97062-7 · doi:10.1007/978-3-642-97062-7 [8] Carayol, Log. Methods Comput. Sci. pp 1– (2006) [9] Aubrun, 26th Int. Symp. on Theoretical Aspects of Computer Science pp 99– (2009) [10] DOI: 10.1017/CBO9780511804090 · Zbl 1193.68112 · doi:10.1017/CBO9780511804090 [11] Simpson, Ergod. Th. & Dynam. Sys. (2012) [12] Rogers, Theory of Recursive Functions and Effective Computability (1987) [13] DOI: 10.1007/BF01418780 · Zbl 0197.46801 · doi:10.1007/BF01418780 [14] DOI: 10.1090/S0002-9947-1921-1501161-8 · doi:10.1090/S0002-9947-1921-1501161-8 [15] DOI: 10.1017/CBO9780511626302 · Zbl 1106.37301 · doi:10.1017/CBO9780511626302 [16] DOI: 10.1090/psapm/060/2078846 · doi:10.1090/psapm/060/2078846 [17] DOI: 10.1016/0012-365X(95)00120-L · Zbl 0861.05017 · doi:10.1016/0012-365X(95)00120-L [18] DOI: 10.1016/S0022-0000(05)80025-X · Zbl 0802.68090 · doi:10.1016/S0022-0000(05)80025-X [19] Knuth, The Art of Computer Programming. Vol. 4 Fasc. 2 (2005) · Zbl 1127.68068 [20] Thue, Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl. 12 (1912) [21] Szelepcsényi, Bull. EATCS 33 pp 96– (1987) [22] DOI: 10.1137/0221036 · Zbl 0761.68067 · doi:10.1137/0221036 [23] DOI: 10.1007/978-3-642-14455-4_23 · Zbl 1250.68109 · doi:10.1007/978-3-642-14455-4_23 [24] DOI: 10.2307/2272354 · Zbl 0288.02021 · doi:10.2307/2272354 [25] Chaitin, Fund. Inform. 86 pp 429– (2008) [26] DOI: 10.1137/0217058 · Zbl 0668.68056 · doi:10.1137/0217058 [27] DOI: 10.1109/TEC.1956.5219803 · doi:10.1109/TEC.1956.5219803
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