Skip to main content Accessibility help

Characterizations of periods of multi-dimensional shifts



We show that the sets of periods of multi-dimensional shifts of finite type are precisely the sets of integers of the complexity class NP. We also show that the functions counting their number are the functions of #P. We also give characterizations of some other notions of periodicity in terms of space complexity. We finish the paper by giving some characterizations for sofic and effective subshifts.



Hide All
[AB09]Arora, S. and Barak, B.. Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge, 2009.
[AS09]Aubrun, N. and Sablik, M.. An order on sets of tilings corresponding to an order on languages. 26th Int. Symp. on Theoretical Aspects of Computer Science (Leibniz Int. Proc. in Informatics, 3). Eds. Albers, S. and Marion, J.-Y.. Schloss Dagstuhl–Leibniz Zentrum für Informatik, Dagstuhl, Germany, 2009, pp. 99110.
[AS13]Aubrun, N. and Sablik, M.. Simulation of effective subshifts by two-dimensional SFT and a generalization. Acta Appl. Math. (2013), to appear.
[BDG88]Balcazar, J., Diaz, J. and Gabarro, J.. Structural Complexity I. Springer, Berlin, 1988.
[Ber64]Berger, R.. The undecidability of the domino problem. PhD Thesis, Harvard University, 1964.
[Ber66]Berger, R.. The Undecidability of the Domino Problem (Memoirs of the American Mathematical Society, 66). American Mathematical Society, Providence, RI, 1966.
[Ber95]Berstel, J.. Axel Thue’s papers on repetitions in words: a translation. Publications du LaCIM 20 (1995).
[Bor08]Borchert, B.. Formal language characterizations of P, NP, and PSPACE. J. Autom. Lang. Comb. 13 (3/4) (2008), 161183.
[Cha08]Chaitin, G.. The halting probability via Wang tiles. Fund. Inform. 86 (4) (2008), 429433.
[CM06]Carayol, A. and Meyer, A.. Context-sensitive languages, rational graphs and determinism. Log. Methods Comput. Sci. (2006), 124.
[DJMM]Durand, A., Jones, N. D., Makowsky, J. A. and More, M.. Fifty years of the spectrum problem: survey and new results. Preprint, arXiv:0907.5495v1.
[DRS10]Durand, B., Romashchenko, A. and Shen, A.. Effective closed subshifts in 1D can be implemented in 2D. Fields of Logic and Computation (Lecture Notes in Computer Science, 6300). Springer, Berlin, 2010, pp. 208226.
[EB90]van Emde Boas, P.. Machine models and simulations. Handbook of Theoretical Computer Science vol A: Algorithms and Complexity. Ed. Leeuwen, J. V.. MIT Press, Cambridge, MA, 1990, pp. 166; Ch. 1.
[Flo56]Flores, I.. Reflected number systems. IRE Trans. Electron. Comput. 5 (2) (1956), 7982.
[HM10]Hochman, M. and Meyerovitch, T.. A characterization of the entropies of multidimensional shifts of finite type. Ann. Math. (2) 171 (3) (2010), 20112038.
[Imm88]Immerman, N.. Nondeterministic space is closed under complementation. SIAM J. Comput. 17 (5) (1988), 935938.
[JS74]Jones, N. D. and Selman, A. L.. Turing machines and the spectra of first-order formulas. J. Symbolic Logic 39 (1) (1974), 139150.
[JV10]Jeandel, E. and Vanier, P.. Periodicity in tilings. Developments in Language Theory (Lecture Notes in Computer Science, 6224). Eds. Gao, Y., Lu, H., Seki, S. and Yu, S.. Springer, Berlin, 2010, pp. 243254.
[Kar92]Kari, J.. The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21 (3) (1992), 571586.
[Kar94]Kari, J.. Reversibility and surjectivity problems of cellular automata. J. Comput. Syst. Sci. 48 (1) (1994), 149182.
[Kar96]Kari, J.. A small aperiodic set of Wang tiles. Discrete Math. 160 (1996), 259264.
[Knu05]Knuth, D. E.. Generating all tuples and permutations. The Art of Computer Programming. Vol. 4 Fasc. 2. Addison-Wesley, Upper Saddle River, NJ, 2005.
[Lin04]Lind, D.. Multidimensional symbolic dynamics. Symbolic Dynamics and Its Applications (Proceedings of Symposia in Applied Mathematics, 60). Ed. Williams, S. G.. American Mathematical Society, San Diego, CA, 2004, pp. 6180.
[Lin96]Lind, D.. A zeta function for ${ \mathbb{Z} }^{d} $-actions. Proceedings of Warwick Symposium on ${ \mathbb{Z} }^{d} $actions (LMS Lecture Notes Series, 228). Cambride University Press, Cambridge, 1996, pp. 433–450.
[LM95]Lind, D. A. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, New York, 1995.
[Mor21]Morse, H. M.. Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc. 22 (1) (1921), 84100.
[Rob71]Robinson, R. M.. Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12 (1971).
[Rog87]Rogers, H.. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, MA, 1987.
[Sim09]Simpson, S. G.. Medvedev degrees of 2-dimensional subshifts of finite type. Ergod. Th. & Dynam. Sys. (2012).
[Sze87]Szelepcsényi, R.. The method of forcing for nondeterministic automata. Bull. EATCS 33 (1987), 96100.
[Thu12]Thue, A.. Über die gegenseitige Lage gleicher Teiler gewisser Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat.-Nat. Kl. 12 (1912), transl. in [Ber95].

Related content

Powered by UNSILO

Characterizations of periods of multi-dimensional shifts



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.