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Normal Sequences with Non-Maximal Automatic Complexity

Authors Liam Jordon , Philippe Moser

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Author Details

Liam Jordon
  • Department of Computer Science, Maynooth University, Maynooth, Ireland
Philippe Moser
  • Department of Computer Science, Maynooth University, Maynooth, Ireland

Funding

  • Jordon, Liam: Supported by the Irish Research Council’s Government of Ireland Postgraduate Scholarship Programme. Grant number: GOIPG/2017/1200.

Cite As Get BibTex

Liam Jordon and Philippe Moser. Normal Sequences with Non-Maximal Automatic Complexity. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.FSTTCS.2021.47

Abstract

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001 [Jeffrey O. Shallit and Ming-wei Wang, 2001]. We demonstrate that there exists a normal sequence T such that I(T) = 0 and S(T) ≤ 1/2, where I(T) and S(T) are the lower and upper automatic complexity rates of T respectively. We furthermore show that there exists a Champernowne sequence C, i.e. a sequence formed by concatenating all strings of length one followed by concatenating all strings of length two and so on, such that S(C) ≤ 2/3.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Automatic Complexity
  • finite-state complexity
  • normal sequences
  • Champernowne sequences
  • de Bruijn strings
  • Kolmogorov complexity

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References

  1. Verónica Becher, Olivier Carton, and Pablo Ariel Heiber. Normality and automata. J. Comput. Syst. Sci., 81(8):1592-1613, 2015. URL: https://doi.org/10.1016/j.jcss.2015.04.007.
  2. Verónica Becher and Pablo Ariel Heiber. Normal numbers and finite automata. Theor. Comput. Sci., 477:109-116, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.019.
  3. Émile Borel. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo, 27(1):247-271, 1909. URL: https://doi.org/10.1007/BF03019651.
  4. Nicolaas Govert De Bruijn. A combinatorial problem. In Proc. Koninklijke Nederlandse Academie van Wetenschappen, volume 49, pages 758-764, 1946. Google Scholar
  5. Cristian S. Calude, Kai Salomaa, and Tania Roblot. Finite state complexity. Theor. Comput. Sci., 412(41):5668-5677, 2011. URL: https://doi.org/10.1016/j.tcs.2011.06.021.
  6. Cristian S. Calude and Ludwig Staiger. Liouville, computable, Borel normal and Martin-löf random numbers. Theory Comput. Syst., 62(7):1573-1585, 2018. URL: https://doi.org/10.1007/s00224-017-9767-8.
  7. Cristian S. Calude, Ludwig Staiger, and Frank Stephan. Finite state incompressible infinite sequences. Inf. Comput., 247:23-36, 2016. URL: https://doi.org/10.1016/j.ic.2015.11.003.
  8. Olivier Carton and Pablo Ariel Heiber. Normality and two-way automata. Inf. Comput., 241:264-276, 2015. URL: https://doi.org/10.1016/j.ic.2015.02.001.
  9. D. G. Champernowne. The construction of decimals normal in the scale of ten. J. Lond. Math. Soc., s1-8(4):254-260, 1933. URL: https://doi.org/10.1112/jlms/s1-8.4.254.
  10. Jack J. Dai, James I. Lathrop, Jack H. Lutz, and Elvira Mayordomo. Finite-state dimension. Theor. Comput. Sci., 310(1-3):1-33, 2004. URL: https://doi.org/10.1016/S0304-3975(03)00244-5.
  11. David Doty and Philippe Moser. Finite-state dimension and lossy decompressors. CoRR, abs/cs/0609096, 2006. URL: http://arxiv.org/abs/cs/0609096.
  12. David Doty and Philippe Moser. Feasible depth. In CiE 2007: Computation and Logic in the Real World - Siena, Italy, volume 4497 of LNCS, pages 228-237. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-73001-9_24.
  13. Harold Fredricksen and Irving J. Kessler. An algorithm for generating necklaces of beads in two colors. Discret. Math., 61(2-3):181-188, 1986. URL: https://doi.org/10.1016/0012-365X(86)90089-0.
  14. Harold Fredricksen and James Maiorana. Necklaces of beads in k colors and k-ary de Bruijn sequences. Discret. Math., 23(3):207-210, 1978. URL: https://doi.org/10.1016/0012-365X(78)90002-X.
  15. Kayleigh Hyde. Nondeterministic finite state complexity. Master’s thesis, University of Hawaii at Manoa, 2013. Accessed: April 20, 2021. URL: http://hdl.handle.net/10125/29507.
  16. Kayleigh Hyde and Bjørn Kjos-Hanssen. Nondeterministic automatic complexity of overlap-free and almost square-free words. Electron. J. Comb., 22(3):P3.22, 2015. URL: https://doi.org/10.37236/4851.
  17. Liam Jordon and Philippe Moser. On the difference between finite-state and pushdown depth. In 46th International Conference on Current Trends in Theory and Practice of Informatics, SOFSEM 2020, Limassol, Cyprus, volume 12011 of LNCS, pages 187-198. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-38919-2_16.
  18. Liam Jordon and Philippe Moser. A normal sequence compressed by PPM* but not by Lempel-Ziv 78. In 47th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2021, Bolzano-Bozen, Italy, volume 12607 of LNCS, pages 389-399. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-67731-2_28.
  19. Bjørn Kjos-Hanssen. Automatic complexity of Fibonacci and Tribonacci words. Discrete Applied Mathematics, 289:446-454, 2021. URL: https://doi.org/10.1016/j.dam.2020.10.014.
  20. Bjørn Kjos-Hanssen. Automatic complexity of shift register sequences. Discrete Mathematics, 341(9):2409-2417, 2018. URL: https://doi.org/10.1016/j.disc.2018.05.015.
  21. Alexander Kozachinskiy and Alexander Shen. Automatic Kolmogorov complexity, normality, and finite-state dimension revisited. J. Comput. Syst. Sci., 118:75-107, 2021. URL: https://doi.org/10.1016/j.jcss.2020.12.003.
  22. James I. Lathrop and Martin Strauss. A universal upper bound on the performance of the Lempel-Ziv algorithm on maliciously-constructed data. In Compression and Complexity of SEQUENCES 1997, Positano, Amalfitan Coast, Salerno, Italy, June 11-13, 1997, Proceedings, pages 123-135. IEEE, 1997. URL: https://doi.org/10.1109/SEQUEN.1997.666909.
  23. M. H. Martin. A problem in arrangements. Bull. Amer. Math. Soc., 40(12):859-864, 1934. URL: https://doi.org/10.1090/S0002-9904-1934-05988-3.
  24. Satyadev Nandakumar and Santhosh Kumar Vangapelli. Normality and finite-state dimension of Liouville numbers. Theory Comput. Syst., 58(3):392-402, 2016. URL: https://doi.org/10.1007/s00224-014-9554-8.
  25. Larry A. Pierce and Paul C. Shields. Sequences incompressible by SLZ (LZW), yet fully compressible by ULZ. In Numbers, Information and Complexity, pages 385-390. Springer, 2000. URL: https://doi.org/10.1007/978-1-4757-6048-4_32.
  26. Joseph J. Rotman. An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Springer, New York, 1995. ISBN: 978-1-4612-8686-8. URL: https://doi.org/10.1007/978-1-4612-4176-8.
  27. Camille Flye Sainte-Marie. Solution to question nr. 48. In L'Intermédiaire des Mathématiciens, volume 1, pages 107-110, 1894. Google Scholar
  28. Claus-Peter Schnorr and H. Stimm. Endliche Automaten und Zufallsfolgen. Acta Inf., 1:345-359, 1972. URL: https://doi.org/10.1007/BF00289514.
  29. Jeffrey O. Shallit and Ming-wei Wang. Automatic complexity of strings. J. Autom. Lang. Comb., 6(4):537-554, 2001. URL: https://doi.org/10.25596/jalc-2001-537.
  30. Michael Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, Boston, Massachusetts, USA, pages 330-335. ACM, 1983. URL: https://doi.org/10.1145/800061.808762.
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