Theory of Computing Systems

, Volume 61, Issue 2, pp 521–535 | Cite as

Some Properties of Antistochastic Strings

Article

Abstract

Algorithmic statistics is a part of algorithmic information theory (Kolmogorov complexity theory) that studies the following task: given a finite object x (say, a binary string), find an ‘explanation’ for it, i.e., a simple finite set that contains x and where x is a ‘typical element’. Both notions (‘simple’ and ‘typical’) are defined in terms of Kolmogorov complexity. It is known that this cannot be achieved for some objects: there are some “non-stochastic” objects that do not have good explanations. In this paper we study the properties of maximally non-stochastic objects; we call them “antistochastic”. In this paper, we demonstrate that the antistochastic strings have the following property (Theorem 6): if an antistochastic string x has complexity k, then any k bit of information about x are enough to reconstruct x (with logarithmic advice). In particular, if we erase all but k bits of this antistochastic string, the erased bits can be restored from the remaining ones (with logarithmic advice). As a corollary we get the existence of good list-decoding codes with erasures (or other ways of deleting part of the information). Antistochastic strings can also be used as a source of counterexamples in algorithmic information theory. We show that the symmetry of information property fails for total conditional complexity for antistochastic strings. An extended abstract of this paper was presented at the 10th International Computer Science Symposium in Russia (Milovanov, 2015).

Keywords

Kolmogorov complexity Algorithmic statistics Stochastic strings Total conditional complexity Symmetry of information 

References

  1. 1.
    Bauwens, B.: Computability in Statistical Hypothesis Testing, and Characterizations of Directed Influence in Time Series Using Kolmogorov Complexity. Ph.D thesis, University of Gent (2010)Google Scholar
  2. 2.
    Bauwens, B., Makhlin, A., Vereshchagin, N., Zimand, M.: Short lists with short programs in short time. Proceedings 28-th IEEE Conference on Computational Complexity (CCC), Stanford CA, 98–108 (2013)Google Scholar
  3. 3.
    Gács, P., Tromp, J., Vitányi, P.M.B.: Algorithmic statistics. IEEE Trans. Inf. Theory 47(6), 2443–2463 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Guruswami, V.: List decoding of error-correcting codes: winning thesis of the 2002 ACM doctoral dissertation competition, Springer (2004)Google Scholar
  5. 5.
    Kolmogorov, A.N.: The complexity of algorithms and the objective definition of randomness. Summary of the talk presented April 16, 1974 at Moscow Mathematical Society. Uspekhi matematicheskich nauk 29(4-178), 155 (1974)Google Scholar
  6. 6.
    Lee, T., Romashchenko, A.: Resource bounded symmetry of information revisited. Theor. Comput. Sci. 345(2–3), 386–405 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov complexity and its applications, 3rd ed., Springer, 2008 (1 ed., 1993; 2 ed., 1997), xxiii+790 pp ISBN 978-0-387-49820-1Google Scholar
  8. 8.
    Longpré, L., Mocas, S.: Symmetry of information and one-way functions. Inf. Process. Lett. 46(2), 95–100 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Longpré, L., Watanabe, O.: On symmetry of information and polynomial time invertibility. Inf. Comput. 121(1), 1–22 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Milovanov, A.: Some properties of antistochastic strings. In: Proceedings of 10th International Computer Science Symposium in Russia (CSR 2015) LNCS, vol. 9139, pp 339–349 (2015)Google Scholar
  11. 11.
    Shen, A.: The concept of (α, β)-stochasticity in the Kolmogorov sense, and its properties. Soviet Mathematics Doklady 271(1), 295–299 (1983)Google Scholar
  12. 12.
    Shen, A.: Game Arguments in Computability Theory and Algorithmic Information Theory. How the World Computes. Turing Centenary Conference and 8th Conference on Computability in Europe, CiE 2012, Cambridge, UK, June 18-23 2012 Proceedings, LNCS 7318, pp 655–666Google Scholar
  13. 13.
    Shen, A.: Around Kolmogorov Complexity: Basic Notions and Results Measures of Complexity. Festschrift for Alexey Chervonenkis Vovk, V., Papadoupoulos, H., Gammerman. A. (eds.) . Springer. ISBN: 978-3-319-21851-9 (2015)Google Scholar
  14. 14.
    Shen, A., Uspensky, V., Vereshchagin, N.: Kolmogorov complexity and algorithmic randomness, MCCME, (Russian). English translation: http://www.lirmm.fr/~ashen/kolmbook-eng.pdf (2013)
  15. 15.
    Vereshchagin, N.: On Algorithmic Strong Sufficient Statistics. In: 9th Conference on Computability in Europe, CiE, Milan, Italy, July 1–5, 2013. Proceedings, LNCS 7921, pp 424–433 (2013)Google Scholar
  16. 16.
    Vereshchagin, N., Vitányi, P.: Kolmogorov’s structure functions with an application to the foundations of model selection. IEEE Trans. Inf. Theory 50(12), 3265–3290 (2004). Preliminary version: Proceedings of 47th IEEE Symposium on the Foundations of Computer Science, 2002, 751–760Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Faculty of Computer ScienceNational Research University Higher School of EconomicsMoscowRussian Federation

Personalised recommendations