Some Properties of Antistochastic Strings

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)

Abstract

Antistochastic strings are those strings that do not have any reasonable statistical explanation. We establish the follow property of such strings: every antistochastic string x is “holographic” in the sense that it can be restored by a short program from any of its part whose length equals the Kolmogorov complexity of x. Further we will show how it can be used for list decoding from erasing and prove that Symmetry of Information fails for total conditional complexity.

Keywords

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

Notes

Acknowledgments

I would like to thank Alexander Shen and Nikolay Vereshchagin for useful discussions, advises and remarks.

References

  1. 1.
    Gács, P., Tromp, J., Vitányi, P.M.B.: Algorithmic statistics. IEEE Trans. Inform. Th. 47(6), 2443–2463 (2001)MATHCrossRefGoogle Scholar
  2. 2.
    Kolmogorov, A.N.: Talk at the Information Theory Symposium in Tallinn, Estonia (1974)Google Scholar
  3. 3.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, New York (2008). (1st edn. 1993; 2nd edn. 1997), pp. xxiii+790, ISBN 978-0-387-49820-1 MATHCrossRefGoogle Scholar
  4. 4.
    Shen, A., Uspensky, V., Vereshchagin, N.: Kolmogorov complexity and algorithmic randomness. MCCME (2013). English translation: http://www.lirmm.fr/~ashen/kolmbook-eng.pdf (Russian)
  5. 5.
    Guruswami, V.: List Decoding of Error-Correcting Codes: Winning Thesis of the 2002 ACM Doctoral Dissertation Competition. LNCS, vol. 3282. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Shen, A.: The concept of \((\alpha, \beta )\)-stochasticity in the Kolmogorov sense, and its properties. Sov. Math. Dokl. 27(1), 295–299 (1983)Google Scholar
  7. 7.
    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: Proceeding of the 47th IEEE Symposium on Foundations of Computer Science, vol. 2002, pp. 751–760 (2004)MATHCrossRefGoogle Scholar
  8. 8.
    Longpré, L., Mocas, S.: Symmetry of information and one-way functions. Inf. Process. Lett. 46(2), 95–100 (1993)MATHCrossRefGoogle Scholar
  9. 9.
    Longpré, L., Watanabe, O.: On symmetry of information and polynomial time invertibility. Inf. Comput. 121(1), 1–22 (1995)CrossRefGoogle Scholar
  10. 10.
    Shen, A.: Game arguments in computability theory and algorithmic information theory. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 655–666. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  11. 11.
    Lee, T., Romashchenko, A.: Resource bounded symmetry of information revisited. Theor. Comput. Sci. 345(2–3), 386–405 (2005)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Vereshchagin, N.: On algorithmic strong sufficient statistics. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 424–433. Springer, Heidelberg (2013) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations