computational complexity

, Volume 27, Issue 1, pp 31–61 | Cite as

Short lists with short programs in short time

  • Bruno BauwensEmail author
  • Anton Makhlin
  • Nikolay Vereshchagin
  • Marius Zimand


Given a machine U, a c-short program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any standard Turing machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a \({\operatorname{O}\bigl({\mathrm{log}}|x|\bigr)}\)-short program for x. We also show that there exists a computable function that maps every x to a list of size |x|2 containing a \({\operatorname{O}\bigl(1\bigr)}\)-short program for x. This is essentially optimal because we prove that for each such function there is a c and infinitely many x for which the list has size at least c|x|2. Finally we show that for some standard machines, computable functions generating lists with 0-short programs must have infinitely often list sizes proportional to 2|x|.


List-approximator Kolmogorov complexity Online matching Expander graph 

Subject Classification

68Q17 68Q30 03D15 03D25 05C70 05C85 


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  1. 1.
    B. Bauwens (2010). Computability in statistical hypotheses testing, and characterizations of independence and directed influences in time series using Kolmogorov complexity. Ph.D. thesis, Ghent University, Faculty of Engineering.Google Scholar
  2. 2.
    B. Bauwens, A. Makhlin, N. Vereshchagin & M. Zimand (2013). Short lists with short programs in short time. In the 28th IEEE Conference in Computational Complexity, Stanford, CA, June 5–7, 2013, 98–108.Google Scholar
  3. 3.
    Bauwens B., Shen A. (2014) Complexity of complexity and strings with maximal plain and prefix Kolmogorov complexity. The Journal of Symbolic Logic 79(02): 620–632MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beigel R., Buhrman H.M., Fejer P., Fortnow L., Grabowski P., Longpre L., Muchnik A., Stephan F., Torenvliet L. (2006) Enumerations of the Kolmogorov Function. The Journal of Symbolic Logic 71(2): 501–528MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. Buhrman, L. Fortnow & S. Laplante (2001). Resource-Bounded Kolmogorov Complexity Revisited. SIAM Journal on Computing 31(3), 887–905.Google Scholar
  6. 6.
    L. Fortnow, J. M. Hitchcock, A. Pavan, N. V. Vinodchandran & F. Wang (2011). Extracting Kolmogorov complexity with applications to dimension zero-one laws. Information and Computation 209(4), 627–636.Google Scholar
  7. 7.
    Gacs P. (1974) On the symmetry of algorithmic information. Soviet Mathematical Doklady 15: 1477–1480zbMATHGoogle Scholar
  8. 8.
    P. Hall (1935). On representatives of subsets. Journal of the London Mathematical Society 1(10), 26–30.Google Scholar
  9. 9.
    J. M. Hitchcock, A. Pavan & N. V. Vinodchandran (2011). Kolmogorov Complexity in Randomness Extraction. Transactions on Computation Theory 3(1), 1.Google Scholar
  10. 10.
    T. Kova̋ri, V.T. Sòs & P. Turàn (1954). On a problem of K. Zarankiewicz. Colloquium Mathematicae 3, 50–57.Google Scholar
  11. 11.
    A. H. Lachlan (1970). On Some Games Which Are Relevant to the Theory of Recursively Enumerable Sets. Annals of Mathematics 91(2), 291–310.
  12. 12.
    Muchnik A.A. (2002) Conditional complexity and codes. Theoretical Computer Science 271(1–2): 97–109MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. A. Muchnik, I. Mezhirov, A. Shen & N. Vereshchagin (2010). Game interpretation of Kolmogorov complexity. Unpublished.Google Scholar
  14. 14.
    D. Musatov (2011). Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive" Derandomization. In International Computer Science Symposium in Russia, 64–76.Google Scholar
  15. 15.
    D. Musatov (2012). Space-Bounded Kolmogorov Extractors. In International Computer Science Symposium in Russia, 266–277.Google Scholar
  16. 16.
    D. Musatov, A. E. Romashchenko & A. Shen (2011). Variations on Muchnik’s Conditional Complexity Theorem. Theory of Computation Systems 49(2), 227–245.Google Scholar
  17. 17.
    J. Radhakrishnan & A. Ta-Shma (2000). Tight bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics 13(1), 2–24.Google Scholar
  18. 18.
    C.P. Schnorr (1975) Optimal Enumerations and Optimal Gödel Numberings. Mathematical Systems Theory 8(2): 182–191MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Shen (2012). Game arguments in computability theory and algorithmic information theory. In Conference on Computability in Europe. Springer, Berlin Heidelberg.Google Scholar
  20. 20.
    M. Sipser (1983). A Complexity Theoretic Approach to Randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, 330–335.Google Scholar
  21. 21.
    F. Stephan (2013). Personal Communication.Google Scholar
  22. 22.
    A. Ta-Shma, C. Umans & D. Zuckerman (2007). Lossless Condensers, Unbalanced Expanders, And Extractors. Combinatorica 27(2), 213–240.Google Scholar
  23. 23.
    J. Teutsch (2014). Short lists for shortest descriptions in short time. Computational Complexity 23(4), 565–583.
  24. 24.
    N. Vereshchagin (2008). Kolmogorov complexity and Games. Bulletin of the European Association for Theoretical Computer Science 94, 51–83.Google Scholar
  25. 25.
    Vereshchagin N. (2016) Algorithmic minimal sufficient statistics: a new approach. Theory of Computing Systems 58(3): 463–481MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. Zimand (2010a). Possibilities and impossibilities in Kolmogorov complexity extraction. SIGACT News 41(4), 74–94.Google Scholar
  27. 27.
    M. Zimand (2010b). Two Sources Are Better than One for Increasing the Kolmogorov Complexity of Infinite Sequences. Theory of Computing Systems 46(4), 707–722.Google Scholar
  28. 28.
    M. Zimand (2011). Symmetry of Information and Bounds on Nonuniform Randomness Extraction via Kolmogorov Extractors. In IEEE Conference on Computational Complexity, 148–156.Google Scholar
  29. 29.
    Zimand M. (2013) Generating Kolmogorov random strings from sources with limited independence. Journal of Logic and Computation 23(4): 909–924MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    M. Zimand (2014). Short Lists with Short Programs in Short Time - A Short Proof. In Language, Life, Limits - 10th Conference on Computability in Europe, CiE 2014, Budapest, Hungary, June 23–27, 2014. Proceedings, 403–408.Google Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Bruno Bauwens
    • 1
    Email author
  • Anton Makhlin
    • 2
  • Nikolay Vereshchagin
    • 1
  • Marius Zimand
    • 3
  1. 1.Department of Computer ScienceNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  3. 3.Department of Computer and Information SciencesTowson UniversityTowsonUSA

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