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computational complexity

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

Short lists with short programs in short time

  • Bruno Bauwens
  • Anton Makhlin
  • Nikolay Vereshchagin
  • Marius Zimand
Article
  • 32 Downloads

Abstract

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|.

Keywords

List-approximator Kolmogorov complexity Online matching Expander graph 

Subject Classification

68Q17 68Q30 03D15 03D25 05C70 05C85 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Bruno Bauwens
    • 1
  • 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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