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Non-reducible Descriptions for Conditional Kolmogorov Complexity

  • Andrej Muchnik
  • Alexander Shen
  • Nikolai Vereshchagin
  • Michael Vyugin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)

Abstract

Let a program p on input a outputs b. We are looking for a shorter program p′ having the same property (p′(a)=b). In addition, we want p′ to be simple conditional to p (this means that the conditional Kolmogorov complexity K (p |p) is negligible). In the present paper, we prove that sometimes there is no such program p′, even in the case when the complexity of p is much bigger than K (b|a). We give three different constructions that use the game approach, probabilistic arguments and algebraic (combinatorial) arguments, respectively.

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References

  1. Muchnik, A.A.: Conditional complexity and codes. Theoretical Computer Science 271, 97–109 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Muchnik, A., Ustinov, M.: Constructing non-reducible programs for given pair of strings (preprint)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Andrej Muchnik
    • 1
  • Alexander Shen
    • 2
    • 3
  • Nikolai Vereshchagin
    • 4
  • Michael Vyugin
    • 4
  1. 1.Institute of New Technologies 
  2. 2.LIF CNRSMarseilleFrance
  3. 3.Poncelet laboratory, CNRSInstitute of Problems of Information TransmissionMoscow
  4. 4.Moscow State University 

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