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Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws

  • Lance Fortnow
  • John M. Hitchcock
  • A. Pavan
  • N. V. Vinodchandran
  • Fengming Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)

Abstract

We apply recent results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α, ε> 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y) > (1–ε)|y|. This result holds for both classical and space-bounded Kolmogorov complexity.

We use the extraction procedure for space-bounded complexity to establish zero-one laws for polynomial-space strong dimension. Our results include:

(i) If Dimpspace(E) > 0, then Dimpspace(E/O(1)) = 1.

(ii) Dim(E/O(1) |ESPACE) is either 0 or 1.

(iii) Dim(E/poly |ESPACE) is either 0 or 1.

In other words, from a dimension standpoint and with respect to a small amount of advice, the exponential-time class E is either minimally complex or maximally complex within ESPACE.

Keywords

Kolmogorov Complexity Annual IEEE Strong Dimension Random String Random Source 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Lance Fortnow
    • 1
  • John M. Hitchcock
    • 2
  • A. Pavan
    • 3
  • N. V. Vinodchandran
    • 4
  • Fengming Wang
    • 3
  1. 1.Department of Computer ScienceUniversity of Chicago 
  2. 2.Department of Computer ScienceUniversity of Wyoming 
  3. 3.Department of Computer ScienceIowa State University 
  4. 4.Department of Computer Science and EngineeringUniversity of Nebraska-Lincoln 

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