Extracting kolmogorov complexity with applications to dimension zero-one laws

  • Authors:

  • Lance Fortnow;John M. Hitchcock;A. Pavan;N. V. Vinodchandran;Fengming Wang

  • Affiliations:

  • Department of Computer Science, University of Chicago;Department of Computer Science, University of Wyoming;Department of Computer Science, Iowa State University;Department of Computer Science and Engineering, University of Nebraska-Lincoln;Department of Computer Science, Iowa State University

  • Venue:
  • ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
  • Year:
  • 2006

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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.