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, USA;Department of Computer Science, University of Wyoming, USA;Department of Computer Science, Iowa State University, USA;Department of Computer Science and Engineering, University of Nebraska-Lincoln, USA;Department of Computer Science, Rutgers University, USA
Venue:
Information and Computation
Year:
2011

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Abstract

We apply results on extracting randomness from independent sources to ''extract'' Kolmogorov complexity. For any @a,@e0, given a string x with K(x)@a|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=@W(|x|), with K(y)(1-@e)|y|. This result holds for both unbounded and space-bounded Kolmogorov complexity. We use the extraction procedure for space-bounded complexity to establish zero-one laws for the strong dimensions of complexity classes within ESPACE. The unbounded extraction procedure yields a zero-one law for the constructive strong dimensions of Turing degrees.