Extracting Kolmogorov complexity with applications to dimension zero-one laws (2006)
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| Venue: | IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING |
| Citations: | 10 - 2 self |
BibTeX
@INPROCEEDINGS{Fortnow06extractingkolmogorov,
author = {Lance Fortnow and John M. Hitchcock and A. Pavan and N. V. Vinodchandran and Fengming Wang},
title = {Extracting Kolmogorov complexity with applications to dimension zero-one laws},
booktitle = {IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING},
year = {2006},
pages = {335--345},
publisher = {Springer-Verlag}
}
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Abstract
We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. 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) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,
