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.
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Fortnow, L., Hitchcock, J.M., Pavan, A., Vinodchandran, N.V., Wang, F. (2006). Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_30
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DOI: https://doi.org/10.1007/11786986_30
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