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On Extracting Space-bounded Kolmogorov Complexity

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Abstract

We present two theorems about extracting space-bounded Kolmogorov complexity. Roughly speaking, they say about extracting complexity from two sources and from one source respectively. Their proofs employ the same “naive” derandomization technique: a random construction is replaced by a pseudo-random one obtained by the Nisan-Wigderson generator. Extracting complexity from two sources is done by an analogue of Kolmogorov extractor. It receives two strings with sufficiently large space-bounded Kolmogorov complexities and sufficiently small “dependency”. From these arguments an extractor produces a string with complexity close to the length. An extractor is strong if complexity is still close to the length even conditional to one of the initial strings. We prove that there exist such functions with near optimal parameters. Extracting complexity from one source uses the ideas of Muchnik’s theorem on conditional complexity. We prove that in space-bounded framework any string a has a “universal code” p such that two properties hold. First, p is simple conditional on a. Second, for any string b of polynomial length a is simple conditional on b and the C s(a|b)-bit prefix of p. It turns out that this prefix has complexity close to its length, causing the term “extraction”.

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Notes

  1. 1 Fortnow et al. do use the term “dependency” in [2] but define it for two distinct space bounds that correspond to s and μ s in our definition.

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Acknowledgments

I want to thank my colleagues and advisors Andrei Romashchenko, Alexander Shen and Nikolay Vereshchagin for stating the problem and many useful comments. I also want to thank four anonymous referees for careful reading and precise comments. I am grateful to participants of seminars in Moscow State University and Moscow Institute of Physics and Technology for their attention and thoughtfulness.

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  1. Moscow Institute of Physics and Technology, Moscow, Russia

    Daniil Musatov

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  1. Daniil Musatov
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Correspondence to Daniil Musatov.

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Supported by ANR Sycomore, NAFIT ANR-08-EMER-008-01, RFBR 09-01-00709-a and RFBR 12-01-00864 grants.

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Musatov, D. On Extracting Space-bounded Kolmogorov Complexity. Theory Comput Syst 56, 643–661 (2015). https://doi.org/10.1007/s00224-014-9563-7

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