Abstract
Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic method does not give “effective” variants of such theorems, i.e. variants for resource-bounded Kolmogorov complexity. We show that a “naive derandomization” approach of replacing these objects by the output of Nisan-Wigderson pseudo-random generator can give polynomial-space variants of such theorems.
Specifically, we improve the preceding polynomial-space analogue of Muchnik’s conditional complexity theorem. I.e., for all a and b there exists a program p of least possible length that transforms a to b and is simple conditional on b. Here all programs work in polynomial space and all complexities are measured with logarithmic accuracy instead of polylogarithmic one in the previous work.
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Acknowledgements
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 five anonymous referees for careful reading and precise comments. I am grateful to participants of seminars in Moscow State University, Moscow Institute of Physics and Technology and St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences for their attention and thoughtfulness.
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Supported by ANR Sycomore, NAFIT ANR-08-EMER-008-01 and RFBR 09-01-00709-a grants.
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Musatov, D. Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive” Derandomization. Theory Comput Syst 55, 299–312 (2014). https://doi.org/10.1007/s00224-012-9432-1
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DOI: https://doi.org/10.1007/s00224-012-9432-1