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Variations on Muchnik’s Conditional Complexity Theorem

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Abstract

Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a program p transforming a to b that has the least possible length and is simple conditional on b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be generalized to prove a version of Muchnik’s theorem for space-bounded Kolmogorov complexity. Another version of Muchnik’s theorem is proven for a resource-bounded variant of Kolmogorov complexity based on Arthur–Merlin protocols.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Daniil Musatov

  2. LIF Marseille, CNRS & Univ. Aix–Marseille, Marseille, France

    Andrei Romashchenko & Alexander Shen

Authors
  1. Daniil Musatov
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  2. Andrei Romashchenko
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  3. Alexander Shen
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Corresponding author

Correspondence to Andrei Romashchenko.

Additional information

This study was supported by ANR Sycomore, NAFIT ANR-08-EMER-008-01 and RFBR 09-01-00709-a grants.

A. Romashchenko and A. Shen on leave from IITP RAS, Moscow, Russia.

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Musatov, D., Romashchenko, A. & Shen, A. Variations on Muchnik’s Conditional Complexity Theorem. Theory Comput Syst 49, 227–245 (2011). https://doi.org/10.1007/s00224-011-9321-z

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  • DOI: https://doi.org/10.1007/s00224-011-9321-z

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