Abstract
Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a short 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.
Supported by ANR Sycomore, NAFIT ANR-08-EMER-008-01 and RFBR 09-01-00709-a grants.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Musatov, D., Romashchenko, A., Shen, A.: Variations on Muchnik’s Conditional Complexity Theorem. Full version, http://arxiv.org/abs/0904.3116
Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics 13(1), 2–24 (2000)
Reingold, O., Shaltiel, R., Wigderson, A.: Extracting randomness via repeated condensing. SIAM Journal on Computing 35(5), 1185–1209 (2006)
Buhrman, H., Fortnow, L., Laplante, S.: Resource bounded Kolmogorov complexity revisited. SIAM Journal on Computing 31(3), 887–905 (2002)
Li, M., Vitanyi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 2nd edn. Springer, Heidelberg (1997)
Musatov, D.: Extractors and an effective variant of Muchnik’s theorem. Diplom (Master thesis). Faculty of Mechanics and Mathematics, MSU (2006) (in Russian), http://arxiv.org/abs/0811.3958
Muchnik, A.A.: On basic structures of the descriptive theory of algorithms. Soviet Math. Dokl. 32, 671–674 (1985)
Muchnik, A.: Conditional complexity and codes. Theoretical Computer Science 271(1-2), 97–109 (2002)
Shen, A.: Combinatorial proof of Muchnik’s theorem. In: Hutter, M., Merkle, W., Vitanyi, P. (eds.) Kolmogorov complexity and applications, Dagstuhl Seminar Proceedings 06051, ISSN 1862–4405, http://drops.dagstuhl.de/opus/volltexte/2006/625
Buhrman, H., Lee, T., van Melkebeek, D.: Language compression and pseudorandom generators. In: Proc. of the 15th IEEE Conference on Computational Complexity. IEEE, Los Alamitos (2004)
Trevisan, L.: Construction of extractors using pseudo-random generators. In: Proc. 45th FOCS, pp. 264–275
Slepian, D., Wolf, J.K.: Noiseless coding of correlated information sources. IEEE Transactions on information Theory 19, 471–480 (1973)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Musatov, D., Romashchenko, A., Shen, A. (2009). Variations on Muchnik’s Conditional Complexity Theorem. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-03351-3_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03350-6
Online ISBN: 978-3-642-03351-3
eBook Packages: Computer ScienceComputer Science (R0)