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Topological Arguments for Kolmogorov Complexity

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Abstract

We present several applications of simple topological arguments (such as non-contractibility of a sphere and similar results) to Kolmogorov complexity. It turns out that discrete versions of these results can be used to prove the existence of strings with prescribed complexity with O(1)-precision (instead of usual O(logn)-precision). In particular, we improve an earlier result of M. Vyugin and show that for every n and for every string x of complexity at least n + O(logn) there exists a string y such that both C(x∣y) and C(y∣x) are equal to n + O(1). We also show that for a given tuple of strings x i (assuming they are almost independent) there exists another string y such that the condition y makes the complexities of all x i twice smaller with O(1)-precision. The extended abstract of this paper was published in [6].

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Notes

  1. The plain Kolmogorov complexity C(x) of a bit string x is defined as the minimal length of a program that outputs x; this definition depends on the choice of a programming language, and we fix some language that makes the complexity minimal. The conditional complexity C(x∣y) is defined as the minimal length of a program that transforms y to x. The sum C(y) + C(x∣y) is equal to the complexity C(x, y) of (the encoding of) the pair (x, y) with logarithmic precision. The mutual information between x and y is defined as C(x) − C(x∣y), or C(y) − C(y∣x), or as C(x) + C(y) − C(x, y); all three quantities coincide with each other with logarithmic precision. Strings x and y are considered as “independent” if I(x : y) is negligible (of course, one has to specify the exact bound when formulating results about independent strings). The information distance between strings x and y is defined as C(x∣y) + C(y∣x); it satisfies the triangle inequality with logarithmic precision.

    We assume that the reader is familiar with all these notions. An introduction to Kolmgorov complexity can be found, e.g., in [2, 7, 9]; see also an extensive textbook on this subject written by Li and Vitányi [3].

  2. A better name would be Lipschitz continuity: if the distance between the images of neighbor grid points is bounded by c, then the distance between the images of arbitrary two points is at most c times bigger than the distance between the points itself (the distance is measured in l 1-sense)

References

  1. Bauwens, B., Makhlin, A., Vereshchagin, N., Zimand, M.: Short lists with short programs in short time. In: Proceedings of the 28th IEEE Conference on Computational Complexity, pp 98–108 (2013)

  2. Gács, P.: Lecture notes on descriptional complexity and randomness, http://www.cs.bu.edu/faculty/gacs/papers/ait-notes.pdf

  3. Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its applications, 3rd edn. Springer Verlag (2008)

  4. Muchnik, A.: Conditional complexity and codes. Chic. J. Theor. Comput. Sci. 271(1–2), 97–109 (2002)

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  5. Muchnik, A., Mezhirov, I., Shen, A., Vereshchagin, N.: Game interpretation of Kolmogorov complexity. arXiv:1003.4712

  6. Romashchenko, A., Shen, A.: Topological arguments for Kolmogorov complexity, Proceedings of the 18th international workshop on cellular automata and 3rd international symposium Journées Automates Cellulaires, EPTCS, 90, p. 127–132, (2012)

  7. Shen, A.: Algorithmic information theory and Kolmogorov complexity, technical report TR2000-034, Uppsala University, http://www.it.uu.se/research/publications/reports/2000-034

  8. Shen, A.: Game arguments in computability theory and algorithmic information theory, Computability in Europe, 2012, LNCS 7318 p.655–666. Extended version (with Andrej Muchnik and Mikhail Vyugin). arXiv: 1204.0198 (2012)

  9. Vereshchagin, N., Uspensky, V., Shen, A. Kolmogorov complexity and algorithmic randomness, 575 p. MCCME Publishers, Moscow (2012). For the electronic version and draft English translation see www.lirmm.fr/~ashen

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  10. Vyugin, M.: Information distances and conditional complexities. Theor. Comput. Sci. 271(1–2), 145–150 (2002)

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Acknowledgments

The authors are grateful to Laurent Bienvenu who suggested to write down this simple argument, Tarik Kaced, and all the colleagues in Escape/NAFIT/Kolmogorov seminar team. We thank the (anonymous) referee for a very detailed review that pointed out many inaccuracies and suggested several improvements.

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  1. CNRS, University of Montpellier, Montpellier, France

    Andrei Romashchenko & Alexander Shen

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  1. Andrei Romashchenko
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  2. Alexander Shen
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Correspondence to Alexander Shen.

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Both authors work at the LIRMM, CNRS & University of Montpellier 2 and are on leave from IITP RAS, Moscow. Supported in part by RFBR 14-01-93107 grant.

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Romashchenko, A., Shen, A. Topological Arguments for Kolmogorov Complexity. Theory Comput Syst 56, 513–526 (2015). https://doi.org/10.1007/s00224-015-9606-8

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  • DOI: https://doi.org/10.1007/s00224-015-9606-8

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