Information Distance Revisited

Author Bruno Bauwens

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Bruno Bauwens
  • National Research University Higher School of Economics, Moscow, Russia


This work was initiated by Alexander (Sasha) Shen, who informed me about the error in [Mahmud, 2009] during a discussion of the paper [Vitányi, 2017]. Afterwards I explained the proof of Theorem 13 to Sasha. He simplified it, and he wrote all of the current manuscript and the proof of Theorem 13, which is available in the ArXiv version of this paper [Bauwens and Shen, 2018]. Later, I added Theorem 14. Only this proof is written by me, and is also available on ArXiv [Bauwens and Shen, 2018]. After this was added, Sasha decided that his contribution was no longer proportional, and decided he did no longer want to remain an author. I am especially grateful for his generous permission to publish this nicely written document, with minor modifications suggested by reviewers. I thank the reviewers for these suggestions. All errors in this document are solely my responsability. I thank Mikhail Andreev for the proof of Proposition 7 and many useful discussions. Finally, I thank Artem Grachev and the participants of the Kolmogorov seminar in Moscow state university for useful discussions.

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Bruno Bauwens. Information Distance Revisited. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We consider the notion of information distance between two objects x and y introduced by Bennett, Gács, Li, Vitanyi, and Zurek [C. H. Bennett et al., 1998] as the minimal length of a program that computes x from y as well as computing y from x, and study different versions of this notion. In the above paper, it was shown that the prefix version of information distance equals max (K(x|y),K(y|x)) up to additive logarithmic terms. It was claimed by Mahmud [Mahmud, 2009] that this equality holds up to additive O(1)-precision. We show that this claim is false, but does hold if the distance is at least logarithmic. This implies that the original definition provides a metric on strings that are at superlogarithmically separated.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Kolmogorov complexity
  • algorithmic information distance


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