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Plain Stopping Time and Conditional Complexities Revisited

Authors Mikhail Andreev, Gleb Posobin, Alexander Shen

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Author Details

Mikhail Andreev
  • IPONWEB, Berlin, Germany
Gleb Posobin
  • National Research University - Higher School of Economics, Moscow, Russia
Alexander Shen
  • LIRMM CNRS / University of Montpellier, France. On leave from IITP RAS, Moscow, Russia

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Mikhail Andreev, Gleb Posobin, and Alexander Shen. Plain Stopping Time and Conditional Complexities Revisited. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 2:1-2:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


In this paper we analyze the notion of "stopping time complexity", the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic [Vovk and Pavlovic, 2016]. It turns out that plain stopping time complexity of a binary string x could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading x on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing x; (c) the conditional complexity C(x|x*) where x in the condition is understood as a prefix of an infinite binary sequence while the first x is understood as a terminated binary string; (d) as a minimal upper semicomputable function K such that each binary sequence has at most 2^n prefixes z such that K(z)<n; (e) as maxC^X(x) where C^X(z) is plain Kolmogorov complexity of z relative to oracle X and the maximum is taken over all extensions X of x. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition y and the object x may differ, and answer an open question from Chernov, Hutter and Schmidhuber [Alexey V. Chernov et al., 2007].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
  • Kolmogorov complexity
  • stopping time complexity
  • structured conditional complexity
  • algorithmic information theory


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  1. Mikhail Andreev, Gleb Posobin, and Alexander Shen. Plain stopping time and conditional complexities revisited. CoRR, abs/1708.08100, 2017. URL:
  2. Mikhail Andreev, Gleb Posobin, and Alexander Shen. Stopping time complexity, abstract. Dagstuhl Reports, Computability Theory, Dagstuhl Seminar 17081, February 2017, page 97, 2017. URL:
  3. Alexey V. Chernov, Marcus Hutter, and Jürgen Schmidhuber. Algorithmic complexity bounds on future prediction errors. Information and Computation, 205(2):242-261, 2007. Google Scholar
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  7. Alexander Shen, Vladimir A. Uspensky, and Nikolay Vereshchagin. Kolmogorov complexity and algorithmic randomness, volume 220 of Mathematical Surveys and Monographs. American Mathematical Society,, 2017.
  8. Vladimir A. Uspensky and Alexander Shen. Relations between varieties of Kolmogorov complexities. Mathematical Systems Theory, 29(3):271-292, Jun 1996. URL:
  9. Vladimir Vovk and Dusko Pavlovic. Universal probability-free conformal prediction. In Alexander Gammerman, Zhiyuan Luo, Jesús Vega, and Vladimir Vovk, editors, Conformal and Probabilistic Prediction with Applications, pages 40-47, Cham, 2016. Springer, see also (March 2016; extended version, April 2017).
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