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Information Distance Revisited
P. 46:1–46:14.
In book
Vol. 154: Leibniz International Proceedings in Informatics (LIPIcs). , Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2020.
Semenov A., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2025 Т. 527 № S С. 7–12
The paper proposes a system of definitions for the basic concepts of computability theory that underlie the mathematics of the digital world: algorithm, computability, calculus, object complexity, close to modern undertnding. Hierarchies of the finite and the problem of consistency are considered. ...
Added: December 6, 2025
Semenov A., Shen A., Vereshchagin N., Theory of Probability and its Applications, USA 2024 Vol. 68 No. 4 P. 582–606
The definition of descriptional complexity of finite objects suggested by Kolmogorov and other authors in the mid-1960s is now well known. In addition, Kolmogorov pointed out some approaches to a more fine-grained classification of finite objects, such as the resource-bounded complexity (1965), structure function (1974), and the notion of $(\alpha,\beta)$-stochasticity (1981). Later it turned out ...
Added: January 16, 2025
Vereshchagin N., / Series arXiv "math". 2024.
The fine approach to measure information dependence is based on the total conditional complexity CT(y|x), which is defined as the minimal length of a total program that outputs y on the input x. It is known that the total conditional complexity can be much larger than than the plain conditional complexity. Such strings x, y ...
Added: August 19, 2024
Bauwens B. F., Zimand M., Journal of the ACM 2023 Vol. 70 No. 2 Article 9
In a lossless compression system with target lengths, a compressor 𝒞 maps an integer m and a binary string x to an m-bit code p, and if m is sufficiently large, a decompressor 𝒟 reconstructs x from p. We call a pair (m,x) achievable for (𝒞,𝒟) if this reconstruction is successful. We introduce the notion ...
Added: March 22, 2023
Bauwens B. F., Gács P., Romashchenko A. et al., Computability 2022 Vol. 11 No. 3-4 P. 165–185
Finding all linear inequalities for entropies remains an important open question in information theory. For a long time the only known inequalities for entropies of tuples of random variables were Shannon (submodularity) inequalities. Only in 1998 Zhang and Yeung 1998 found the first inequality that cannot be represented as a convex combination of Shannon inequalities, and ...
Added: December 23, 2022
Vereshchagin N., Theoretical Computer Science 2023 Vol. 940 P. 108–122
We consider the network consisting of three nodes 1, 2, 3 connected by two open channels
1 → 2 and 1 → 3. The information present in the node 1 consists of four strings x , y , z , w.
The nodes 2, 3 know x , w and need to know y , z, respectively. ...
Added: December 19, 2022
Milovanov A., , in: Computer Science – Theory and Applications: 16th International Computer Science Symposium in Russia, CSR 2021, Sochi, Russia, June 28–July 2, 2021, Proceedings.: Springer, 2021. Ch. 17 P. 283–295.
We combine Solomonoff’s approach to universal prediction with algorithmic statistics and suggest to use the computable measure that provides the best “explanation” for the observed data (in the sense of algorithmic statistics) for prediction. In this way we keep the expected sum of squares of prediction errors bounded (as it was for the Solomonoff’s predictor) ...
Added: August 11, 2021
Bauwens B. F., Blinnikov I., , in: Computer Science – Theory and Applications 15th International Computer Science Symposium in Russia, CSR 2020, Yekaterinburg, Russia, June 29 – July 3, 2020, ProceedingsVol. 12159.: Springer, 2020. P. 130–141.
It is known that the normalized algorithmic information distance is not computable and not semicomputable. We show that for all 𝜀<1/2, there exist no semicomputable functions that differ from N by at most 𝜀. Moreover, for any computable function f such that |lim𝑡𝑓(𝑥,𝑦,𝑡)−N(𝑥,𝑦)|≤𝜀 and for all n, there exist strings x, y of length n such that ...
Added: February 5, 2021
Vereshchagin N., Theoretical Computer Science 2020 Vol. 809 P. 531–537
The purpose of this paper is to answer two questions left open in [B. Durand, A. Shen, and N. Vereshchagin, Descriptive Complexity of Computable Sequences, Theoretical Computer Science 171 (2001), pp. 47--58]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence $\alpha$: $C^{0'}(\alpha )$, defined as ...
Added: January 17, 2020
Beaudou L., Coupechoux P., Dailly A. et al., Theoretical Computer Science 2018 Vol. 746 P. 19–35
Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore.
We extend the definition of octal games to play them ...
Added: April 11, 2019
Shen A., Vereshchagin N., , in: Computability and Complexity.: Berlin: Springer, 2017. P. 669–737.
Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there ...
Added: October 26, 2018
Milovanov A., Theory of Computing Systems 2019 Vol. 63 No. 4 P. 833–848
Algorithmic statistics looks for models of observed data that are good in the following sense: a model is simple (i.e., has small Kolmogorov complexity) and captures all the algorithmically discoverable regularities in the data. However, this idea can not be used in practice as is because Kolmogorov complexity is not computable. In this paper we ...
Added: October 17, 2018
Posobin G. I., Shen A., Andreev M., , in: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)Vol. 117.: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2018. P. 1–24.
In this paper we analyze the notion of "stopping time complexity", informally defined as the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic (2016). It turns out that plain stopping time complexity of a binary string x could be equivalently defined as (a) ...
Added: October 11, 2018
Rubtsov A. A., , in: Developments in Language Theory 22nd International Conference, DLT 2018, Tokyo, Japan, September 10-14, 2018, Proceedings.: Cham: Springer, 2018. P. 553–565.
We present a new structural lemma for deterministic con- text free languages. From the first sight, it looks like a pumping lemma, because it is also based on iteration properties, but it has significant distinctions that makes it much easier to apply. The structural lemma is a combinatorial analogue of KC-DCF-Lemma (based on Kolmogorov complexity), ...
Added: September 12, 2018
Milovanov A., , in: Sailing Routes in the World of Computation.: Springer, 2018. P. 287–296.
Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple i.e. should have small Kolmogorov complexity and capture all the algorithmically discoverable regularities in the data. However this idea can not be used in practice as is because Kolmogorov complexity is not computable.
In recent years resource-bounded ...
Added: September 4, 2018
Milovanov A., , in: Computer Science – Theory and Applications: 12th International Computer Science Symposium in Russia (CSR 2017)Vol. 10304.: Luxemburg: Springer, 2017. P. 232–244.
Algorithmic statistics studies explanations of observed data that are good in the algorithmic sense: an explanation should be simple i.e. should have small Kolmogorov complexity and capture all the algorithmically discoverable regularities in the data. However this idea can not be used in practice because Kolmogorov complexity is not computable.
In this paper we develop algorithmic ...
Added: October 15, 2017
Shen A., Uspensky V. A., Vereshchagin N., American Mathematical Society, 2017.
Added: October 12, 2017
Vereshchagin N., Milovanov A., , in: 32nd Computational Complexity Conference.: Вадерн: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, 2017. P. 1–18.
A fundamental notion in Algorithmic Statisticsis that of a stochastic object, i.e., an object having a simple plausible explanation. Informally, a probability distribution is a plausible explanation for x if it looks likely that x was drawn at random with respect to that distribution. In this paper, we suggest three definitions of a plausible statistical ...
Added: October 12, 2017
Vereshchagin N., Bauwens B. F., Makhlin A. et al., Computational Complexity 2018 Vol. 27 No. 1 P. 31–61
Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any universal machine, it is possible to compute in polynomial time on input $x$ a list of ...
Added: April 22, 2017
Vereshchagin N., Milovanov A., / Series Technical report "Electronic Colloquium on Computational Complexity". 2017. No. TR17-043.
Added: March 7, 2017
Vereshchagin N., Shen A., Lecture Notes in Computer Science 2017 Vol. 10010 P. 669–737
Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there ...
Added: February 13, 2017
Bienvenu L., Desfontaines D., Shen A., Logical Methods in Computer Science 2016 Vol. 12 No. 2
The halting problem is undecidable --- but can it be solved for "most" inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the ...
Added: February 7, 2017