36,710 research outputs found

    Large Alphabets and Incompressibility

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    We briefly survey some concepts related to empirical entropy -- normal numbers, de Bruijn sequences and Markov processes -- and investigate how well it approximates Kolmogorov complexity. Our results suggest â„“\ellth-order empirical entropy stops being a reasonable complexity metric for almost all strings of length mm over alphabets of size nn about when nâ„“n^\ell surpasses mm

    Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks

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    We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application

    Kolmogorov complexity spectrum for use in analysis of UV-B radiation time series

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    We have used the Kolmogorov complexity and sample entropy measures to estimate the complexity of the UV-B radiation time series in the Vojvodina region (Serbia) for the period 1990-2007. We defined the Kolmogorov complexity spectrum and have introduced the Kolmogorov complexity spectrum highest value (KLM). We have established the UV-B radiation time series on the basis of their daily sum (dose) for seven representative places in this region using (i) measured data, (ii) data calculated via a derived empirical formula and (iii) data obtained by a parametric UV radiation model. We have calculated the Kolmogorov complexity (KL) based on the Lempel-Ziv Algorithm (LZA), KLM and Sample Entropy (SE) values for each time series. We have divided the period 1990-2007 into two sub-intervals: (a) 1990-1998 and (b)1999-2007 and calculated the KL, KLM and SE values for the various time series in these sub-intervals. It is found that during the period 1999-2007, there is a decrease in the KL, KLM and SE, comparing to the period 1990-1998. This complexity loss may be attributed to (i) the increased human intervention in the post civil war period causing increase of the air pollution and (ii) the increased cloudiness due to climate changes.Comment: 10 pages, 1 figure, 1 table. arXiv admin note: substantial text overlap with arXiv:1301.2039; This paper has been accepted in Modern Physics Letters B on Aug 14, 201

    Zipf's law and L. Levin's probability distributions

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    Zipf's law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation. In this paper I suggest that at least in certain situations, Zipf's law can be explained as a special case of the a priori distribution introduced and studied by L. Levin. The Zipf ranking corresponding to diminishing probability appears then as the ordering determined by the growing Kolmogorov complexity. One argument justifying this assertion is the appeal to a recent interpretation by Yu. Manin and M. Marcolli of asymptotic bounds for error--correcting codes in terms of phase transition. In the respective partition function, Kolmogorov complexity of a code plays the role of its energy. This version contains minor corrections and additions.Comment: 19 page

    Martingales, Efficient Market Hypothesis and Kolmogorov’s Complexity Theory

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    Efficient market theory states that financial markets can process information instantly. Empirical observations have challenged the stricter form of the efficient market hypothesis (EMH). These empirical observations and theoretical considerations show that price changes are difficult to predict if one starts from the time series of price changes. This paper provides an explanation in terms of algorithmic complexity theory of Kolmogorov that makes a clearer connection between the efficient market hypothesis and the unpredictable character of stock returns
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