968 research outputs found

    Around Kolmogorov complexity: basic notions and results

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    Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author

    Algorithmic Identification of Probabilities

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    TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target) belongs to a computably enumerable (c.e.) or co-computably enumerable (co-c.e.) set of computable probability mass functions, then there is an algorithm to almost surely identify the target in the limit. The technical tool is the strong law of large numbers. If the set is finite and the elements of the sequence are dependent while the sequence is typical in the sense of Martin-L\"of for at least one measure belonging to a c.e. or co-c.e. set of computable measures, then there is an algorithm to identify in the limit a computable measure for which the sequence is typical (there may be more than one such measure). The technical tool is the theory of Kolmogorov complexity. We give the algorithms and consider the associated predictions.Comment: 19 pages LaTeX.Corrected errors and rewrote the entire paper. arXiv admin note: text overlap with arXiv:1208.500

    Algorithmic Thermodynamics

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    Algorithmic entropy can be seen as a special case of entropy as studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of 'microstates', and treat any function on X as an 'observable'. For any collection of observables, we can study the Gibbs ensemble that maximizes entropy subject to constraints on expected values of these observables. We illustrate this by taking the log runtime, length, and output of a program as observables analogous to the energy E, volume V and number of molecules N in a container of gas. The conjugate variables of these observables allow us to define quantities which we call the 'algorithmic temperature' T, 'algorithmic pressure' P and algorithmic potential' mu, since they are analogous to the temperature, pressure and chemical potential. We derive an analogue of the fundamental thermodynamic relation dE = T dS - P d V + mu dN, and use it to study thermodynamic cycles analogous to those for heat engines. We also investigate the values of T, P and mu for which the partition function converges. At some points on the boundary of this domain of convergence, the partition function becomes uncomputable. Indeed, at these points the partition function itself has nontrivial algorithmic entropy.Comment: 20 pages, one encapsulated postscript figur

    Kolmogorov's Structure Functions and Model Selection

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    In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal (Kolmogorov) complexity. The ``structure function'' of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. We show that the structure function determines all stochastic properties of the data: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the ``true'' model is in the model class considered or not. In this setting, this happens {\em with certainty}, rather than with high probability as is in the classical case. We precisely quantify the goodness-of-fit of an individual model with respect to individual data. We show that--within the obvious constraints--every graph is realized by the structure function of some data. We determine the (un)computability properties of the various functions contemplated and of the ``algorithmic minimal sufficient statistic.''Comment: 25 pages LaTeX, 5 figures. In part in Proc 47th IEEE FOCS; this final version (more explanations, cosmetic modifications) to appear in IEEE Trans Inform T
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