Kolmogorov Complexity Computational Complexity Course Report
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Algorithmic Information Theory and Kolmogorov Complexity
This document contains lecture notes of an introductory course on Kolmogorov complexity. They cover basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), notion of randomness (Martin-Löf randomness, Mises–Church randomness), Solomonoff universal a priori probability and their properties (symmetry of information, connection between a priori probabil...
متن کاملAround Kolmogorov complexity: basic notions and results
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devoted to this theory [4, 1, 5, 2, 7] where one can find the detailed exposition of many difficult results as well as historical references.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 basi...
متن کاملBounds on the Kolmogorov complexity function for infinite words
The Kolmogorov complexity function of an infinite word ξ maps a natural number to the complexity K(ξ n) of the n-length prefix of ξ. We investigate the maximally achievable complexity function if ξ is taken from a constructively describable set of infinite words. Here we are interested in linear upper bounds where the slope is the Hausdorff dimension of the set. As sets we consider Π1-definable...
متن کاملThe pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
We continue an investigation into resource-bounded Kolmogorov complexity [ABK06], which highlights the close connections between circuit complexity and Levin’s time-bounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introd...
متن کاملComputational power of neural networks: a characterization in terms of Kolmogorov complexity
The computational power of recurrent neural networks is shown to depend ultimately on the complexity of the real constants (weights) of the network. The complexity, or information contents, of the weights is measured by a variant of resource-bounded Kolmogorov complexity, taking into account the time required for constructing the numbers. In particular, we reveal a full and proper hierarchy of ...
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