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Tropiącym fałszywe i malowanych sama skrywała się refleksji do trochę a low spatial frequency the vertical direction and the image is scanned columns because these coefficients are more likely to be correlated with their neighbors the vertical direction. If u 0. There can be at most a finite number of more likely descriptions, and therefore infinite number of less likely descriptions. Of the latter, there can be at most a finite number that are shorter than M. Therefore there must be infinite number of less likely descriptions that are longer than M, for all M. The question remains whether all strings found the real world are created by computable or finitely describable processes. This must be true for finite strings, but there are known to exist, at least mathematics, infinite length strings such as Chaitin's constant Ω that are not computable. fact, the vast majority of infinite length strings do not have finite length descriptions. Could there exist phenomena the real world that have infinite length descriptions that are not compressible? For example, would it be possible to take infinite number of measurements or observations, or to measure something with infinite precision? Do there exist infinite sources of random data? The laws of physics say no. At one time it was believed that the universe could be infinitely large and made up of matter that was infinitely divisible. The discoveries of the expanding universe and of atoms showed otherwise. The universe has a finite age, T, about 13 billion years. Because information cannot travel faster than the speed of light, c, our observable universe is limited to apparent 13 billion light years, although the furthest objects we can have since moved further away. Its mass is limited by the gravitational constant, G, to a value that prevents the universe from collapsing on itself. A complete description of the universe could therefore consist of a description of the exact positions and velocities of a finite number of particles. But quantum mechanics limits any combination of these two quantities to discrete multiples of Planck's constant, h. Therefore the universe, and everything it, must have a finite description length. The entropy nats bits 1 bits) is given by the Bekenstein bound as 1 of the area of the event horizon Planck units of area hG 2πc 3, a square of 1 x 10 meters on a side. For a sphere of radius Tc 13 billion light years, the bound is 2 x 10 bits. We now make two observations. First, if the universe were divided into regions the size of bits, then each volume would be about the size of a proton or neutron. This is rather remarkable because the number is derived only from T and the physical constants c, h, and G, which are unrelated to the properties of any particles. Second, if the universe were squashed flat, it would form a sheet about one neutron thick. Occam's Razor, which the computability of physics makes true, suggests that these two observations are not coincidences. I thank Ivo Danihelka, Glenn Szymon Grabowski, Aki Jäntti, Harri Hirvola, Mattern, Ilia Muraviev, Newman, Ondrus, Carol Rhatushnyak, Friedrich Regen, Steve Richfield, Sami Runsas, A. Shelwien, Ali Imran Khan Shirani, Yan Yin, and Bulat Ziganshin for helpful comments on this book. Thanks to Osman Turan for convering this book to XHTML compliant. Please send corrections or comments to mattmahoneyfl at gmail com. T. I. H. Witten. J. G. Cleary Modeling for Text Compression, ACM Computing Surveys 4, pp. 557. G. Chaitin On the length of programs for computing finite binary sequences. Journal of the ACM, G. Cleary, W. J. Teahan Experiments on the zero frequency problem, Proc. Data Compression Conference, 480. G. Cormack, N. Horspool Data compression using dynamic Markov modeling, Computer Journal 30 S. Grabowski, J. Swacha Language-independent word-based text compression with fast decompression, Proc. MEMSTECH, Polyana, Ukraine. M. Hutter artificial intelligence: Sequential decisions based on algorithmic probability. Springer, Berlin. H. Itoh, H. Tanaka efficient method for memory construction of suffix arrays, Proceedings of the IEEE String Processing and Information Retrieval Symposium, pp. 81. P. Ko, S. Aluru Space-efficient linear time construction of suffix arrays, Proceedings of the 14th Annual Symposium on Combinatorial Pattern Matching, pp. 200. A. Kolmogorov Three approaches to the quantitative definition of information. Problems Inform. Transmission, 1-7. L. A. Levin sequential search problems. Problems of Information Transmission, 9--266. L. A. Levin Randomness Conservation Inequalities: Information and Independence Mathematical Theories. Information and Control, S. J. Puglisi Exposition and analysis of a suffix sorting algorithm, Technical Report Number CAS-05-WS Dept of Computing and Software, McMaster University, Hamilton, Ontario, Canada. S. J. Puglisi, M. A. Maniscalco Faster lightweight suffix