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# Topic: Kolmogorov complexity

###### In the News (Tue 23 Apr 19)

 Kolmogorov complexity - Wikipedia, the free encyclopedia In computer science, the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity) of an object such as a piece of text is a measure of the computational resources needed to specify the object. Strings whose Kolmogorov complexity is small relative to the string's size are not considered to be complex. The notion of Kolmogorov complexity is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem. en.wikipedia.org /wiki/Kolmogorov_complexity   (1960 words)

 Andrey Kolmogorov   (Site not responding. Last check: 2007-11-07) Kolmogorov was one of the broadest of this century's mathematicians. Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 1941 on turbulence that even today are of fundamental importance. Kolmogorov's notion of complexity is a measure of randomness, one that is closely related to Claude Shannon's entropy rate of an information source. www.exploratorium.edu /complexity/CompLexicon/kolmogorov.html   (330 words)

 Greg Harfst's Home Page: Kolmogorov Complexity   (Site not responding. Last check: 2007-11-07) Put simply, the Kolmogorov Complexity of an object is the length of the shortest computer program that runs on a computer and outputs that object. It can be shown that objects with high Kolmogorov Complexity are random, in the sense that it will pass all tests of randomness. Kolmogorov Complexity also gives a way of analyzing the complexity of a finite string. nms.lcs.mit.edu /~gch/kolmogorov.html   (593 words)

 PlanetMath: Kolmogorov complexity   (Site not responding. Last check: 2007-11-07) Kolmogorov Complexity provides an answer to these questions in the form of a measure of information content in individual objects. See Kolmogorov complexity function and invariance theorem for more details. This is version 6 of Kolmogorov complexity, born on 2003-07-01, modified 2005-01-02. planetmath.org /encyclopedia/KolmogorovComplexity.html   (158 words)

 [No title] Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. `Kolmogorov' complexity was earlier proposed by Ray Solomonoff in a Zator Technical Report in 1960 and in a long paper in Information and Control in 1964. homepages.cwi.nl /~paulv/kolmogorov.html   (824 words)

 Kolmogorov Complexity I   (Site not responding. Last check: 2007-11-07) Kolmogorov complexity allows us to describe the fact that some strings are more regular in some sense than are others. Kolmogorov Complexity tells us how compressible a string is. The higher the Kolmogorov complexity, the less we can squeeze it. A long string with low Kolmogorov complexity can be described by a smaller string -- therefore it could be replaced with a smaller string and we would still have the same information. www.oswego.edu /~delancey/309_DIR/LLT_LECTURES/13_kolmogorov_1_out.html   (616 words)

 Cybernetics and Information Theory in the United States, France and the Soviet Union [72] Kolmogorov was also among the first mathematicians to appreciate the significance of Shannon’s “Mathematical Theory of Communication.” [73] Kolmogorov and his students developed a rigorous mathematical foundation of information theory, providing precise definitions and meticulous proofs of major theorems. [93] Kolmogorov was particularly pleased to remark (in private) that from the viewpoint of information theory (Soviet) newspapers were less informative than poetry since political discourse employed a large number of stock phrases and was highly predictable in its content. Kolmogorov’s poetic studies had a surprising outcome, leading to the elaboration of an original mathematical theory of complexity related to the concepts of information and entropy. www.infoamerica.org /documentos_word/shannon-wiener.htm   (12630 words)

 Nick Szabo -- Introduction to Algorithmic Information Theory This field is also known by its main result, Kolmogorov complexity. Kolmogorov complexity gives us a new way to grasp the mathematics of information, which is used to describe the structures of the world. The description method that meets these criteria is the Kolmogorov complexity: the size of the shortest program (in bits) that without additional data, computes the string and terminates. szabo.best.vwh.net /kolmogorov.html   (1710 words)

 Citebase - Shannon Information and Kolmogorov Complexity We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. In classical information theory the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the string itself. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0410002   (866 words)

 Kolmogorov Complexity and Applications, Autumn 2000   (Site not responding. Last check: 2007-11-07) Kolmogorov Complexity is a central concept and a powerful tool in the understanding of the quantitative nature of information and its processing and transmission. We use theory of algorithms to define the complexity of an individual string (instead of random variables as in Shannon information theory) and the notion of individual random object. An Introduction to Kolmogorov Complexity and its Applications. user.it.uu.se /~vorobyov/Courses/KC/2000   (232 words)

 GENERALIZED KOLMOGOROV COMPLEXITY The algorithmic information or Kolmogorov complexity of a bitstring x is the length of the shortest program that computes x and halts. This leads to generalizations of the concept of Kolmogorov complexity, and has consequences for Solomonoff's theory of algorithmic probability and universal prediction. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the ``true'' information content of some (possibly infinite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. www.idsia.ch /~juergen/kolmogorov.html   (580 words)

 Kolmogorov Complexity and Solomonoff Induction   (Site not responding. Last check: 2007-11-07) Kolmogorov defined the complexity of a string x as the length of its shortest description p on a universal Turing machine U (K(x)=min{l(p):U(p)=x}). A gentle tour of Kolmogorov complexity and its applications, presented by Chris Hillman at UIUC, April 1999, in PostScript. Kolmogorov Complexity and its Applications : lecture notes by Alexander Shen. www.idsia.ch /~marcus/kolmo.htm   (1308 words)

 NICTA Course: Introduction to Kolmogorov complexity and applications We treat the central ideas and their applications of Kolmogorov complexity---a modern notion of randomness dealing with the quantity of information in individual objects. Kolmogorov complexity is known variously as 'algorithmic information', 'algorithmic entropy', 'Kolmogorov-Chaitin complexity', 'descriptional complexity', 'shortest program length', 'algorithmic randomness', and others. This series of talks is based on the (text)book by Ming Li and Paul Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, Springer-Verlag, New York, 2nd Edition, 1997. www.cse.unsw.edu.au /news/vitanyi.html   (383 words)

 Kolmogorov Complexity II One exciting feature of Kolmogorov complexity is that it has an incompleteness result as a relatively straightforward consequence. Consider for example a string S that is Chaitin random and is of Kolmogorov complexity k. Therefore, the Kolmogorov complexity of S is at most n (since a program P of size n can print S!). www.oswego.edu /~delancey/309_DIR/LLT_LECTURES/14_kolmogorov_2_out.html   (1219 words)

 Algorithmic information theory - Wikipedia, the free encyclopedia According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously." Algorithmic information theory principally studies Kolmogorov complexity and other complexity measures on strings (or other data structures). Since most mathematical objects can be described in terms of strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including the integers and real numbers. en.wikipedia.org /wiki/Algorithmic_information_theory   (363 words)

 Chaitin-Kolmogorov Complexity   (Site not responding. Last check: 2007-11-07) A program is a procedure for forming the organization; it might be the set of instructions that are given to a negotiator to secure a trade agreement, or it might be a description of the procedures a movie producer follows to ``put a deal together''. Intuitively, more complex organizations are more difficult to form and this is revealed by the length of the shortest program that will turn the inputs into the required organization. Hence, the Chaitin-Kolmogorov complexity of the consistent coalitions is less than or equal to N + c. www.beje.decon.ufpe.br /vany/node11.html   (306 words)

 Dembski Intelligent Design Presentation (Skeptical Inquirer November 2002) A logically consistent definition of complexity is given, for example, in the algorithmic theory of randomness-probability-complexity (and is often referred to as Kolmogorov complexity). The Kolmogorov complexity of a perfectly spherical piece of stone is much lower than it is for any other pebble having irregular shape and non-uniform distribution of density and color. He has authored nearly 300 papers in various fields of physics and electrochemistry, as well as several books; he holds a number of patents and is a recipient of several prizes and awards for his research, including one from the Royal Society of London for the discovery and study of the photodeposition of semiconductors. www.csicop.org /si/2002-11/dembski.html   (2551 words)

 Computational Complexity: More on Kolmogorov Complexity   (Site not responding. Last check: 2007-11-07) Computational complexity and other fun stuff in math and computer science as viewed by Lance Fortnow. Kolmogorov formalized this idea by measuring the randomness of a string x, denoted C(x), by the smallest program that generates x. At this Dagstuhl workshop there are many talk on a number of variations of Kolmogorov complexity. weblog.fortnow.com /2003/04/more-on-kolmogorov-complexity.html   (234 words)

 Quantum Kolmogorov Complexity - Berthiaume, van Dam, Laplante (ResearchIndex)   (Site not responding. Last check: 2007-11-07) In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the shortest quantum input to a universal quantum Turing machine that produces the initial qubit string with... citeseer.ist.psu.edu /367854.html   (482 words)

 Amazon.com: An Introduction to Kolmogorov Complexity and Its Applications (Texts in Computer Science): Books: Ming ...   (Site not responding. Last check: 2007-11-07) The theory of Kolmogorov complexity is slowly making its way into applications, these being coding theory and computational intelligence, and network performance optimization, and this book serves as a fine reference for those readers interested in these applications. The basic idea behind Kolmogorov complexity is straighforward: a good measure of the complexity of an object is the length of the shortest computer program which will construct that object. What Kolmogorov complexity provides is the so-called "universal" distribution, which is guarunteed to be a "good" initial distirbution. www.amazon.com /exec/obidos/tg/detail/-/0387948686?v=glance   (2538 words)

 Publications Li and P.M.B. Vitanyi, A brief introduction to Kolmogorov complexity and its applications, in: Chinese Mathematics into the 21st Century, Wu Wen-tsun and Cheng Min-de, Eds, Peking University Press, Peking, China, 1991, pp. Li and P.M.B. Vitanyi, Kolmogorov complexity and its applications, Japanese version, Chapter IV in: Handbook for Theoretical Computer Science, (J. van Leeuwen, Editor), Maruzen Publishing Co., Oyama, Japan, 1994. P.M.B. Vitanyi, Andrei Nikolaevich Kolmogorov, CWI Quarterly, 1(1988), pp. homepages.cwi.nl /~paulv/kolmcompl.html   (1936 words)

 Complexity 2003 - Kolmogorov day In honor of the one hundreth anniversary of the birth of Andrei N. Kolmogorov, the Conference on Computational Complexity is holding a special day of talks on the topic of Kolmogorov complexity and its applications to theoretical computer science and computational complexity. Andrei Nikolaevich Kolmogorov was born on April 25, 1903 in Tambov, Russia, and died 20 Oct 1987 in Moscow, Russia. He made fundamental contributions to several areas of mathematics, including probability theory, mathematical statistics, information theory, dynamical systems, topology, logic, as well as Hilbert's 6th and 13th problems. www.lri.fr /~laplante/kolmogorov.htm   (214 words)

 Topics in Kolmogorov Complexity Kolgmogorov complexity is a deep and sophisticated theory that provides the means to measure the intrinsic information in objects via their algorithmic description length. The main text for this seminar will be An introduction to Kolmogorov complexity and its applications by Li and Vitanyi. All section references in the list below are to the course book: An Introduction to Kolmogorov Complexity and its applications, by Li and Vitanyi, Second edition. www.cs.technion.ac.il /~tzach/Kolmogorov   (376 words)

 Measures of Complexity: SubIndex on KOLMOGOROV - the philosophy of complexity per se with application to some examples in evolution, To be published in: F. Heylighen & Aerts (eds.): The Evolution of Complexity, Kluwer, Dordrecht. Cover,TM; 1983, Kolmogorov Complexity, Data Compression, and Inference, in The Impact of Data Processing Techniques on Communications, Eds.Durand,H; di Lullo,M; Sinclair,C;, Nijhoff, Dordrecht, pages 23-33 Zvonkin,AK; Levin,LA; 1970, The Complexity of Finite Objects and the Development of the Concepts of Information and Randomness by Means of the Theory of Algorithms, Russian Mathematics Surveys, 256, 83-124 bruce.edmonds.name /combib/kolmogorov.html   (952 words)

 An Introduction to Kolmogorov Complexity and Its Applications An Introduction to Kolmogorov Complexity and Its Applications (1 cr) Together with Ming Li they pioneered applications of Kolmogorov complexity and co-authored ``An Introduction to Kolmogorov Complexity and its Applications,'' Springer-Verlag, New York, 1993 (2nd Edition 1997), parts of which have been translated into Chinese, Russian and Japanese. Recommended book: Ming Li and Paul Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, Springer-Verlag, New York, 2nd Edition, 1997. www.tcs.hut.fi /~pkaski/kca   (454 words)

 CompLearn NCD The method is the outcome of a mathematical theoretical developments based on Kolmogorov complexity. NGD Normalized Google Distance when Google is used to determine a probability density function over search terms that yields a Shannon-Fano code length by taking the negative log of the probability of a term. An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, New York, 2nd Edition, 1997. www.complearn.org /ncd.html   (636 words)

 Kolmogorov complexity   (Site not responding. Last check: 2007-11-07) Special issue of the Computer Journal on Kolmogorov Complexity (1999). Tributes to, pictures of, bibliography of, etc. Andrei Nikolaevich Kolmogorov. Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "Kolmogorov complexity", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. www.nist.gov /dads/HTML/kolmogorov.html   (152 words)

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