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Kolmogorov complexity

Measure of algorithmic complexity / From Wikipedia, the free encyclopedia

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In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 [1][2] and is a generalization of classical information theory.

This image illustrates part of the Mandelbrot set fractal. Simply storing the 24-bit color of each pixel in this image would require 23 million bytes, but a small computer program can reproduce these 23 MB using the definition of the Mandelbrot set and the corner coordinates of the image. Thus, the Kolmogorov complexity of this image is much less than 23 MB in any pragmatic model of computation. PNG's general-purpose image compression only reduces it to 1.6 MB, smaller than the raw data but much larger than the Kolmogorov complexity.

The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than P's own length (see section § Chaitin's incompleteness theorem); hence no single program can compute the exact Kolmogorov complexity for infinitely many texts.