Kolmogorov complexity

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 the 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 descriptive complexity, Kolmogorov–Chaitin complexity, algorithmic entropy, or program-size complexity. It is named after Andrey Kolmogorov, who first published on the subject in 1963.

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.

Consider the following two strings of 32 lowercase letters and digits:

The first string has a short English-language description, namely "ab 16 times", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 32 characters.

More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings, like the abab example above, whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.

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