Kolmogorov Complexity

In algorithmic information theory (a subfield of computer science), the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object. It is named after the Andrey Kolmogorov who first published on the subject in 1963.

Kolmogorov complexity is also known as "descriptive complexity" (not to be confused with descriptive complexity theory), Kolmogorov–Chaitin complexity, algorithmic entropy, or program-size complexity.

For example, consider the following two strings of length 64, each containing only lowercase letters and digits:

The first string has a short English-language description, namely "ab 32 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 64 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 whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.