Editing Kolmogorov complexity (section)

=== Intuition ===
Consider the following two [[string (computer science)|strings]] of 32 lowercase letters and digits:

:  <code>abababababababababababababababab</code> , and
:  <code>4c1j5b2p0cv4w1x8rx2y39umgw5q85s7</code>

The first string has a short English-language description, namely "write ab 16 times", which consists of '''17''' characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, i.e., "write 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" which has '''38''' characters. Hence the operation of writing the first string can be said to have "less complexity" than writing the second.

More formally, the [[complexity]] of a string is the length of the shortest possible description of the string in some fixed [[Turing complete|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.

The Kolmogorov complexity can be defined for any mathematical object, but for simplicity the scope of this article is restricted to strings. We must first specify a description language for strings.  Such a description language can be based on any computer programming language, such as [[Lisp programming language|Lisp]], [[Pascal (programming language)|Pascal]], or [[Java (programming language)|Java]].  If '''P''' is a program which outputs a string ''x'', then '''P''' is a description of ''x''. The length of the description is just the length of '''P''' as a character string, multiplied by the number of bits in a character (e.g., 7 for [[ASCII]]).

We could, alternatively, choose an encoding for [[Turing machine]]s, where an ''encoding'' is a function which associates to each Turing Machine '''M''' a bitstring <'''M'''>. If '''M''' is a Turing Machine which, on input ''w'', outputs string ''x'', then the concatenated string <'''M'''> ''w'' is a description of ''x''. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. In this article, an informal approach is discussed.

Any string ''s'' has at least one description. For example, the second string above is output by the [[pseudo-code]]:

 '''function''' GenerateString2()
     '''return''' "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7"

whereas the first string is output by the (much shorter) pseudo-code:

 '''function''' GenerateString1()
     '''return''' "ab" × 16

If a description ''d''(''s'') of a string ''s'' is of minimal length (i.e., using the fewest bits), it is called a '''minimal description''' of ''s'', and the length of ''d''(''s'') (i.e. the number of bits in the minimal description) is the '''Kolmogorov complexity''' of ''s'', written ''K''(''s''). Symbolically,

:''K''(''s'') = |''d''(''s'')|.

The length of the shortest description will depend on the choice of description language; but the effect of changing languages is bounded (a result called the ''invariance theorem'').