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Kolmogorov randomness

Kolmogorov randomness defines a string (usually of bits) as being random if and only if every computer program that can produce that string is at least as long as the string itself. To make this precise, a universal computer (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program that is shorter than the string itself. For every universal computer, there is at least one algorithmically random string of each length. Whether a particular string is random, however, depends on the specific universal computer that is chosen. This is because a universal computer can have a particular string hard-coded in itself, and a program running on this universal computer can then simply refer to this hard-coded string using a short sequence of bits (i.e. much shorter than the string itself).

This definition can be extended to define a notion of randomness for infinite sequences from a finite alphabet. These algorithmically random sequences can be defined in three equivalent ways. One way uses an effective analogue of measure theory; another uses effective martingales. The third way defines an infinite sequence to be random if the prefix-free Kolmogorov complexity of its initial segments grows quickly enough — there must be a constant c such that the complexity of an initial segment of length n is always at least n − c. This definition, unlike the definition of randomness for a finite string, is not affected by which universal machine is used to define prefix-free Kolmogorov complexity.