The Definition of Random Sequences

Kolmogorov has defined tile conditional complexity of an object y when the object x is already given to us as the minimal length of a binary program which by means of x computes y on a certain asymptotically optimal machine. On the basis of this definition he has proposed to consider those elements of a given large finite population to be random whose complexity is maximal. Almost all elements of the population have a complexity which is close to the maximal value. In this paper it is shown that the random elements as defined by Kolmogorov possess all conceivable statistical properties of randomness. They can equivalently be considered as the elements which withstand a certain universal stochasticity test. The definition is extended to infinite binary sequences and i t is shown that the non random sequences form a maximal constructive null set. Finally, the Kollektivs introduced by yon Mises obtain a definition which seems to satisfy all intuitive requirements.