Increasing the Gap between Descriptional Complexity and Algorithmic Probability

In this paper we seek to analyze the relationship between two different ways of understanding the intrinsic algorithmic randomness of strings and real numbers. In the setting of discrete spaces, a cornerstone of the theory of prefix-free Kolmogorov complexity is the coding theorem. This theorem states that the negative logarithm of the probability that a string is the output of a universal prefix-free machine coincides, within some additive constant, with the length of its shortest description. It has been suggested that this ties two fundamental principles together: Bayes´ theorem and Occam´s razor. Certainly, the result lends support to the belief that prefix-free complexity is a natural measure of the algorithmic randomness of strings.