The Kolmogorov complexity of random reals

We investigate the initial segment complexity of random reals. Let K(σ) denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K(α ↾ n) and K(β ↾ n). It is well-known that a real α is 1-random iff there is a constant c such that for all n, K(α ↾ n)⩾n−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K(α ↾ n) for random α. Following work of Downey, Hirschfeldt and LaForte, we say that α⩽K β iff there is a constant O(1) such that for all n, K(α ↾ n)⩽K(β ↾ n)+O(1). We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K(α ↾ n)−K(Ω ↾ n)=∞ where Ω is Chaitinʹs random real. One is based upon (unpublished) work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ({β : β⩽K α})=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets (namely Ω∅(n−1) and Ω∅(m−1)) are different. The techniques introduced in this paper have already found a number of other applications.