The Kolmogorov complexity of random reals

Abstract

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 sup n K(α ↾ n)−K(Ω ↾ n)=∞ where Ω is Chaitin's 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 One is based upon (unpublished) work of Solovay, and the other exploits a new idea.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.