Oscillation in the initial segment complexity of random reals

Abstract

We study oscillation in the prefix-free complexity of initial segments of 1-random reals. For upward oscillations, we prove that P n∈ω 2 −g(n)

Solovay functions and their applications in algorithmic randomness

“…Another consequence of the construction performed in this proof is the dual version of Theorem 3.1 stated in Proposition 3.3. The first part of the proposition has been obtained earlier on and in different ways by Miller and Yu [30,Corollary 3.2], and in fact with the weaker hypothesis that h is unbounded. Proposition 3.3 There exists no function h : N → N that tends to infinity and such that…”

Kolmogorov complexity and computably enumerable sets

“…A natural 2 At this point we would like to draw a parallel between the study of the K degrees of c.e. sets and the K degrees of Martin-Löf random sets that was the object of study in [MY08,MY10]. One of the main open questions in this study was whether there is a maximal element in the K degrees of random reals, which is an analogue of question (3.5).…”

Pricing complexity options

“…In this case, however, sup D n (X ↾ n) will be ∞ for almost all X ∈ 2 ω . In fact Li and Vitányi showed D n (X ↾ n) > log n for infinitely many n for almost all X. Solovay showed that lim inf D n (X ↾ n) will be finite [MY11]. V = ∞ in this case since we can simply wait for a sufficiently high D n value.…”