Descriptive complexity of computable sequences revisited

The purpose of this paper is to answer two questions left open in Durand et al. (2001) [2]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence alpha: C-0' (alpha), defined as the minimal length of a program with oracle 0' that prints alpha, and M-infinity(alpha), defined as lim sup C(alpha(1:n)vertical bar n), where alpha(1:n) denotes the length-n prefix of alpha and C(x vertical bar y) stands for conditional Kolmogorov complexity. We show that C-0'(alpha) <= M-infinity (alpha)+ 0(1) and M-infinity (alpha) is not bounded by any computable function of C-0' (alpha), even on the domain of computable sequences. (C) 2020 Elsevier B.V. All rights reserved.