On the number of infinite sequences with trivial initial segment complexity

The sequences which have trivial prefix-free initial segment complexity are known as K -trivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of K -trivial sets in the various levels of the hierarchy is ∆3. This answers a question of Downey/Miller/Yu (see Downey (2010) [7, Section 10.1.4]) which also appears in Nies (2009) [17, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the computable oracles. In both cases the classification ∆3 is sharp. © 2011 Elsevier B.V. All rights reserved.