On the number of infinite sequences with trivial initial segment complexity

Authors:
George Barmpalias;T. F. Sterkenburg
Affiliations:
State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, P.O. Box 8718, China;Institute for Logic, Language and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

The sequences which have trivial prefix-free initial segment complexity are known as K-trivial sets, and form a cumulative hierarchy of length @w. We show that the problem of finding the number of K-trivial sets in the various levels of the hierarchy is @D"3^0. 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 @D"3^0 is sharp.