Analogues of Chaitin's Omega in the computably enumerable sets

Authors:
G. Barmpalias;R. HöLzl;A. E. M. Lewis;W. Merkle
Affiliations:
State Key Lab of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, P.O. Box 8718, China;Bureau 6A35 LIAFA, 175 rue du Chevaleret, 75013 Paris, France;School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom;Institut für Informatik, Ruprecht-Karls-Universität, Heidelberg, Germany
Venue:
Information Processing Letters
Year:
2013

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Abstract

We show that there are computably enumerable (c.e.) sets with maximum initial segment Kolmogorov complexity amongst all c.e. sets (with respect to both the plain and the prefix-free version of Kolmogorov complexity). These c.e. sets belong to the weak truth table degree of the halting problem, but not every weak truth table complete c.e. set has maximum initial segment Kolmogorov complexity. Moreover, every c.e. set with maximum initial segment prefix-free complexity is the disjoint union of two c.e. sets with the same property; and is also the disjoint union of two c.e. sets of lesser initial segment complexity.