Kolmogorov complexity and instance complexity of recursively enumerable sets
Logic
2009-09-25 v1
Abstract
We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2log n upper bound on the Kolmogorov complexity of initial segments of r.e.\ sets is optimal and characterize the T-degrees of r.e. sets which attain this bound. The main part of the paper is concerned with instance complexity of r.e. sets. We construct a nonrecursive r.e. set with instance complexity logarithmic in the Kolmogorov complexity. This refutes a conjecture of Ko, Orponen, Sch"oning, and Watanabe. In the other extreme, we show that all wtt-complete set and all Q-complete sets have infinitely many hard instances.
Cite
@article{arxiv.math/9405208,
title = {Kolmogorov complexity and instance complexity of recursively enumerable sets},
author = {Martin Kummer},
journal= {arXiv preprint arXiv:math/9405208},
year = {2009}
}