When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2009.1810
URN: urn:nbn:de:0030-drops-18107
URL: http://drops.dagstuhl.de/opus/volltexte/2009/1810/
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### Kolmogorov Complexity and Solovay Functions

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### Abstract

Solovay (1975) proved that there exists a computable upper bound~$f$ of the prefix-free Kolmogorov complexity function~$K$ such that $f(x)=K(x)$ for infinitely many~$x$. In this paper, we consider the class of computable functions~$f$ such that $K(x) \leq f(x)+O(1)$ for all~$x$ and $f(x) \leq K(x)+O(1)$ for infinitely many~$x$, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.

### BibTeX - Entry

@InProceedings{bienvenu_et_al:LIPIcs:2009:1810,
author =	{Laurent Bienvenu and Rod Downey},
title =	{{Kolmogorov Complexity and Solovay Functions}},
booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
pages =	{147--158},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-09-5},
ISSN =	{1868-8969},
year =	{2009},
volume =	{3},
editor =	{Susanne Albers and Jean-Yves Marion},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},