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Theory of Computing Systems

, Volume 52, Issue 1, pp 28–47 | Cite as

On the Gap Between Trivial and Nontrivial Initial Segment Prefix-Free Complexity

  • Martijn Baartse
  • George BarmpaliasEmail author
Article

Abstract

An infinite sequence X is said to have trivial (prefix-free) initial segment complexity if the prefix-free Kolmogorov complexity of each initial segment of X is the same as the complexity of the sequence of 0s of the same length, up to a constant. We study the gap between the minimum complexity K(0 n ) and the initial segment complexity of a nontrivial sequence, and in particular the nondecreasing unbounded functions f such that for a nontrivial sequence X, where K denotes the prefix-free complexity. Our first result is that there exists a \(\varDelta^{0}_{3}\) unbounded nondecreasing function f which does not have this property. It is known that such functions cannot be \(\varDelta^{0}_{2}\) hence this is an optimal bound on their arithmetical complexity. Moreover it improves the bound \(\varDelta^{0}_{4}\) that was known from Csima and Montalbán (Proc. Amer. Math. Soc. 134(5):1499–1502, 2006).

Our second result is that if f is \(\varDelta^{0}_{2}\) then there exists a non-empty \(\varPi^{0}_{1}\) class of reals X with nontrivial prefix-free complexity which satisfy (⋆). This implies that in this case there uncountably many nontrivial reals X satisfying (⋆) in various well known classes from computability theory and algorithmic randomness; for example low for Ω, non-low for Ω and computably dominated reals. A special case of this result was independently obtained by Bienvenu, Merkle and Nies (STACS, pp. 452–463, 2011).

Keywords

Kolmogorov complexity Initial segment prefix-free complexity K-triviality Low for Ω 

Notes

Acknowledgements

Barmpalias was supported by a research fund for international young scientists No. 611501-10168 and an International Young Scientist Fellowship number 2010-Y2GB03 from the Chinese Academy of Sciences. Partial support was also obtained by the Grand project: Network Algorithms and Digital Information of the Institute of Software, Chinese Academy of Sciences.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Computer Science InstituteTechnical University CottbusCottbusGermany
  2. 2.State Key Lab of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingPeople’s Republic of China

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