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

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## Abstract

*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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