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Theory of Computing Systems

, Volume 52, Issue 1, pp 113–132 | Cite as

Characterization of Kurtz Randomness by a Differentiation Theorem

  • Kenshi Miyabe
Article

Abstract

Brattka, Miller and Nies (2012) showed that some major algorithmic randomness notions are characterized via differentiability. The main goal of this paper is to characterize Kurtz randomness by a differentiation theorem on a computable metric space. The proof shows that integral tests play an essential part and shows that how randomness and differentiation are connected.

Keywords

Differentiability The differentiation theorem Kurtz randomness Integral test Computable metric space 

Notes

Acknowledgements

The author thanks Jason Rute and André Nies for the useful comments. The author also appreciates the anonymous referees for their careful reading and many comments. This work was partly supported by GCOE, Kyoto University and JSPS KAKENHI 23740072.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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