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Demuth’s Path to Randomness

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Computation, Physics and Beyond (WTCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7160))

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Abstract

Osvald Demuth (1936–1988) studied constructive analysis in the Russian style. For this he introduced notions of effective null sets which, when phrased in classical language, yield major algorithmic randomness notions. He proved several results connecting constructive analysis and randomness that were rediscovered only much later.

We give an overview in mostly chronological order. We sketch a proof that Demuth’s notion of Denjoy sets (or reals) coincides with computable randomness. We show that he worked with a test notion that is equivalent to Schnorr tests relative to the halting problem. We also discuss the invention of Demuth randomness, and Demuth’s and Kučera’s work on semigenericity.

A. Kučera is partially supported by the Research project of the Ministry of Education of the Czech Republic MSM0021620838. Nies is partially supported by the Marsden Foundation of New Zealand under grant 09-UOA-184. We wish to dedicate this paper to Cris Calude on the occasion of his 60th birthday. We are very grateful to Cris because he is the person who started the study of algorithmic randomness in New Zealand. Today it is one of the most active fields in logic and computer science. Contributions by New Zealand-based researchers were essential.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Antonín Kučera

  2. Department of Computer Science, University of Auckland, Auckland, New Zealand

    André Nies

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  1. Antonín Kučera
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  2. André Nies
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Editors and Affiliations

  1. Department of Computer Science, University of Auckland, Private Bag 92019, 1142, Auckland, New Zealand

    Michael J. Dinneen , Bakhadyr Khoussainov  & André Nies ,  & 

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Kučera, A., Nies, A. (2012). Demuth’s Path to Randomness. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_12

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27653-8

  • Online ISBN: 978-3-642-27654-5

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