Demuth’s Path to Randomness

  • Antonín Kučera
  • André Nies
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7160)

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.

Keywords

Computable Function Random Real Computable Sequence Turing Degree Arithmetical Real 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Antonín Kučera
    • 1
  • André Nies
    • 2
  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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