Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

2 results

  • 7 - Biased algorithmic randomness

  • Edited by Johanna N. Y. Franklin, Hofstra University, New York, Christopher P. Porter, Drake University, Iowa
  • Book:
    Algorithmic Randomness
    Published online:
    07 May 2020
    Print publication:
    07 May 2020, pp 206-231

  • Summary

    In this survey, we lay out the central results in the study of algorithmic randomness with respect to biased probability measures. The first part of the survey covers biased randomness with respect to computable measures. The central technique in this area is the transformation of random sequences via certain randomness-preserving Turing functionals, which can be used to induce non-uniform probability measures. The second part of the survey covers biased randomness with respect to non-computable measures, with an emphasis on the work of Reimann and Slaman on the topic, as well as the contributions of Miller and Day in developing Levin's notion of a neutral measure. We also discuss blind randomness as well as van Lambalgen's theorem for both computable and non-computable measures. As there is no currently-available source covering all of these topics, this survey fills a notable gap in the algorithmic randomness literature.

  • RANK AND RANDOMNESS

  • RUPERT HÖLZL, CHRISTOPHER P. PORTER
  • Journal:
    The Journal of Symbolic Logic / Volume 84 / Issue 4 / December 2019
    Published online by Cambridge University Press:
    19 September 2019, pp. 1527-1543
    Print publication:
    December 2019

  • We show that for each computable ordinal $\alpha > 0$ it is possible to find in each Martin-Löf random ${\rm{\Delta }}_2^0 $ degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R’s rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.