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3 results

  • 5 - Relativization in 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 134-174

  • Summary

    We discuss the different contexts in which relativization occurs in randomness and the effect that the relativization chosen has on the results we can obtain. We study several characterizations of the K-trivials in terms of concepts ranging from cuppability to density, and we consider a uniform relativization for randomness that gives us more natural results for computable randomness, Schnorr randomness, and Kurtz randomness than the classical relativization does (the relativization for Martin-Löf randomness is unaffected by this change). We then evaluate the relativizations we have considered and suggest some avenues for further work.

  • STRONG JUMP-TRACEABILITY

  • NOAM GREENBERG, DAN TURETSKY
  • Journal:
    Bulletin of Symbolic Logic / Volume 24 / Issue 2 / June 2018
    Published online by Cambridge University Press:
    07 August 2018, pp. 147-164
    Print publication:
    June 2018

  • We review the current knowledge concerning strong jump-traceability. We cover the known results relating strong jump-traceability to randomness, and those relating it to degree theory. We also discuss the techniques used in working with strongly jump-traceable sets. We end with a section of open questions.