Counting the Changes of Random \({\Delta^0_2}\) Sets

  • Santiago Figueira
  • Denis Hirschfeldt
  • Joseph S. Miller
  • Keng Meng Ng
  • André Nies
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6158)

Abstract

Consider a Martin-Löf random \({\Delta^0_2}\) set Z. We give lower bounds for the number of changes of \(Z_s \upharpoonright n\) for computable approximations of Z. We show that each nonempty \({\Pi^0_1}\) class has a low member Z with a computable approximation that changes only o(2n) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that \(Z_s\upharpoonright n\) changes more than c 2n times.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Santiago Figueira
    • 1
  • Denis Hirschfeldt
    • 2
  • Joseph S. Miller
    • 3
  • Keng Meng Ng
    • 3
  • André Nies
    • 4
  1. 1.Dept. of Computer Science, FCEyNUniversity of Buenos Aires and CONICET 
  2. 2.Dept. of MathematicsThe University of Chicago 
  3. 3.Dept. of MathematicsUniversity of Wisconsin—Madison 
  4. 4.Dept. of Computer ScienceThe University of Auckland 

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