Pseudojump Operators and $\Pi^0_1$ Classes

  • Authors:

  • Douglas Cenzer;Geoffrey Laforte;Guohua Wu

  • Affiliations:

  • Department of Mathematics, University of Florida P.O. Box 118105, Gainesville, Florida 32611, USA;Department of Computer Science, University of West Florida Pensacola, Florida 32514, USA;School of Physical and Mathematical Sciences, Nanyang Technological University, 639798, Singapore

  • Venue:
  • CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
  • Year:
  • 2007

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Abstract

For a pseudojump operator VXand a$\Pi^0_1$ class P, we consider properties of the set{VX: XεP}. We show that there always exists XεPwith $V^X \leq_T {\mathbf 0'}$ and that if PisMedvedev complete, then there exists XεPwith $ V^X \equiv_T {\mathbf 0'}$. We examine theconsequences when VXis Turingincomparable with VYfor X≄ Yin Pand when $W_e^X = W_e^Y$ for allX,Yε P. Finally, we give acharacterization of the jump in terms of $\Pi^0_1$ classes.