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Immunity for Closed Sets

  • Douglas Cenzer
  • Rebecca Weber
  • Guohua Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5635)

Abstract

The notion of immune sets is extended to closed sets and \(\Pi^0_1\) classes in particular. We construct a \(\Pi^0_1\) class with no computable member which is not immune. We show that for any computably inseparable sets A and B, the class S(A,B) of separating sets for A and B is immune. We show that every perfect thin \(\Pi^0_1\) class is immune. We define the stronger notion of prompt immunity and construct an example of a \(\Pi^0_1\) class of positive measure which is promptly immune. We show that the immune degrees in the Medvedev lattice of closed sets forms a filter. We show that for any \(\Pi^0_1\) class P with no computable element, there is a \(\Pi^0_1\) class Q which is not immune and has no computable element, and which is Medvedev reducible to P. We show that any random closed set is immune.

Keywords

Computability \(\Pi^0_1\) Classes 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Douglas Cenzer
    • 1
  • Rebecca Weber
    • 2
  • Guohua Wu
    • 3
  1. 1.Department of MathematicsUniversity of FloridaGainesville
  2. 2.Department of MathematicsDartmouth CollegeHanover
  3. 3.School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore

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