Immunity for Closed Sets

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


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.


Computability \(\Pi^0_1\) Classes 


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  1. 1.
    Ambos-Spies, K., Jockusch, C.G., Shore, R.A., Soare, R.I.: An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. Amer. Math. Soc. 281, 109–128 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barmpalias, G., Brodhead, P., Cenzer, D., Dashti, S., Weber, R.: Algorithmic randomness of closed sets. J. Logic Comp. 17, 1041–1062 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brodhead, P., Cenzer, D., Dashti, S.: Random closed sets. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 55–64. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Binns, S.: A splitting theorem for the Medvedev and Muchnik lattices. Math. Logic Q. 49, 327–335 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Binns, S.: Small \(\Pi^0_1\) classes. Arch. Math. Logic 45, 393–410 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Binns, S.: Hyperimmunity in 2. Notre Dame J. Formal Logic 4, 293–316 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cenzer, D., Downey, R., Jockusch, C.J.: Countable Thin \(\Pi^0_1\) Classes. Ann. Pure Appl. Logic 59, 79–139 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cenzer, D., Nies, A.: Initial segments of the lattice of \(\Pi^0_1\) classes. Journal of Symbolic Logic 66, 1749–1765 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cholak, P., Coles, R., Downey, R., Hermann, E.: Automorphisms of the Lattice of \(\Pi^0_1\) Classes. Transactions Amer. Math. Soc. 353, 4899–4924 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cenzer, D., Remmel, J.B.: \(\Pi^0_1\) classes in mathematics. In: Ershov, Y., Goncharov, S., Nerode, A., Remmel, J. (eds.) Handbook of Recursive Mathematics, Part Two. Elsevier Studies in Logic, vol. 139, pp. 623–821 (1998)Google Scholar
  11. 11.
    Cenzer, D., Remmel, J.B.: Effectively Closed Sets. ASL Lecture Notes in Logic (in preparation)Google Scholar
  12. 12.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity (in preparation),
  13. 13.
    Jockusch, C.G.: \(\Pi^0_1\) classes and boolean combinations of recursively enumerable sets. J. Symbolic Logic 39, 95–96 (1974)CrossRefzbMATHGoogle Scholar
  14. 14.
    Jockusch, C.G., Soare, R.: Degrees of members of \(\Pi^0_1\) classes. Pacific J. Math. 40, 605–616 (1972)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jockusch, C.G., Soare, R.: \(\Pi^0_1\) classes and degrees of theories. Trans. Amer. Math. Soc. 173, 35–56 (1972)zbMATHGoogle Scholar
  16. 16.
    Medvedev, Y.T.: Degrees of difficulty of the mass problem. Dokl. Akad. Nauk SSSR (N.S.) 104, 501–504 (1955) (in Russian)MathSciNetGoogle Scholar
  17. 17.
    Muchnik, A.A.: On strong and weak reducibilities of algorithmic problems. Sibirsk. Mat. Ž. 4, 1328–1341 (1963) (in Russian)Google Scholar
  18. 18.
    Simpson, S.: Mass problems and randomness. Bull. Symb. Logic 11, 1–27 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Simpson, S.: \(\Pi^0_1\) classes and models of \(\text{WKL}_0\). In: Simpson, S. (ed.) Reverse Mathematics 2001. Association for Symbolic Logic: Lecture Notes in Logic, vol. 21, pp. 352–378 (2005)Google Scholar
  20. 20.
    Simpson, S.: An extension of the recursively enumerable Turing degrees. J. London Math. Soc. 75, 287–297 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Soare, R.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar

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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