Skip to main content
Log in

Immunity properties and strong positive reducibilities

  • Original Article
  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has \({\overline{K}\not\le_{\rm ss} B}\) (respectively, \({\overline{K}\not\le_{\overline{\rm s}} B}\)): here \({\le_{\overline{\rm s}}}\) is the finite-branch version of s-reducibility, ≤ss is the computably bounded version of \({\le_{\overline{\rm s}}}\), and \({\overline{K}}\) is the complement of the halting set. Restriction to \({\Sigma^0_2}\) sets provides a similar characterization of the \({\Sigma^0_2}\) hyperhyperimmune sets in terms of s-reducibility. We also show that no \({A \geq_{\overline{\rm s}}\overline{K}}\) is hyperhyperimmune. As a consequence, \({\deg_{\rm s}(\overline{K})}\) is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cooper S.B.: Enumeration reducibility using bounded information: counting minimal covers. Z. Math. Logik Grundlag. Math. 33, 537–560 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cooper S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton (2003)

    Google Scholar 

  3. Friedberg R.M., Rogers H. Jr.: Reducibility and completeness for sets of integers. Z. Math. Logik Grundlag. Math. 5, 117–125 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gill J.T. III, Morris P.H.: On subcreative sets and S-reducibility. J. Symb. Logic 39(4), 669–677 (1974)

    Article  MathSciNet  Google Scholar 

  5. McEvoy, K.: The Structure of the Enumeration Degrees. PhD thesis, School of Mathematics, University of Leeds (1984)

  6. Omanadze R.Sh.: On the upper semilattice of recursively enumerable sQ-degrees. Algebra Logic 30, 265–271 (1992) (English translation)

    Article  MathSciNet  Google Scholar 

  7. Omanadze R.Sh., Sorbi A.: A characterization of the \({\Delta^0_2}\) hyperhyperimmune sets. J. Symbolic Logic 73(4), 1407–1415 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Omanadze R.Sh., Sorbi A.: Immunity properties of the s-degrees. Georgian Math. J. 17(3), 563–579 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Rogers H. Jr.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)

    MATH  Google Scholar 

  10. Soare R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, Omega Series. Springer, Heidelberg (1987)

    Google Scholar 

  11. Soare R.I.: Computability and recursion. Bull. Symb. Logic 2, 284–321 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Solov’ev V.D.: Q-reducibility and hyperhypersimple sets. Veroyatn. Metod. i Kibern. 10–11, 121–128 (1974) Russian

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrea Sorbi.

Additional information

The research of the second author was supported by the Georgian National Science Foundation (Grants #GNSF/ST07/3-178 and #GNSF/ST08/3-391).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chitaia, I.O., Omanadze, R.S. & Sorbi, A. Immunity properties and strong positive reducibilities. Arch. Math. Logic 50, 341–352 (2011). https://doi.org/10.1007/s00153-010-0216-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-010-0216-5

Keywords

Mathematics Subject Classification (2000)

Navigation