Skip to main content
SpringerLink
Log in

Minimal generalized computable enumerations and high degrees

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We establish that the set of minimal generalized computable enumerations of every infinite family computable with respect to a high oracle is effectively infinite. We find some sufficient condition for enumerations of the infinite families computable with respect to high oracles under which there exist minimal generalized computable enumerations that are irreducible to the enumerations.

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

Access this article

Log in via an institution

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. Goncharov S. S. and Sorbi A., “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, vol. 36, no. 6, 359–369 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  2. Badaev S. A. and Goncharov S. S., “Rogers semilattices of families of arithmetic sets,” Algebra and Logic, vol. 40, no. 5, 283–291 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  3. Badaev S. A. and Podzorov S. Yu., “Minimal coverings in the Rogers semilattices of Σn 0-computable numberings,” Sib. Math. J., vol. 43, no. 4, 616–622 (2002).

    Article  MATH  Google Scholar 

  4. Badaev S. A., Goncharov S. S., and Sorbi A., “Elementary theories for Rogers semilattices,” Algebra and Logic, vol. 44, no. 3, 143–147 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  5. Badaev S. A., Goncharov S. S., and Sorbi A., “Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy,” Algebra and Logic, vol. 45, no. 6, 361–370 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ershov Yu. L., Theory of Numberings [Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  7. Ershov Yu. L., “Theory of numberings,” in: Handbook of Computability Theory (E. R. Griffor, ed.), Elsevier, Amsterdam, 1999, 473–503 (Stud. Logic Found. Math.; vol. 140).

    Chapter  Google Scholar 

  8. Soare R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, and Tokyo (1987).

    Book  MATH  Google Scholar 

  9. Faizrahmanov M. Kh., “Universal generalized computable numerations and hyperimmunity,” Algebra and Logic (to be published).

Download references

Author information

Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kazan, Russia

    M. Kh. Faizrahmanov

Authors
  1. M. Kh. Faizrahmanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Kh. Faizrahmanov.

Additional information

Original Russian Text Copyright © 2017 Faizrahmanov M.Kh.

The author was supported by the subsidy of the government task for Kazan (Volga Region) Federal University (Grant 1.1515.2017/PP) and the Russian Foundation for Basic Research (Grant 15–01–08252).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faizrahmanov, M.K. Minimal generalized computable enumerations and high degrees. Sib Math J 58, 553–558 (2017). https://doi.org/10.1134/S0037446617030181

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446617030181

Keywords

Search

Navigation