Skip to main content
SpringerLink
Log in

On the Relation of Σ-Reducibility Between Admissible Sets

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Reducibility on admissible sets is studied which is a stronger version of the usual Σ-presentability of models. One of its informal prototypes is the interpretability of one computational device in the other. We obtain criteria of reducibility for recursively listed and pure sets, introduce the notion of jump, and prove exact boundaries for the ordinals of jumps. We also show that this reducibility is lifted to \(\mathbb{H}\mathbb{Y}\mathbb{P}\)-superstructures. Several results are proven on the relations of this reducibility to some known reducibilities.

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 Sviridenko D. I., “Logical programming in a board sense,” Algorithm Theory and Its Applications (Vychislitel'nye Sistemy), 129, 3-48 (1989).

    Google Scholar 

  2. Goncharov S. S. and Sviridenko D. I., “Mathematical foundations of semantic programming,” Sov. Phys. Dokl., 301, 608-610 (1986).

    Google Scholar 

  3. Sviridenko D. I. Goncharov S. S., and Ershov Yu. L., “Semantic programming,” Information Processing, Proc. IFIP 10th World Comput. Congr., Dublin/Ireland, 1113-1120 (1986).

  4. Rogers, H., Theory of Recursive Functions and Effective Computability, McGraw-Hill Book Company, New York; St. Louis; San Francisco; Toronto; London; Sydney (1967).

    Google Scholar 

  5. Morozov A. S., “∑-Definability of admissible sets,” Proceedings of Logic Colloquium-98. Praha, 334-351 (2000).

  6. Ershov Yu. L., Computability and Definability, Plenum Publ. Co., New York (1996).

    Google Scholar 

  7. Khisamiev A. N., “Degrees of algebraic structures and ∑-definability,” in: Logic Colloquium'2002: Abstracts of Contributed Papers, Münster, Germany, 2002, p. 39.

  8. Barwise J., Admissible Sets and Structures, Springer-Verlag, Berlin; Göttingen; Heidelberg (1975).

    Google Scholar 

  9. Richter L. J., “Degrees of structures,” J. Symbolic Logic, 46,No. 4, 723-731 (1981).

    Google Scholar 

  10. Slaman T. A., “Relative to any nonrecursive set,” Proc. Amer. Math. Soc., 126,No. 7, 2117-2122 (1998).

    Google Scholar 

  11. Morozov A. S. and Puzarenko V. G., “On ∑-subsets of natural numbers,” Algebra and Logic, 2004 (to appear).

Download references

Authors
  1. A. S. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Morozov, A.S. On the Relation of Σ-Reducibility Between Admissible Sets. Siberian Mathematical Journal 45, 522–535 (2004). https://doi.org/10.1023/B:SIMJ.0000028617.17064.08

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:SIMJ.0000028617.17064.08

Search

Navigation