Skip to main content
SpringerLink
Log in

A Generic relation on Recursively Enumerable Sets

  • Published:
Algebra and Logic Aims and scope

Abstract

We introduce the concept of a generic relation for algorithmic problems, which preserves the property of being decidable for a problem for almost all inputs and possesses the transitive property. As distinct from the classical m-reducibility relation, the generic relation under consideration does not possess the reflexive property: we construct an example of a recursively enumerable set that is generically incomparable with itself. We also give an example of a set that is complete with respect to the generic relation in the class of recursively enumerable sets.

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. I. Kapovich, A. Myasnikov, P. Schupp, and V. Shpilrain, “Generic-case complexity, decision problems in group theory, and random walks,” J. Alg., 264, No. 2, 665-694 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  2. I. Kapovich, A. Myasnikov, P. Schupp, and V. Shpilrain, “Average-case complexity and decision problems in group theory,” Adv. Math., 190, No. 2, 343-359 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. D. Hamkins and A. Miasnikov, “The halting problem is decidable on a set of asymptotic probability one,” Notre Dame J. Form. Log., 47, No. 4, 515-524 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. S. A. Cook, “The complexity of theorem-proving procedures,” ACM, Proc. 3d Ann. ACM Sympos. Theory Computing (Shaker Heights, Ohio 1971) (1971), pp. 151-158.

  6. C. G. jun. Jockusch, and P. E. Schupp, “Generic computability, Turing degrees, and asymptotic density,” J. London Math. Soc., II. Ser., 85, No. 2, 472-490 (2012).

  7. G. Igusa, “The generic degrees of density-1 sets, and a characterization of the hyperarithmetic reals,” J. Symb. Log., 80, No. 4, 1290-1314 (2015).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, ul. Pevtsova 13, Omsk, 644099, Russia

    A. N. Rybalov

  2. Omsk State Technical University, pr. Mira 11, Omsk, 644050, Russia

    A. N. Rybalov

Authors
  1. A. N. Rybalov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Rybalov.

Additional information

Supported by Russian Science Foundation, project 14-11-00085.

Translated from Algebra i Logika, Vol. 55, No. 5, pp. 587-596, September-October, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rybalov, A.N. A Generic relation on Recursively Enumerable Sets. Algebra Logic 55, 387–393 (2016). https://doi.org/10.1007/s10469-016-9410-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-016-9410-9

Keywords

Search

Navigation