Skip to main content
SpringerLink
Log in

A Hierarchy below the Halting Problem for Additive Machines

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We answer the question posed by Klaus Meer and Martin Ziegler in “Uncomputability below the Real Halting Problem” whether the set of rational numbers is strictly easier than the Halting Problem with respect to additive machines. We define a hierarchy below the Halting Problem such that the set of rational numbers is strictly easier than the new problems. Moreover, every problem of the hierarchy is strictly easier than its successor.

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. Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Am. Math. Soc. 21, 1–46 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998)

    Google Scholar 

  3. Cucker, F., Koiran, P.: Computing over the reals with addition and order: higher complexity classes. J. Complex. 11, 358–376 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fournier, H., Koiran, P.: Lower bounds are not easier over the reals. In: ICALP’2000. Lecture Notes in Comput. Sci., vol. 1853, pp. 832–843 (2000)

  5. Koiran, P.: Computing over the reals with addition and order. Theor. Comput. Sci. 133, 35–47 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  6. Meer, K., Ziegler, M.: An explicit solution to Post’s problem over the reals. In: 15th International Symposium on Fundamentals of Computation Theory. Lecture Notes in Comput. Sci., vol. 3623, pp. 456–467 (2005)

  7. Meer, K., Ziegler, M.: On the word problem for a class of groups with infinite presentations. Preprint (2006)

  8. Meer, K., Ziegler, M.: Uncomputability below the real Halting Problem. In: CiE2006. Lecture Notes in Comput. Sci., vol. 3988, pp. 368–377 (2006)

  9. Meer, K., Ziegler, M.: An explicit solution to Post’s problem over the reals. J. Complex. 23 (2007, to appear). Available online 20 April 2007

  10. Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc. 50, 284–316 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many-sorted algebras. In: Abramskz, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 5, pp. 317–523. Oxford University Press (2000)

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, F.-L.-Jahn-Straße 15 a, 17487, Greifswald, Germany

    Christine Gaßner

Authors
  1. Christine Gaßner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christine Gaßner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gaßner, C. A Hierarchy below the Halting Problem for Additive Machines. Theory Comput Syst 43, 464–470 (2008). https://doi.org/10.1007/s00224-007-9020-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-007-9020-y

Keywords

Search

Navigation