Skip to main content
SpringerLink
Log in

δ-Approximable Functions

  • Conference paper
  • First Online:
Computability and Complexity in Analysis (CCA 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2064))

Included in the following conference series:

  • 404 Accesses

Abstract

In this paper we study several notions of approximability of functions in the framework of the BSS model. Denoting with ϕ δM the function computed by a BSS machine M when its comparisons are against −δ rather than 0, we study classes of functions f for which ϕ δM → f in some sense (pointwise, uniformly, etc.). The main equivalence results show that this notion coincides with Type 2 computability when the convergence speed is recursively bounded. Finally, we study the possibility of extending these results to computations over Archimedean fields.

The last two authors have been partially supported by the ESPRIT Working Group EP 27150, Neural and Computational Learning II (NeuroCOLT II).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Paolo Boldi and Sebastiano Vigna. δ-uniform BSS machines. J. Complexity, 14(2):234–256, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  3. Paolo Boldi and Sebastiano Vigna. The Turing closure of an Archimedean field. Theoret. Comput. Sci., 231:143–156, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  4. Vasco Brattka. Personal electronic communication, 2001.

    Google Scholar 

  5. Vasco Brattka and Peter Hertling. Feasible real random access machines. J. Complexity, 14(4):490–526, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  6. Thomas Chadzelek and Günter Hotz. Analytic machines. Theoret. Comput. Sci., 219(1-2):151–167, 1999.

    MATH  MathSciNet  Google Scholar 

  7. Abbas Edalat and Philipp Sünderhauf. A domain-theoretic approach to computability on the real line. Theoret. Comput. Sci., 210(1):73–98, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  8. Rudolf Freund. Real functions and numbers defined by Turing machines. Theoret. Comput. Sci., 23(3):287–304, May 1983.

    Article  MATH  MathSciNet  Google Scholar 

  9. Armin Hemmerling. On approximate and algebraic computability over the real numbers. Theoret. Comput. Sci., 219:185–223, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  10. Ker-I Ko. Reducibilities on real numbers. Theoret. Comput. Sci., 31(1_2):101–123, May 1984.

    Article  MATH  MathSciNet  Google Scholar 

  11. Joseph R. Shoenfield. Degrees of Unsolvability. North-Holland, Amsterdam, 1971.

    Book  MATH  Google Scholar 

  12. Robert I. Soare. Recursion theory and Dedekind cuts. Trans. Amer. Math. Soc., 140:271–294, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  13. John V. Tucker and Jeffery I. Zucker. Computation by ‘While’ programs on topological partial algebras. Theoret. Comput. Sci., 219(1-2):379–420, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  14. Sebastiano Vigna. On the relations between distributive computability and the BSS model. Theoret. Comput. Sci., 162:5–21, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  15. Robert F.C. Walters. An imperative language based on distributive categories. Math. Struct. Comp. Sci., 2:249–256, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  16. Klaus Weihrauch. Introduction to Computable Analysis. Texts in Theoretical Computer Science. Springer-Verlag, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. École Normale Supérieure de Lyon, France

    Charles Meyssonnier

  2. Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, Italy

    Paolo Boldi & Sebastiano Vigna

Authors
  1. Charles Meyssonnier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Boldi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebastiano Vigna
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK

    Jens Blanck

  2. Informatikzentrum, FernUniversität Hagen, 58084, Hagen, Germany

    Vasco Brattka  & Peter Hertling  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Meyssonnier, C., Boldi, P., Vigna, S. (2001). δ-Approximable Functions. In: Blanck, J., Brattka, V., Hertling, P. (eds) Computability and Complexity in Analysis. CCA 2000. Lecture Notes in Computer Science, vol 2064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45335-0_12

Download citation

  • DOI: https://doi.org/10.1007/3-540-45335-0_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42197-9

  • Online ISBN: 978-3-540-45335-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation