Advertisement

δ-Approximable Functions

  • Charles Meyssonnier
  • Paolo Boldi
  • Sebastiano Vigna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2064)

Abstract

In this paper we study several notions of approximability of functions in the framework of the BSS model. Denoting with ϕ M δ 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 M δ 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.

Keywords

Strict Inequality Recursive Function Computable Function Approximable Function Partial Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Paolo Boldi and Sebastiano Vigna. δ-uniform BSS machines. J. Complexity, 14(2):234–256, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Paolo Boldi and Sebastiano Vigna. The Turing closure of an Archimedean field. Theoret. Comput. Sci., 231:143–156, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Vasco Brattka. Personal electronic communication, 2001.Google Scholar
  5. 5.
    Vasco Brattka and Peter Hertling. Feasible real random access machines. J. Complexity, 14(4):490–526, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Thomas Chadzelek and Günter Hotz. Analytic machines. Theoret. Comput. Sci., 219(1-2):151–167, 1999.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Abbas Edalat and Philipp Sünderhauf. A domain-theoretic approach to computability on the real line. Theoret. Comput. Sci., 210(1):73–98, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Rudolf Freund. Real functions and numbers defined by Turing machines. Theoret. Comput. Sci., 23(3):287–304, May 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Armin Hemmerling. On approximate and algebraic computability over the real numbers. Theoret. Comput. Sci., 219:185–223, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ker-I Ko. Reducibilities on real numbers. Theoret. Comput. Sci., 31(1_2):101–123, May 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Joseph R. Shoenfield. Degrees of Unsolvability. North-Holland, Amsterdam, 1971.zbMATHCrossRefGoogle Scholar
  12. 12.
    Robert I. Soare. Recursion theory and Dedekind cuts. Trans. Amer. Math. Soc., 140:271–294, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sebastiano Vigna. On the relations between distributive computability and the BSS model. Theoret. Comput. Sci., 162:5–21, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Robert F.C. Walters. An imperative language based on distributive categories. Math. Struct. Comp. Sci., 2:249–256, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Klaus Weihrauch. Introduction to Computable Analysis. Texts in Theoretical Computer Science. Springer-Verlag, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Charles Meyssonnier
    • 1
  • Paolo Boldi
    • 2
  • Sebastiano Vigna
    • 2
  1. 1.École Normale Supérieure de LyonFrance
  2. 2.Dipartimento di Scienze dell’InformazioneUniversità degli Studi di MilanoItaly

Personalised recommendations