On The Equivalence of Some Approaches to Computability on the Real Line

  • Dieter Spreen
  • Holger Schulz
Part of the Semantic Structures in Computation book series (SECO, volume 1)

Abstract

There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch’s Type Two Theory of Effectivity or on domain-theoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysis-oriented line of research is based on Grzegorczyk’s work. All these approaches have turned out to be equivalent.

In this paper it is shown that real numbers as well as real-valued functions are computable in Weihrauch’s sense if and only if they are definable in Escardó’s functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this domain.

For defining the semantics of the language Real. PCF a full subcategory of the category of bounded-complete ω-continuous directed complete partial orders is introduced and it is defined when a domain in this category is effectively given. The subcategory of effectively given domains contains the interval domain and is Cartesian closed.

Keywords

Computable analysis Real PCF interval domain type two theory of effectivity effectively given domains 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Abramsky, A. Jung, Domain theory, in: S. Abramsky et el., eds., Handbook of Logic in Computer Science, vol. 3 (Clarendon Press, Oxford, 1994) 1–168.Google Scholar
  2. 2.
    U. Berger, Totale Objekte und Mengen in der Bereichstheorie, Ph.D. Thesis, Universität München, München, 1990.Google Scholar
  3. 3.
    U. Berger, Total sets and objects in domain theory, Ann. Pure Appl. Logic 60 (1993) 91–117.CrossRefGoogle Scholar
  4. 4.
    T. Deil, Darstellung und Berechenbarkeit reeller Zahlen, Ph.D. Thesis, FernUniversität Hagen, Hagen, 1983.Google Scholar
  5. 5.
    T. Erker, M.H. Escardó, K. Keimel, The way-below relation of function spaces over semantic domains, Topology Appl. 89 (1998) 61–74.CrossRefGoogle Scholar
  6. 6.
    P. Di Gianantonio, A functional approach to computability on real numbers, Ph.D. Thesis TD-6/93, Università Pisa-Genova-Udine, Pisa, 1993.Google Scholar
  7. 7.
    P. Di Gianantonio, Real number computability and domain theory, Inform. and Comput. 127 (1996) 11–25.CrossRefGoogle Scholar
  8. 8.
    A. Edalat, P. Sünderhauf, A domain-theoretic approach to computability on the real line, Theoret. Comput. Sci. 210 (1999) 73–98.CrossRefGoogle Scholar
  9. 9.
    A. Edalat, Domains for computation in mathematics, physics and real arithmetic, Bull. Symbolic Logic 3 (1997) 401–452.CrossRefGoogle Scholar
  10. 10.
    M.H. Escardó, PCF extended with real numbers, Theoret. Comput. Sci. 162 (1996) 79–115.CrossRefGoogle Scholar
  11. 11.
    M.H. Escardó, Real PCF extended with 3 is universal, in: A. Edalat et al., eds., Advances in Theory and Formal Methods of Computing: Proc. of the Third Imperial College Workshop, April 1996 (IC Press, 1996) 13–24.Google Scholar
  12. 12.
    M.H. Escardó, PCF extended with real numbers: a domain-theoretic approach to higher-order exact real number computation, Ph.D. Thesis, University of London, London, 1997.Google Scholar
  13. 13.
    A. Grzegorczyk, Computable functionals, Fund. Math. 42 (1955) 168–202.Google Scholar
  14. 14.
    A. Grzegorczyk, On the definitions of computable real continuous functions, Fund. Math. 44 (1957) 61–74.Google Scholar
  15. 15.
    C. Kreitz, K. Weihrauch, Theory of representations, Theoret. Comp. Sci. 38 (1985) 35–53.CrossRefGoogle Scholar
  16. 16.
    J.C. Mitchell, Foundations for Programming Languages (The MIT Press, Cambridge, MA, 1996).Google Scholar
  17. 17.
    G.D. Plotkin, LCF considered as a programming language, Theoret. Comput. Sci. 5 (1977) 223–255.CrossRefGoogle Scholar
  18. 18.
    M.B. Pour-El, J.I. Richards, Computability in Analysis and Physics (Springer, Berlin, 1989).Google Scholar
  19. 19.
    H. Schulz, Berechenbarkeit auf reellen Zahlen - ein Vergleich, Diploma Thesis, Universität Siegen, Siegen, 1997.Google Scholar
  20. 20.
    D. Scott, Outline of a mathematical theory of computation, in: Proc. 4th Ann. Princeton Conference on Information Science and Systems, 1970, 169–176.Google Scholar
  21. 21.
    V. Stoltenberg-Hansen, J.V. Tucker, Effective algebras, in: S. Abramsky et al., eds., Handbook of Logic in Computer Science, vol. 4 (Clarendon Press, Oxford, 1995) 357–526.Google Scholar
  22. 22.
    V. Stoltenberg-Hansen, J.V. Tucker, Concrete models of computation for topological algebras, Theoret. Comput. Sci. 219 (1999) 347–378.CrossRefGoogle Scholar
  23. 23.
    A. Tang, Recursion theory and descriptive set theory in effectively given To spaces, Ph.D. Thesis, University of Princeton, Princeton, MA, 1974.Google Scholar
  24. 24.
    A.M. Turing, On computable numbers, with an application to the “Entscheidungsproblem”, Proc. London Math. Soc., Ser. 2 42 (1936) 230–265; corr. ibid. 43 (1937) 544–546.Google Scholar
  25. 25.
    K. Weihrauch, Computability (Springer, Berlin, 1987).CrossRefGoogle Scholar
  26. K. Weihrauch, A simple introduction to computable analysis, Informatik Berichte 171, FernUniversität Hagen, Hagen, 1995.Google Scholar
  27. 27.
    K. Weihrauch, A foundation for computable analysis, in: D.S. Bridges et al., eds., Combinatorics, Complexity, and Logic, Proc. DMTCS’96 (Springer, Singapore, 1997) 66–89.Google Scholar
  28. 28.
    K. Weihrauch, Computable Analysis, An Introduction (Springer, Berlin, 2000).CrossRefGoogle Scholar
  29. 29.
    K. Weihrauch, T. Deil, Berechenbarkeit auf cpo’s, Schriften zur Angewandten Mathematik und Informatik Nr. 63, Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, 1980.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Dieter Spreen
    • 1
  • Holger Schulz
    • 1
  1. 1.Fachbereich Mathematik AG Theoretische InformatikUniversität SiegenSiegenGermany

Personalised recommendations