Real number computability and domain theory

  • Pietro Di Gianantonio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 711)


We propose a possible implementation, using lazy functional programming, of the exact computation on real numbers. Using domain theory we can analyze this kind of computation and give a definition of computability for the functions on the real number. This definition turns out to be equivalent to other definitions given in the literature using different methods.

Domain theory is a useful tool to study higher order computability on the real numbers. An interesting connection between Scott Topology and the topologies on the real line and on the space of the real functions is stated. The main original result in this work is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology.


Computable Function Canonical Representation Domain Theory Compact Open Topology Real Continuous Function 
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.


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  1. 1.
    L.E.J. Brouwer, “Beweis, dass jede volle Funktion gleichmässig stetig ist” Proc. Amsterdam 27 (1924) 189–194.Google Scholar
  2. 2.
    M.J. Beeson, “Foundation of Constructive Mathematics” Spriger-Verlag, Berlin, 1985.Google Scholar
  3. 3.
    H.-J. Boehm, R. Cartwright, M. Riggle, and M.J. O'Donell, “Exact Real Arithmetic: A Case Study in Higher Order Programming.” 1986 ACM Symposium on Lisp and Functional Programming.Google Scholar
  4. 4.
    H.-J. Boehm, R. Cartwright, “Exact Real Arithmetic: Formulating Real Numbers as Functions” in “Research Topics in Functional Programming” David Turner editor, Addison-Wesley, 1990, pp. 43–64Google Scholar
  5. 5.
    D. Bridges and E. Bishop, “Constructive Analysis”. Springer-Verlag, Berlin, 1985.Google Scholar
  6. 6.
    A. Grzegorczyk, “On the Definition of Computable Real Continuous Functions”. Fund. Math. 44 (1957) 61–77.Google Scholar
  7. 7.
    K. Ko and Friedmann, “Computational Complexity of Real Functions”. Theoret. Comput. Sci. 20 (1982) 323–352.Google Scholar
  8. 8.
    D. Lacombe, “Quelques procédés de définitions en topologie recursif.” in: Constructivity in Mathematics, North-Holland (1959) 129–158.Google Scholar
  9. 9.
    P. Martin-Löf, “Note on Constructive Mathematics”. Almqvist and Wiksell, Stockholm (1970).Google Scholar
  10. 10.
    J. Myhill, “What is a Real Number?” America Mathematical Monthly (1979) 748–754.Google Scholar
  11. 11.
    H.G. Rice, “Recursive Real Numbers”. Proc. Amer. Math. Soc 5 (1954) 784–791.Google Scholar
  12. 12.
    D. Scott, “Outline of the Mathematical Theory of Computation”. Proc. 4th Princeton Conference on Information Science (1970).Google Scholar
  13. 13.
    D. Scott, “Data Types as Lattices”. SIAM J. Comput. 5 (1976) 522–587.Google Scholar
  14. 14.
    A.S. Troelstra and D. van Dalen, “Constructivism in Mathematics”. North-Holland, Amsterdam (1988).Google Scholar
  15. 15.
    A.M. Turing, “On Computable Numbers, with an Application to the Entscheidungs Problem”. Proc. London Math. Soc. 42 (1937) 230–265.Google Scholar
  16. 16.
    J. Vuillemin, “Exact Real Computer Arithmetic with Continued Fraction”. Proc. A.C.M. conference on Lisp and functional Programming (1988) 14–27.Google Scholar
  17. 17.
    K. Weihrauch, U. Schreiber, “Embeding Metric Spaces into cpo's” Theoret. Comp. Sci. 16 (1981) 5–34.Google Scholar
  18. 18.
    K. Weihrauch and C. Kreitz, “Representation of the Real Numbers and of the Open Subsets of the Set of Real Numbers”. Annals of Pure and Applied Logic 35 (1987) 247–260.Google Scholar
  19. 19.
    K. Weihrauch, “Computability”. Springer-Verlag, Berlin, 1987.Google Scholar
  20. 20.
    E. Wiedmer, “Computing with Infinite Objects”. Theoret. Comp. Sci. 10 (1980) 133–155.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Pietro Di Gianantonio
    • 1
  1. 1.Dip. di Matematica e InformaticaUniversità di UdineUdineItaly

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