Lazy functional algorithms for exact real functionals

  • Alex K. Simpson
Contributed Papers Programming
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1450)

Abstract

We show how functional languages can be used to write programs for real-valued functionals in exact real arithmetic. We concentrate on two useful functionals: definite integration, and the functional returning the maximum value of a continuous function over a closed interval. The algorithms are a practical application of a method, due to Berger, for computing quantifiers over streams. Correctness proofs for the algorithms make essential use of domain theory.

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References

  1. 1.
    J.-C. Bajard, D. Michelucci, J.-M. Moreau, and J.-M. Muller. Introduction to special issue: “Real Numbers and Computers”. Journal of Universal Computer Science, 1(7):436–438, 1995.Google Scholar
  2. 2.
    U. Berger. Totale Objecte und Mengen in Bereichstheorie. PhD Thesis, University of Munich, 1990.Google Scholar
  3. 3.
    U. Berger. Total objects and sets in domain theory. Journal of Pure and Applied Logic, 60:91–117, 1993.MATHCrossRefGoogle Scholar
  4. 4.
    H.J. Boehm and R. Cartwright. Exact real arithmetic: Formulating real numbers as functions. In D. Turner, editor, Research Topics in Functional Programming, pages 43–64. Adison-Wesley, 1990.Google Scholar
  5. 5.
    H.J. Boehm, R. Cartwright, M. Riggle, and M.J. O'Donnel. Exact real arithmetic: a case study in higher order programming. In ACM Symposium on LISP and Functional Programming, 1986.Google Scholar
  6. 6.
    P. Di Gianantonio. A Functional Approach to Computability on Real Numbers. PhD Thesis, University of Pisa, 1993.Google Scholar
  7. 7.
    P. Di Gianantonio. An abstract data type for real numbers. In Proceedings of ICALP-97, pages 121–131. Springer LNCS 1256, 1997.Google Scholar
  8. 8.
    A. Edalat and M.H. Escardó. Integration in Real PCF. Information and Computation, To appear, 1998.Google Scholar
  9. 9.
    A. Edalat and P.J. Potts. Exact Real Computer Arithmetic. Presented at workshop: New Paradigms for Computation on Classical Spaces, Birmingham, 1997.Google Scholar
  10. 10.
    M.H. Escardó. PCF extended with real numbers. Theoretical Computer Science, 162(1):79–115, 1996.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    M.H. Escardó. Properly injective spaces and function spaces. Topology and its Applications, To appear, 1998.Google Scholar
  12. 12.
    C.A. Gunter. Semantics of Programming. MIT Press, 1992.Google Scholar
  13. 13.
    Ker-I Ko. Complexity Theory of Real Functions. Birkhauser, Boston, 1991.MATHGoogle Scholar
  14. 14.
    V. Menissier-Morain. Arbitrary precision real arithmetic: Design and algorithms. Journal of Symbolic Computation, Submitted, 1996.Google Scholar
  15. 15.
    G.D. Plotkin. LCF considered as a programming language. Theoretical Computer Science, 5(1):223–255, 1977.MathSciNetCrossRefGoogle Scholar
  16. 16.
    G.D. Plotkin. Full abstraction, totality and PCF. Math. Struct. in Comp. Sci., To appear, 1998.Google Scholar
  17. 17.
    J. Vuillemin. Exact real arithmetic with continued fractions. IEEE Transactions on Computers, 39(8):1087–1105, 1990.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Alex K. Simpson
    • 1
  1. 1.LFCS, Department of Computer ScienceUniversity of EdinburghEdinburghScotland

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