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Formalising Exact Arithmetic in Type Theory

  • Milad Niqui
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)

Abstract

In this work we focus on a formalisation of the algorithms of lazy exact arithmetic á la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output.

Keywords

Type Theory Productive Element Expression Tree Exact Arithmetic Polynomial Functor 
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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References

  1. 1.
    Potts, P.J., Edalat, A.: Exact real computer arithmetic. Technical Report DOC 97/9, Department of Computing, Imperial College (1997)Google Scholar
  2. 2.
    Edalat, A., Potts, P.J., Sünderhauf, P.: Lazy computation with exact real numbers. In: Berman, M., Berman, S. (eds.) Proceedings of the third ACM SIGPLAN International Conference on Functional Programming (ICFP 1998), Baltimore, Maryland, USA, September 27-29, 1998. ACM SIGPLAN Notices, vol. 34(1), pp. 185–194. ACM Press, New York (1999)Google Scholar
  3. 3.
    Niqui, M.: Formalising Exact Arithmetic: Representations, Algorithms and Proofs. PhD thesis, Radboud Universiteit Nijmegen (2004)Google Scholar
  4. 4.
    Giménez, E.: Un Calcul de Constructions Infinies et son Application a la Verification des Systemes Communicants. PhD thesis PhD 96-11, Laboratoire de l’Informatique du Parallélisme, Ecole Normale Supérieure de Lyon (1996)Google Scholar
  5. 5.
    The Coq Development Team: The Coq Proof Assistant Reference Manual, Version 8.0. INRIA (2004), http://coq.inria.fr/doc/main.html, (cited 31st January 2005)
  6. 6.
    Hagino, T.: A Categorical Programming Language. PhD thesis CST-47-87, Laboratory for Foundations of Computer Science, Dept. of Computer Science, Univ. of Edinburgh (1987)Google Scholar
  7. 7.
    Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. Elsevier Science Publishers, Amsterdam (1998)CrossRefGoogle Scholar
  8. 8.
    Martin-Löf, P.: Intuitionistic Type Theory, Biblioplois, Napoli. Notes of Giovanni Sambin on a series of lectures given in Padova (1984)Google Scholar
  9. 9.
    Coquand, T., Paulin, C.: Inductively defined types (preliminary version). In: Martin-Löf, P., Mints, G. (eds.) COLOG 1988. LNCS, vol. 417, pp. 50–66. Springer, Heidelberg (1990)Google Scholar
  10. 10.
    Potts, P.J.: Exact Real Arithmetic using Möbius Transformations. PhD thesis, University of London, Imperial College (1998)Google Scholar
  11. 11.
    Saïbi, A.: Typing algorithm in type theory with inheritance. In: POPL 1997: Proceedings of the 24th ACM SIGPLAN-SIGACT symposium on Principles of programming languages. ACM SIGACT and SIGPLAN, pp. 292–301. ACM Press, New York (1997)CrossRefGoogle Scholar
  12. 12.
    Sijtsma, B.A.: On the productivity of recursive list definitions. ACM Trans. Program. Lang. Syst (TOPLAS) 11, 633–649 (1989)CrossRefGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. An introduction, p. 285. Springer, Heidelberg (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Milad Niqui
    • 1
  1. 1.Institute for Computing and Information SciencesRadboud University NijmegenThe Netherlands

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