A Proof of Strong Normalisation of the Typed Atomic Lambda-Calculus

  • Tom Gundersen
  • Willem Heijltjes
  • Michel Parigot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8312)


The atomic lambda-calculus is a typed lambda-calculus with explicit sharing, which originates in a Curry-Howard interpretation of a deep-inference system for intuitionistic logic. It has been shown that it allows fully lazy sharing to be reproduced in a typed setting. In this paper we prove strong normalization of the typed atomic lambda-calculus using Tait’s reducibility method.


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  1. 1.
    Abadi, M., Cardelli, L., Curien, P.-L., Lévy, J.-J.: Explicit substitutions. Journal of Functional Programming 1(4), 375–416 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Accattoli, B., Kesner, D.: The structural λ-calculus. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 381–395. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Ariola, Z.M., Felleisen, M., Maraist, J., Odersky, M., Wadler, P.: A call-by-need lambda calculus. In: POPL (1995)Google Scholar
  4. 4.
    Asperti, A., Guerrini, S.: The Optimal Implementation of Functional Programming Languages. Cambridge University Press (1998)Google Scholar
  5. 5.
    Balabonski, T.: A unified approach to fully lazy sharing. In: POPL (2012)Google Scholar
  6. 6.
    Brünnler, K., Tiu, A.F.: A local system for classical logic. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 347–361. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Coppo, M., Dezani-Ciancaglini, M.: An extension of the basic functionality theory for the λ-calculus. Notre Dame Journal of Formal Logic 21(4), 685–693 (1980)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    David, R., Guillaume, B.: A λ-calculus with explicit weakening and explicit substitution. MSCS 11(1), 169–206 (2001)MathSciNetMATHGoogle Scholar
  9. 9.
    Cosmo, R.D., Kesner, D., Polonovski, E.: Proof nets and explicit substitutions. In: MSCS (2003)Google Scholar
  10. 10.
    Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press (1989)Google Scholar
  11. 11.
    Guglielmi, A., Gundersen, T., Parigot, M.: A proof calculus which reduces syntactic bureaucracy. In: RTA, pp. 135–150 (2010)Google Scholar
  12. 12.
    Gundersen, T., Heijltjes, W., Parigot, M.: Atomic lambda-calculus: a typed lambda-calculus with explicit sharing. In: LICS (2013)Google Scholar
  13. 13.
    Gundersen, T., Heijltjes, W., Parigot, M.: Un lambda-calcul atomique. Journées Francophones des Langages Applicatifs (2013)Google Scholar
  14. 14.
    Hughes, R.J.M.: Super-combinators: a new implementation method for applicative languages. In: ACM Symposium on Lisp and Functional Programming, pp. 1–10 (1982)Google Scholar
  15. 15.
    Kesner, D., Lengrand, S.: Resource operators for lambda-calculus. Information and Computation 205(4), 419–473 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Krivine, J.-L.: Lambda-calculus types and models. Ellis Horwood, Chichester, UK (1993)Google Scholar
  17. 17.
    Lamping, J.: An algorithm for optimal lambda calculus reduction. In: POPL, pp. 16–30 (1990)Google Scholar
  18. 18.
    Lescanne, P.: From lambda-sigma to lambda-upsilon, a journey through calculi of explicit substitutions. In: POPL (1994)Google Scholar
  19. 19.
    Pottinger, G.: A type assignment for the strongly normalizable λ-terms. In: To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 561–577. Academic Press, London (1980)Google Scholar
  20. 20.
    Tait, W.W.: Intensional interpretations of functionals of finite type I. The Journal of Symbolic Logic 32(2), 198–212 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    van Oostrom, V., van de Looij, K.-J., Zwitserlood, M.: Lambdascope: another optimal implementation of the lambda-calculus. In: Workshop on Algebra and Logic on Programming Systems (2004)Google Scholar
  22. 22.
    Wadsworth, C.P.: Semantics and Pragmatics of the Lambda-Calculus. PhD thesis, University of Oxford (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tom Gundersen
    • 1
  • Willem Heijltjes
    • 2
  • Michel Parigot
    • 1
  1. 1.Laboratoire Preuves, Programmes, Systèmes CNRS & Université Paris DiderotFrance
  2. 2.University of BathUK

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