Simple Proofs of Characterizing Strong Normalization for Explicit Substitution Calculi

  • Kentaro Kikuchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4533)

Abstract

We present a method of lifting to explicit substitution calculi some characterizations of the strongly normalizing terms of λ-calculus by means of intersection type systems. The method is first illustrated by applying to a composition-free calculus of explicit substitutions, yielding a simpler proof than the previous one by Lengrand et al. Then we present a new intersection type system in the style of sequent calculus, and show that it characterizes the strongly normalizing terms of Dyckhoff and Urban’s extension of Herbelin’s explicit substitution calculus.

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References

  1. 1.
    Abadi, M., Cardelli, L., Curien, P.-L., Lévy, J.-J.: Explicit substitutions. J. Funct. Program. 1, 375–416 (1991)MATHCrossRefGoogle Scholar
  2. 2.
    Barendregt, H., Coppo, M., Dezani-Ciancaglini, M.: A filter lambda model and the completeness of type assignment. J. Symb. Log. 48, 931–940 (1983)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bloo, R.: Preservation of Termination for Explicit Substitution. PhD thesis, Eindhoven University of Technology (1997)Google Scholar
  4. 4.
    Bloo, R., Geuvers, H.: Explicit substitution: On the edge of strong normalization. Theor. Comput. Sci. 211, 375–395 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bloo, R., Rose, K.H.: Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection. In: Proceedings of CSN 1995 (Computing Science in the Netherlands), pp. 62–72 (1995)Google Scholar
  6. 6.
    Bonelli, E.: Perpetuality in a named lambda calculus with explicit substitutions. Math. Structures Comput. Sci. 11, 47–90 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Curien, P.-L., Herbelin, H.: The duality of computation. In: Proceedings of ICFP 2000, pp. 233–243 (2000)Google Scholar
  8. 8.
    Dershowitz, N.: Orderings for term-rewriting systems. Theor. Comput. Sci. 17, 279–301 (1982)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dougherty, D., Lescanne, P.: Reductions, intersection types, and explicit substitutions. Math. Structures Comput. Sci. 13, 55–85 (2003)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dougherty, D., Ghilezan, S., Lescanne, P.: Characterizing strong normalization in a language with control operators. In: Proceedings of PPDP 2004, pp. 155–166 (2004)Google Scholar
  11. 11.
    Dyckhoff, R., Urban, C.: Strong normalization of Herbelin’s explicit substitution calculus with substitution propagation. J. Log. Comput. 13, 689–706 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Herbelin, H.: A λ-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 61–75. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    Kesner, D., Lengrand, S.: Resource operators for the λ-calculus. Inform. and Comput. 205, 419–473 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Khasidashvili, Z., Ogawa, M., van Oostrom, V.: Uniform normalisation beyond orthogonality. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 122–136. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Kikuchi, K.: A direct proof of strong normalization for an extended Herbelin’s calculus. In: Kameyama, Y., Stuckey, P.J. (eds.) FLOPS 2004. LNCS, vol. 2998, pp. 244–259. Springer, Heidelberg (2004)Google Scholar
  16. 16.
    Lengrand, S., Lescanne, P., Dougherty, D., Dezani-Ciancaglini, M., van Bakel, S.: Intersection types for explicit substitutions. Inform. and Comput. 189, 17–42 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Melliès, P.-A.: Typed λ-calculi with explicit substitutions may not terminate. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 328–334. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  18. 18.
    Polonovski, E.: Strong normalization of \(\overline{\lambda}\mu\tilde{\mu}\)-calculus with explicit substitutions. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 423–437. Springer, Heidelberg (2004)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, San Diego (1980)Google Scholar
  20. 20.
    van Raamsdonk, F., Severi, P.: On normalisation. Technical Report CS-R9545, CWI (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Kentaro Kikuchi
    • 1
  1. 1.RIEC, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577Japan

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