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A λ-calculus à la de Bruijn with explicit substitutions

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Programming Languages: Implementations, Logics and Programs (PLILP 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 982))

Abstract

The aim of this paper is to present the λs-calculus which is a very simple λ-calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of λ-terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the λσ-calculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the λs-calculus is obtained by the “interpretation method” ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the λv-calculus (cf. [BBLRD95]). The relation between λs and λv is also studied.

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References

  1. M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lévy. Explicit Substitutions. Journal of Functional Programming, 1(4):375–416, 1991.

    Google Scholar 

  2. H. Barendregt. The Lambda Calculus: Its Syntax and Semantics (revised edition). North Holland, 1984.

    Google Scholar 

  3. Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. λv, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

    Google Scholar 

  4. P.-L. Curien, T. Hardin, and J.-J. Lévy. Confluence properties of weak and strong calculi of explicit substitutions. Technical Report RR 1617, INRIA, Rocquencourt, 1992. To appear in the JACM.

    Google Scholar 

  5. P.-L. Curien, T. Hardin, and A. Ríos. Strong Normalization of Substitutions in Proceedings of MFCS'92. In I.M. Havel and V. Koubek, editors, Lecture Notes in Computer Science 629, pages 209–217, Prague, 1992. Springer-Verlag.

    Google Scholar 

  6. P.-L. Curien and A. Ríos. Un résultat de Complétude pour les substitutions explicites. Comptes Rendus de l'Académie des Sciences, 312, I:471–476, 1991.

    Google Scholar 

  7. P.-L. Curien. Categorical Combinators, Sequential Algorithms and Functional Programming. Pitman, 1986. Revised edition: Birkhäuser (1993).

    Google Scholar 

  8. N. de Bruijn. Lambda-Calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser Theorem. Indag. Mat., 34(5):381–392, 1972.

    Google Scholar 

  9. N. de Bruijn. Lambda-Calculus notation with namefree formulas involving symbols that represent reference transforming mappings. Indag. Mat., 40:348–356, 1978.

    Google Scholar 

  10. N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

    Google Scholar 

  11. T. Hardin. Confluence Results for the Pure Strong Categorical Logic CCL: λ-calculi as Subsystems of CCL. Theoretical Computer Science, 65(2):291–342, 1989.

    Google Scholar 

  12. T. Hardin and A. Laville. Proof of Termination of the Rewriting System SUBST on CCL. Theoretical Computer Science, 46:305–312, 1986.

    Google Scholar 

  13. T. Hardin and J.-J. Lévy. A Confluent Calculus of Substitutions. France-Japan Artificial Intelligence and Computer Science Symposium, December 1989.

    Google Scholar 

  14. G. Huet. Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems. Journal of the Association for Computing Machinery, 27:797–821, October 1980.

    Google Scholar 

  15. D. Knuth and P. Bendix. Simple Word Problems in Universal Algebras. In J. Leech, editor, Computational Problems in Abstract Algebra, pages 263–297. Pergamon Press, 1970.

    Google Scholar 

  16. F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197–240, 1993.

    Google Scholar 

  17. M. Mauny. Compilation des langages fonctionnels dans les combinateurs catégoriques. Application au langage ML. PhD thesis, Université Paris VII, Paris, France, 1985.

    Google Scholar 

  18. P.-A. Melliès. Typed λ-calculi with explicit substitutions may not terminate in Proceedings of TLCA'95. Lecture Notes in Computer Science, 902, 1995.

    Google Scholar 

  19. A. Ríos. Contribution à l'étude des λ-calculs avec substitutions explicites. PhD thesis, Université de Paris 7, 1993.

    Google Scholar 

  20. H. Zantema. Termination of term rewriting: interpretation and type elimination. J. Symbolic Computation, 17(1):23–50, 1994.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing Science, University of Glasgow, 17 Lilybank Gardens, G12 8QQ, Glasgow, Scotland

    Fairouz Kamareddine & Alejandro Ríos

Authors
  1. Fairouz Kamareddine
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  2. Alejandro Ríos
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Manuel Hermenegildo S. Doaitse Swierstra

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© 1995 Springer-Verlag Berlin Heidelberg

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Kamareddine, F., Ríos, A. (1995). A λ-calculus à la de Bruijn with explicit substitutions. In: Hermenegildo, M., Swierstra, S.D. (eds) Programming Languages: Implementations, Logics and Programs. PLILP 1995. Lecture Notes in Computer Science, vol 982. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0026813

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  • DOI: https://doi.org/10.1007/BFb0026813

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60359-7

  • Online ISBN: 978-3-540-45048-1

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