Skip to main content
SpringerLink
Log in
Book cover

International Conference on Rewriting Techniques and Applications

RTA 1996: Rewriting Techniques and Applications pp 184–199Cite as

Confluence properties of extensional and non-extensional λ-calculi with explicit substitutions (extended abstract)

  • Regular Papers
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1103))

Abstract

This paper studies confluence properties of extensional and non-extensional λ-calculi with explicit substitutions, where extensionality is interpreted by η-expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many well-known calculi such as λσ, λσ⇑ and λv, or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fining the scheme, such as λs, how to reason about confluence properties of their extensional and non-extensional versions.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Abadi, L. Cardelli, P-L. Curien, and J-J. Lévy. Explicit substitutions. JFP, 4(1):375–416, 1991.

    Google Scholar 

  2. Y. Akama. On mints' reductions for ccc-calculus. In TLCA, LNCS 664, 1993.

    Google Scholar 

  3. H. Barendregt. The Lambda Calculus; Its syntax and Semantics (revised edition). North Holland, 1984.

    Google Scholar 

  4. Z-E-A. Benaissa and D. Briaud and P. Lescanne and J. Rouyer-Degli. λv, a calculus of explicit substitutions which preserves strong normalisation. Available from http://www.loria.fr/lescanne/publications.html. 1995.

    Google Scholar 

  5. D. Briaud. An explicit eta rewrite rule. In TLCA, LNCS 902, 1995.

    Google Scholar 

  6. P-L. Curien and R. Di Cosmo. A confluent reduction system for the λ-calculus with surjective pairing and terminal object. In ICALP, LNCS 510, 1991.

    Google Scholar 

  7. P-L. Curien, T. Hardin, and J-J. Lévy. Confluence properties of weak and strong calculi of explicit substitutions. Technical Report 1617, INRIA Rocquencourt, 1992.

    Google Scholar 

  8. P-L. Curien, T. Hardin, and A. Ríos. Strong normalisation of substitutions. In MFCS'92, LNCS 629, 1992.

    Google Scholar 

  9. N. de Bruijn. Lambda-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem. Indag. Mat., 5(35):381–392, 1972.

    Google Scholar 

  10. N. de Bruijn. Lambda-calculus notation with namefree formulas involving symbols that represent reference transforming mappings. Indag. Mat., (40):384–356, 1978.

    Google Scholar 

  11. Roberto Di Cosmo and Delia Kesner. A confluent reduction for the extensional typed λ-calculus with pairs, sums, recursion and terminal object. In ICALP, LNCS 700, 1993.

    Google Scholar 

  12. Roberto Di Cosmo and Delia Kesner. Combining first order algebraic rewriting systems, recursion and extensional typed lambda calculi. In ICALP, LNCS 820, 1994.

    Google Scholar 

  13. R. Di Cosmo and D. Kesner. Rewriting with extensional polymorphic λ-calculus. In CSL, 1995.

    Google Scholar 

  14. R. Di Cosmo and A. Pipemo. Expanding extensional polymorphism. In TLCA, LNCS 902, 1995.

    Google Scholar 

  15. D. Dougherty. Some lambda calculi with categorical sums and products. In RTA, LNCS 690, 1993.

    Google Scholar 

  16. Thomas Ehrhard. Une sémantique catégorique des types dépendants. Application au calcul des constructions. Thèse de doctorat, Université de Paris VII, 1988.

    Google Scholar 

  17. J-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1990.

    Google Scholar 

  18. T. Hardin. Résultats de confluence pour les règles fortes de la logique combinatoire catégorique et liens avec les lambda-calculs. Thèse de doctorat, Université de Paris VII, 1987.

    Google Scholar 

  19. T. Hardin. η-reduction for explicit substitutions. In ALP'92, LNCS 632, 1992.

    Google Scholar 

  20. T. Hardin. Eta-conversion for the languages of explicit substitutions. AAECC, 5:317–341, 1994.

    Article  Google Scholar 

  21. T. Hardin and A. Laville. Proof of termination of the rewriting system subst on c.c.l. TCS, 1986.

    Google Scholar 

  22. T. Hardin and J-J. Lévy. A confluent calculus of substitutions. In France-Japan Art. Int. and Comp. Sci. Symp., 1989.

    Google Scholar 

  23. G. Huet. Résolution d'équations dans les langages d'ordre 1,2,...,ω. Thèse de doctorat d'état, Université Paris VII, 1976.

    Google Scholar 

  24. C. Barry Jay and N. Ghani. The virtues of eta-expansion. JFP, 5(2):135–154, 1995.

    Google Scholar 

  25. D. Kesner. Confluence properties of extensional and non-extensional λ-calculi with explicit substitutions, 1995. Available as ftp://ftp.lri.fr/lri/articles/kesner/explicit.ps.gz.

    Google Scholar 

  26. F. Kamareddine and A. Ríos. The confluence of the λs e -calculus via a generalized interpretation method, 1995. Draft.

    Google Scholar 

  27. F. Kamareddine and A. Ríos. A λ-calculus à la de bruijn with explicit substitutions. In PULP, LNCS 982, 1995.

    Google Scholar 

  28. P. Lescanne. Personal Communication.

    Google Scholar 

  29. P. Lescanne. From λσ to λv, a journey through calculi of explicit substitutions. In POPL, pages 60–69. ACM, 1994.

    Google Scholar 

  30. P. Lescanne and J. Rouyer-Degli. The calculus of explicit substitutions λv. Technical Report, INRIA, Lorraine, 1994.

    Google Scholar 

  31. C. Marché. Réécriture modulo une théorie présentée par un système convergent et décidabilité des problèmes du mot dans certains classes de théories équationnelles. PhD thesis, Université Paris-Sud, Orsay, 1993.

    Google Scholar 

  32. P-A. Mellies. Typed λ-calculi with explicit substitutions may not terminate. In TLCA, LNCS 902, 1995.

    Google Scholar 

  33. G. Mints. Closed categories and the theory of proofs. Zap. Nauch. Semin. Leningradskogo, 68:83–114, 1977.

    Google Scholar 

  34. G. Mints. Teorija categorii i teoria dokazatelstv.l. Aktualnye problemy logiki i metodologii nauky, pages 252–278, 1979.

    Google Scholar 

  35. G. Pottinger. The Church Rosser Theorem for the Typed lambda-calculus with Surjective Pairing. Notre Dame Jour. of Formal Logic, 22(3):264–268, 1981.

    Google Scholar 

  36. D. Prawitz. Ideas and results in proof theory. Proceedings of the 2nd Scandinavian Logic Symposium, pages 235–307, 1971.

    Google Scholar 

  37. A. Ríos. Contribution à l'étude des λ-calculus avec substitutions explicites. Thèse de doctorat, Université de Paris VII, 1993.

    Google Scholar 

  38. V. van Oostrom and F. van Raamsdonk. Weak orthogonality implies confluence: the higher-order case. In LICS, 1994

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS and LRI, Bât 490, Université Paris-Sud, 91405, Orsay Cedex, France

    Delia Kesner

Authors
  1. Delia Kesner
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Harald Ganzinger

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kesner, D. (1996). Confluence properties of extensional and non-extensional λ-calculi with explicit substitutions (extended abstract). In: Ganzinger, H. (eds) Rewriting Techniques and Applications. RTA 1996. Lecture Notes in Computer Science, vol 1103. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61464-8_52

Download citation

  • DOI: https://doi.org/10.1007/3-540-61464-8_52

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61464-7

  • Online ISBN: 978-3-540-68596-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation