Modular properties of algebraic type systems

  • Gilles Barthe
  • Herman Geuvers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1074)


We introduce the framework of algebraic type systems, a generalisation of pure type systems with higher order rewriting à la Jouannaud-Okada, and initiate a generic study of the modular properties of these systems. We give a general criterion for one system of this framework to be strongly normalising. As an application of our criterion, we recover all previous strong normalisation results for algebraic type systems.


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  1. 1.
    T. Altenkirch. Constructions, inductive types and strong normalisation. PhD thesis, Laboratory for the Foundations of Computer Science, University of Edinburgh, 1994.Google Scholar
  2. 2.
    F. Barbanera and M. Fernandez. Combining first and higher order rewrite systems with type assignment systems. In M.Bezem and Groote [9], pages 60–74.Google Scholar
  3. 3.
    F. Barbanera, M. Fernandez, and H. Geuvers. Modularity of strong normalisation and confluence in the algebraic λ-cube. In Proceedings of LICS'94, pages 406–415. IEEE Press, 1994.Google Scholar
  4. 4.
    H.P. Barendregt. Lambda calculi with types. In S. Abramsky, D. M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 117–309. Oxford Science Publications, 1992.Google Scholar
  5. 5.
    G. Barthe. Combining dependent type theories with equational term-rewriting. Manuscript, 1995.Google Scholar
  6. 6.
    G. Barthe. η-reduction and algebraic rewriting in λ-calculus. Manuscript, 1995.Google Scholar
  7. 7.
    G. Barthe. Towards modular proofs of termination for algebraic type systems. Manuscript, submitted for publication, 1995.Google Scholar
  8. 8.
    G. Barthe and H. Geuvers. Congruence types. Presented at CSL'95. Submitted for publication in the proceedings, 1995.Google Scholar
  9. 9.
    M. Bezem and J-F. Groote, editors. Proceedings of TLCA'93, volume 664 of Lecture Notes in Computer Science. Springer-Verlag, 1993.Google Scholar
  10. 10.
    V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings of LICS'88, pages 82–90. IEEE Press, 1988.Google Scholar
  11. 11.
    V. Breazu-Tannen and J. Gallier. Polymorphic rewriting conserves algebraic strong normalisation. Theoretical Computer Science, 83:3–28, 1990.Google Scholar
  12. 12.
    M. Fernandez. Modèles de calcul multiparadigmes fondés sur la réécriture. PhD thesis, Université Paris-Sud Orsay, 1993.Google Scholar
  13. 13.
    M. Fernandez and J-P.Jouannaud. Modularity of termination of term-rewriting systems revisited. In Recent Trends in Data Type Specification, volume 906 of Lecture Notes in Computer Science, pages 255–272. Springer-Verlag, 1994.Google Scholar
  14. 14.
    J. Gallier. On Girard's “candidats de réducibilité”. In P. Odifreddi, editor, Logic and Computer Science, pages 123–203. Academic Press, 1990.Google Scholar
  15. 15.
    H. Geuvers and M.J. Nederhof. A modular proof of Strong Normalization for the Calculus of Constructions, Journal of Functional Programming 1, 2 (1991), 155–189.Google Scholar
  16. 16.
    H. Geuvers. Logics and type systems. PhD thesis, University of Nijmegen, 1993.Google Scholar
  17. 17.
    H. Geuvers. A short and flexible proof of strong normalisation for the calculus of constructions. In P. Dybjer, B. Nordström, and J. Smith, editors, Proceedings of TYPES'94, volume 996 of Lecture Notes in Computer Science, pages 14–38. Springer-Verlag, 1995.Google Scholar
  18. 18.
    J-Y. Girard. Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre supérieur. PhD thesis, Université Paris 7, 1972.Google Scholar
  19. 19.
    J-P. Jouannaud and M. Okada. Executable higher-order algebraic specification languages. In Proceedings of LICS'91, pages 350–361. IEEE Press, 1991.Google Scholar
  20. 20.
    Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Number 11 in International Series of Monographs on Computer Science. Oxford University Press, 1994.Google Scholar
  21. 21.
    A. Middeldorp. Modular properties of term-rewriting systems. PhD thesis, Department of Computer Science, Vrije Universiteit, Amsterdam, 1990.Google Scholar
  22. 22.
    F. Müller. Confluence of the lambda calculus with left-linear algebraic rewriting. Information Processing Letters, 41:293–299, 1992.Google Scholar
  23. 23.
    C. Paulin-Mohring. Inductive definitions in the system Coq. Rules and properties. In Bezem and Groote [9], pages 328–345.Google Scholar
  24. 24.
    W. Tait. A realisability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium 73, volume 453 of Lectures Notes in Mathematics, pages 240–251, 1975.Google Scholar
  25. 25.
    J. Terlouw. Strong normalisation in type systems: a model-theoretical approach. In Dirk van Dalen Festschrift, pages 161–190. University of Utrecht, 1993. To appear in Annals of Pure and Applied Logic.Google Scholar
  26. 26.
    Y. Toyama. On the Church-Rosser property for the direct sum of term rewriting systems. Journal of the ACM, 34(1):128–143, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Gilles Barthe
    • 1
  • Herman Geuvers
    • 2
    • 3
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Faculty of Mathematics and InformaticsUniversity of NijmegenThe Netherlands
  3. 3.Fac. of Math. and InformaticsTechn. Univ. of EindhovenThe Netherlands

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