Advertisement

A de Bruijn Notation for Higher-Order Rewriting

(Extended Abstract)
  • Eduardo Bonelli
  • Delia Kesner
  • Alejandro Ríos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1833)

Abstract

We propose a formalism for higher-order rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higher-order rewrite systems. We give formal translations from higher-order rewriting with names to higher-order rewriting with de Bruijn indices, and vice-versa. These translations can be viewed as an interface in programming languages based on higher-order rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and that confluence is preserved when translating one formalism into the other.

Keywords

SERS Formalism Path Condition Parameter Path Index Notation Rewrite Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics, Revised edn. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1984)Google Scholar
  2. 2.
    Bloo, R.: Preservation of Termination for Explicit Substitution. PhD thesis, Eindhoven University (1997) Google Scholar
  3. 3.
    Bloo, R., Rose, K.: Combinatory reduction systems with explicit substitution that preserve strong normalisation. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 169–183. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Bonelli, E., Kesner, D., Ríos, A.: A de Bruijn notation for higher-order rewriting (2000), Available as ftp://ftp.lri.fr/LRI/articles/kesner/dBhor.ps.gz
  5. 5.
    Curien, P.-L.: Categorical combinators, sequential algorithms and functional programming, 2nd edn. Progress in Theoretical Computer Science. Birkhaüser, Basel (1993)zbMATHGoogle Scholar
  6. 6.
    Curien, P.-L., Hardin, T., Lévy, J.-J.: Confluence properties of weak and strong calculi of explicit substitutions. Journal of the ACM 43(2), 362–397 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    de Bruijn, N.: 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
  8. 8.
    Dowek, G.: La part du calcul. Univesité de Paris VII, Thèse d’Habilitation à diriger des recherches (1999)Google Scholar
  9. 9.
    Dowek, G., Hardin, T., Kirchner, C.: Higher-order unification via explicit substitutions. In: LICS (1995)Google Scholar
  10. 10.
    Hindley, R., Seldin, J.P.: Introduction to Combinators and λ-calculus. London Mathematical Society (1980)Google Scholar
  11. 11.
    Kamareddine, F., Ríos, A.: Bridging de bruijn indices and variable names in explicit substitutions calculi. Logic Journal of the Interest Group of Pure and Applied Logic (IGPL) 6(6), 843–874 (1998)zbMATHGoogle Scholar
  12. 12.
    Khasidashvili, Z.: Expression reduction systems. In: Proceedings of I. Vekua Institute of Applied Mathematics, Tbilisi, vol. 36 (1990)Google Scholar
  13. 13.
    Khasidashvili, Z., van Oostrom, V.: Context-sensitive Conditional Expression Reduction Systems. In: Proceedings of the Joint COMPUGRAPH/ SEMAGRAPH Workshop on Graph Rewriting and Computation (SEGRAGRA 1995), ENTCS, Volterra, vol. 2 (1995)Google Scholar
  14. 14.
    Klop, J.W.: Combinatory Reduction Systems. Mathematical Centre Tracts, vol. 127. CWI, Amsterdam (1980), PhD ThesiszbMATHGoogle Scholar
  15. 15.
    Nipkow, T.: Higher order critical pairs. In: LICS, pp. 342–349 (1991)Google Scholar
  16. 16.
    Pagano, B.: Des Calculs de Susbtitution Explicite et leur application à la compilation des langages fonctionnels. PhD thesis, Université Paris VI (1998) Google Scholar
  17. 17.
    Pollack, R.: Closure under alpha-conversion. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, Springer, Heidelberg (1994)Google Scholar
  18. 18.
    Rehof, J., Sørensen, M.H.: The λ Δ calculus. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 516–542. Springer, Heidelberg (1994)Google Scholar
  19. 19.
    Rose, K.: Explicit cyclic substitutions. In: Rusinowitch, M., Remy, J.-L. (eds.) CTRS 1992. LNCS, vol. 656, pp. 36–50. Springer, Heidelberg (1993)Google Scholar
  20. 20.
    van Oostrom, V., van Raamsdonk, F.: Weak orthogonality implies confluence: the higher-order case. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 379–392. Springer, Heidelberg (1994)Google Scholar
  21. 21.
    van Raamsdonk, F.: Confluence and Normalization for higher-Order Rewriting. PhD thesis, Amsterdam University, The Netherlands (1996) Google Scholar
  22. 22.
    Wolfram, D.: The Clausal Theory of Types. Cambridge Tracts in Theoretical Computer Science, vol. 21. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Eduardo Bonelli
    • 1
    • 2
  • Delia Kesner
    • 2
  • Alejandro Ríos
    • 1
  1. 1.Departamento de Computación – Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos Aires, Pabellón IBuenos AiresArgentina
  2. 2.LRI (UMR 8623) - Bât 490Université de Paris-SudOrsay CedexFrance

Personalised recommendations