Encoding the Pure Lambda Calculus into Hierarchical Graph Rewriting

  • Kazunori Ueda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5117)

Abstract

Fine-grained reformulation of the lambda calculus is expected to solve several difficulties with the notion of substitutions—definition, implementation and cost properties. However, previous attempts including those using explicit substitutions and those using Interaction Nets were not ideally simple when it came to the encoding of the pure (as opposed to weak) lambda calculus. This paper presents a novel, fine-grained, and highly asynchronous encoding of the pure lambda calculus using LMNtal, a hierarchical graph rewriting language, and discusses its properties. The major strength of the encoding is that it is significantly simpler than previous encodings, making it promising as an alternative formulation, rather than just the encoding, of the pure lambda calculus. The membrane construct of LMNtal plays an essential role in encoding colored tokens and operations on them. The encoding has been tested using the publicly available LMNtal implementation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abadi, M., Cardelli, L., Curien, P.-L., Lévy, J.-J.: Explicit Substitutions. Journal of Functional Programming 1(4), 375–416 (1991)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bloo, R., Geuvers, H.: Explicit Substitution: on the Edge of Strong Normalization. Theoretical Computer Science 211(1–2), 375–395 (1999)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Berry, G., Boudol, G.: The Chemical Abstract Machine. In: Proc. POPL 1990, pp. 81–94. ACM, New York (1990)CrossRefGoogle Scholar
  4. 4.
    Curien, P.-L., Hardin, T., Lévy, J.-J.: Confluence Properties of Weak and Strong Calculi of Explicit Substitutions. J. ACM 43(2), 362–397 (1996)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Fernández, M., Mackie, I., Sinot, F.-R.: Closed Reduction: Explicit Substitutions without α-conversion. Mathematical Structures in Computer Science 15(2), 343–381 (2005)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Lafont, Y.: Interaction Nets. In: Proc. POPL 1990, pp. 95–108. ACM, New York (1990)CrossRefGoogle Scholar
  7. 7.
    Lamping, J.: An Algorithm for Optimal Lambda-Calculus Reductions. In: Proc. POPL 1990, pp. 16–30. ACM, New York (1990)CrossRefGoogle Scholar
  8. 8.
    Lang, F.: Modèles de la β-réduction pour les implantations. Ph.D. Thesis, Ècole Normale Supérieure de Lyon (1998)Google Scholar
  9. 9.
    Mackie, I.: YALE: Yet Another Lambda Evaluator Based on Interaction Nets. In: Proc. ICFP 1998, pp. 117–128. ACM, New York (1998)CrossRefGoogle Scholar
  10. 10.
    Mackie, I.: Efficient λ-Evaluation with Interaction Nets. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 155–169. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    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
  12. 12.
    Sinot, F.-R.: Call-by-Name and Call-by-Value as Token-Passing Interaction Nets. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 386–400. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Peyton Jones, S.L.: The Implementation of Functional Programming Languages. Prentice-Hall, Englewood Cliffs (1987)MATHGoogle Scholar
  14. 14.
    Ueda, K., Kato, N.: LMNtal: a Language Model with Links and Membranes. In: Mauri, G., Păun, G., Jesús Pérez-Jímenez, M., Rozenberg, G., Salomaa, A. (eds.) WMC 2004. LNCS, vol. 3365, pp. 110–125. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Ueda, K.: Encoding Distributed Process Calculi into LMNtal. Electronic Notes in Theoretical Computer Science 209, 187–200 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Kazunori Ueda
    • 1
  1. 1.Dept. of Computer Science and EngineeringWaseda UniversityTokyoJapan

Personalised recommendations