A Lambda-Calculus with Constructors

  • Ariel Arbiser
  • Alexandre Miquel
  • Alejandro Ríos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4098)


We present an extension of the λ(η)-calculus with a case construct that propagates through functions like a head linear substitution, and show that this construction permits to recover the expressiveness of ML-style pattern matching. We then prove that this system enjoys the Church-Rosser property using a semi-automatic ‘divide and conquer’ technique by which we determine all the pairs of commuting subsystems of the formalism (considering all the possible combinations of the nine primitive reduction rules). Finally, we prove a separation theorem similar to Böhm’s theorem for the whole formalism.


Normal Form Operational Semantic Reduction Rule Critical Pair Separation Theorem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arbiser, A., Miquel, A., Ríos, A.: A λ-calculus with constructors. Available from the web pages of the authors (manuscript, 2006)Google Scholar
  2. 2.
    Baader, F., Nipkow, T.: Rewriting and All That. Addison-Wesley, Reading (1999)MATHGoogle Scholar
  3. 3.
    Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and The Foundations of Mathematics, vol. 103. North-Holland, Amsterdam (1984)MATHGoogle Scholar
  4. 4.
    Böhm, C., Dezani-Ciancaglini, M., Peretti, P., Ronchi Della Rocha, S.: A discrimination algorithm inside lambda-beta-calculus. Theoretical Computer Science 8(3), 265–291 (1979)CrossRefGoogle Scholar
  5. 5.
    Cerrito, S., Kesner, D.: Pattern matching as cut elimination. In: Logics In Computer Science (LICS 1999), pp. 98–108 (1999)Google Scholar
  6. 6.
    Church, A.: The calculi of lambda-conversion. Annals of Mathematical Studies, vol. 6. Princeton University Press, Princeton (1941)Google Scholar
  7. 7.
    Cirstea, H., Kirchner, C.: Rho-calculus, the rewriting calculus. In: 5th International Workshop on Constraints in Computational Logics (1998)Google Scholar
  8. 8.
    Girard, J.-Y.: Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11(3), 301–506 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Barry Jay, C.: The pattern calculus. ACM Transactions on Programming Languages and Systems 26(6), 911–937 (2004)CrossRefGoogle Scholar
  10. 10.
    Peyton Jones, S., et al.: The Revised Haskell 98 Report. Cambridge Univ. Press, Cambridge (2003), Also on: Google Scholar
  11. 11.
    Kahl, W.: Basic Pattern Matching Calculi: A Fresh View on Matching Failure. In: Kameyama, Y., Stuckey, P.J. (eds.) FLOPS 2004. LNCS, vol. 2998, pp. 276–290. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Milner, R., Tofte, M., Harper, R.: The definition of Standard ML. MIT Press, Cambridge (1990)Google Scholar
  13. 13.
    The Objective Caml language,
  14. 14.
    van Oostrom, V.: Lambda calculus with patterns. Technical Report IR-228, Vrije Universiteit, Amsterdam (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ariel Arbiser
    • 1
  • Alexandre Miquel
    • 2
  • Alejandro Ríos
    • 1
  1. 1.Departamento de Computación – Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresArgentina
  2. 2.PPS & Université Paris 7 – Case 7014PARISFrance

Personalised recommendations