Conservativeness of Λ over λσ-calculus

  • Masahiko Sato
  • Yukiyoshi Kameyama
Part of the Lecture Notes in Computer Science book series (LNCS, volume 792)


Λ is a unique functional programming language which has the facility of the encapsulated assignment, without losing referential transparency. The let-construct in Λ can be considered as an environment, which has a close relationship to substitution in λσ-calculus. This paper discusses the relationship between these two calculi; we first define a slightly modified version of Λ-calculus which adopts de Bruijn's index notation. We then define an injective map from λσ-calculus to Λ, and show that the Beta-reduction and the σ-reductions in λσ-calculus correspond to the β-reduction and let-reductions in Λ-calculus, respectively. Finally, we prove that, as equality theories, Λ is conservative over the λσ-calculus.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadi, M., L. Cardelli, P.-L. Curien, and J.-J. Levy: Explicit Substitutions, 17th Annual ACM Symp. on Principles of Programming Languages, pp. 31–46, 1990.Google Scholar
  2. 2.
    Curien, P.-L.: Categorical Combinators, Information and Control, 69, pp. 188–254, 1986.CrossRefGoogle Scholar
  3. 3.
    de Bruijn, N. G., Lambda-calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, Indag. Mat., 34, pp. 381–392, 1972.Google Scholar
  4. 4.
    Sato, M: A Purely Functional Language with Encapsulated Assignment, to appear in Proc. of Second Intl Symp. on Theoretical Aspects of Computer Software, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Masahiko Sato
    • 1
  • Yukiyoshi Kameyama
    • 1
  1. 1.Research Institute of Electrical CommunicationTohoku UniversitySendaiJapan

Personalised recommendations