Advertisement

Confluence of Untyped Lambda Calculus via Simple Types

  • Silvia Ghilezan
  • Viktor Kunčak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2202)

Abstract

We present a new proof of confluence of the untyped lambda calculus by reducing the confluence of β-reduction in the untyped lambda calculus to the confluence of β-reduction in the simply typed lambda calculus. This is achieved by embedding typed lambda terms into simply typed lambda terms. Using this embedding, an auxiliary reduction, and β-reduction on simply typed lambda terms we define a new reduction on all lambda terms. The transitive closure of the reduction defined is β-reduction on all lambda terms. This embedding allows us to use the confluence of β-reduction on simply typed lambda terms and thus prove the confluence of the reduction defined. As a consequence we obtain the confluence of β-reduction in the untyped lambda calculus.

Keywords

Transitive Closure Simple Type Type Assignment Lambda Calculus Lambda Term 
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. North-Holland, Amsterdam (1984).MATHGoogle Scholar
  2. 2.
    Barendregt, H. P.: Lambda calculi with types. In: Abramsky, S., Gabbay, D. M. and T. S. E. Maibaum (eds.): Handbook of Logic in Computer Science, Vol. 2. Oxford University Press, Oxford (1992) 117–309.Google Scholar
  3. 3.
    Dershowitz, N. and J. P. Jounnaud: Rewrite Systems. In: Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Elsevier Science Publishers B. V. (1990).Google Scholar
  4. 4.
    Ghilezan, S.: Application of typed lambda calculi in the untyped lambda calculus. In: Nerode, A. and Yu. Matiyasevich (eds.): Logical Foundations of Computer Science’ 94. Lecture Notes in Computer Science 813, Springer-Verlag, Berlin (1994) 129–139.Google Scholar
  5. 5.
    Ghilezan, S.: Generalized finiteness of developments in typed lambda calculi. Journal of Automata, Languages and Combinatorics 4 (1996) 247–257.MathSciNetGoogle Scholar
  6. 6.
    Klop, J. W., V. van Oostrom, and R. de Vrijer: A geometric proof of confluence by decreasing diagrams. Journal of Logic and Computation 10(3) (2000) 437–460.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Koletsos, G.: Church-Rosser theorem for typed functionals. Journal of Symbolic Logic 50 (1985) 782–790.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Koletsos, G. and G. Stavrinos: Church-Rosser theorem for conjunctive type systems. In: Kakas, A. K. and A. Sinachopoulos (eds.): Proceedings of the First Panhellenic Logic Symposium. Nicosia (1997) 25–37.Google Scholar
  9. 9.
    Krivine, J. L.: Lambda-calcul types et modèles. Masson, Paris (1990)MATHGoogle Scholar
  10. 10.
    Meyer, A. M.: What is a model of lambda calculus? Information and control 122 (1982) 52–87.Google Scholar
  11. 11.
    Mitchell, J.: Type Systems for Programming Languages. In: Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Elsevier Science Publishers B. V. (1990) 365–458.Google Scholar
  12. 12.
    Newman, M. H. A.: On theories with a combinatorial definition of ‘equivalence’. Annals of Mathematics 43 (1942) 223–243.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Pfenning, F.: A Proof of the Church-Rosser theorem and its representation in a Logical Framework, CMU-CS-92-186, (September 1992), forthocoming in Journal of Authomated Reasoning.Google Scholar
  14. 14.
    Scott, D.: Relating theories of the lambda calculus. In: Seldin, J. P. and J. R. Hindley: To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London (1980) 403–450.Google Scholar
  15. 15.
    Statman, R.: Logical relations and the simply typed lambda calculus. Information and Control 65 (1985) 85–97.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Takahashi, M.: Parallel Reductions in λ-Calculus. Journal of Symbolic Computation 7 (1989) 113–123.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    van Oostrom, V.: Confluence by decreasing diagrams. Theoretical Computer Science 126 (1994) 259–280.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    van Oostrom, V. and F. van Raamsdonk: Weak orthogonality implies confluence: the higher order case. In: Nerode, A. and Yu. Matiyasevich (eds.): Logical Foundations of Computer Science’ 94. Lecture Notes in Computer Science 813, Springer-Verlag, Berlin (1994) 379–392.Google Scholar
  19. 19.
    Wadsworth, C. P.: The relation between computational and denotational properties for Scott’s D∞-models of the lambda calculus. SIAM Journal of Computing 5(3) (1976) 488–521.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Silvia Ghilezan
    • 1
    • 2
  • Viktor Kunčak
    • 3
  1. 1.Faculty of EngineeringUniversity of Novi SadYugoslavia
  2. 2.Computing Science DepartmentCatholic UniversityNijmegenThe Netherlands
  3. 3.Laboratory of Computer ScienceMITCambridgeUSA

Personalised recommendations