Revisiting Zucker’s Work on the Correspondence Between Cut-Elimination and Normalisation

  • Christian UrbanEmail author
Part of the Trends in Logic book series (TREN, volume 39)


Zucker showed that in the fragment of intuitionistic logic whose formulae are build up from \(\wedge \), \(\supset \) and \(\forall \) only, every reduction sequence in natural deduction corresponds to a reduction sequence in the sequent calculus and vice versa. Unfortunately, the technical machinery in Zucker’s work is rather cumbersome and complicated. One contribution of this chapter is to greatly simplify his arguments. For example he defined a cut-elimination procedure modulo an equivalence relation; our cut-elimination procedure will be a simple term-rewriting system instead. Zucker also showed that the correspondence breaks down when the connectives \(\vee \) or \(\exists \) are included. We shall show that this negative result is not because cut-elimination fails to be strongly normalising for these connectives, as asserted by Zucker, rather it is because certain cut-elimination reductions do not correspond to any normalisation reduction.


Intuitionistic logic Sequent calculus Cut-elimination Natural deduction Normalisation 


  1. 1.
    Altenkirch, T., Dybjer, P., Hofmann, M., & Scott, P. (2001). Normalization by Evaluation for typed lambda calculus with coproducts. Proceedings of Logic in Computer Science (pp. 203–210).Google Scholar
  2. 2.
    Barbanera, F. & Berardi, S. (1994). A symmetric lambda calculus for “classical” program extraction. Theoretical Aspects of Computer Software, volume 789 of LNCS (pp. 495–515). Springer.Google Scholar
  3. 3.
    Barendregt, H., & Ghilezan, S. (2000). Theoretical pearls: Lambda terms for natural deduction, sequent calculus and cut elimination. Journal of Functional Programming, 10(1), 121–134.CrossRefGoogle Scholar
  4. 4.
    Bloo, R. (1997). Preservation of termination for explicit substitution. Ph.D. thesis. Eindhoven University of Technology.Google Scholar
  5. 5.
    Espírito Santo, J. C. (2000). Revisiting the correspondence between cut elimination and normalisation. Proceedings of ICALP 2000, volume 1853 of LNCS (pp. 600–611). Springer.Google Scholar
  6. 6.
    Gallier, J. (1993). Constructive logics. Part I: A tutorial on proof systems and typed \(\lambda \)-calculi. Theoretical Computer Science, 110(2), 239–249.CrossRefGoogle Scholar
  7. 7.
    Gentzen, G. (1935). Untersuchungen über das logische Schließen I and II. Mathematische Zeitschrift, 39(176–210), 405–431.CrossRefGoogle Scholar
  8. 8.
    Girard, J.-Y., Lafont, Y., & Taylor, P. (1989). Proofs and types, volume 7 of cambridge tracts in theoretical computer science. Cambridge: Cambridge University Press.Google Scholar
  9. 9.
    Herbelin, H. (1994). A \(\lambda \)-calculus structure isomorphic to sequent calculus structure. Computer Science Logic, volume 933 of LNCS (pp. 67–75). Springer.Google Scholar
  10. 10.
    Kreisel, G. (1971). A survey of proof theory II. Proceedings of the 2nd Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics pp. 109–170. North-Holland.Google Scholar
  11. 11.
    Laird, J. (2001). A deconstruction of non-deterministic cut elimination. Proceedings of the 5th International Conference on Typed Lambda Calculi and Applications, Volume 2044 of LNCS pp. 268–282. Springer.Google Scholar
  12. 12.
    Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  13. 13.
    Pottinger, G. (1977). Normalisation as homomorphic image of cut-elimination. Annals of Mathematical Logic, 12, 323–357.CrossRefGoogle Scholar
  14. 14.
    Prawitz, D. (1965). Natural deduction: A proof-theoretical study. Stockholm: Almquist and Wiksell.Google Scholar
  15. 15.
    Troelstra, A. S., & Schwichtenberg, H. (2000). Basic proof theory. Cambridge tracts in theoretical computer science (2\(^{nd}\) ed.). Cambridge: Cambridge University Press.Google Scholar
  16. 16.
    Ungar, A. M. (1992). Normalisation, cut-elimination and the theory of proofs, volume 28 of CLSI Lecture Notes. Stanford: Center for the Study of Language and Information.Google Scholar
  17. 17.
    Urban, C. (2000). Classical logic and computation. Ph.D. Thesis. Cambridge University.Google Scholar
  18. 18.
    Urban, C., & Bierman, G. M. (2001). Strong normalisation of cut-elimination in classical logic. Fundamenta Informaticae, 45(1–2), 123–155.Google Scholar
  19. 19.
    Zucker, J. (1974). The correspondence between cut-elimination and normalisation. Annals of Mathematical Logic, 7, 1–112.CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of InformaticsKing’s College LondonLondonUK

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