Confluence of Cut-Elimination Procedures for the Intuitionistic Sequent Calculus

  • Kentaro Kikuchi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


We prove confluence of two cut-elimination procedures for the implicational fragment of a standard intuitionistic sequent calculus. One of the cut-elimination procedures uses global proof transformations while the other consists of local ones. Both of them include permutation of cuts to simulate β-reduction in an isomorphic image of the λ-calculus. We establish the confluence properties through a conservativity result on the cut-elimination procedures.


Sequent calculus Cut-elimination Confluence λ-calculus Explicit substitution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bloo, R., Geuvers, H.: Explicit substitution: On the edge of strong normalization. Theoretical Computer Science 211, 375–395 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Danos, V., Joinet, J.-B., Schellinx, H.: A new deconstructive logic: Linear logic. The Journal of Symbolic Logic 62, 755–807 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Espírito Santo, J.: Revisiting the correspondence between cut elimination and normalisation. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 600–611. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Gentzen, G.: Untersuchungen über das logische Schliessen. Mathematische Zeitschrift, 39: pp. 176–210, pp. 405–431, English translation in [9 pp. 68–131] (1935)Google Scholar
  5. 5.
    Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hardin, T.: Résultats de confluence pour les règles fortes de la logique combinatoire catégorique et liens avec les lambda-calculs. Thèse de doctorat, Université de Paris VII (1987)Google Scholar
  7. 7.
    Howard, W.A.: The formulae-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pp. 479–490. Academic Press, San Diego (1980)Google Scholar
  8. 8.
    Kikuchi, K.: On a local-step cut-elimination procedure for the intuitionistic sequent calculus. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 120–134. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Szabo, M.E. (ed.): The Collected Papers of Gerhard Gentzen. North-Holland, Amsterdam (1969)MATHGoogle Scholar
  10. 10.
    Takahashi, M.: Parallel reductions in λ-calculus. Information and Computation 118, 120–127 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Urban, C., Bierman, G.M.: Strong normalisation of cut-elimination in classical logic. Fundamenta Informaticae 45, 123–155 (2001)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Kentaro Kikuchi
    • 1
  1. 1.RIEC, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577Japan

Personalised recommendations