Strong Normalisation of Cut-Elimination That Simulates β-Reduction

  • Kentaro Kikuchi
  • Stéphane Lengrand
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4962)


This paper is concerned with strong normalisation of cut-elimination for a standard intuitionistic sequent calculus. The cut- elimination procedure is based on a rewrite system for proof-terms with cut-permutation rules allowing the simulation of β-reduction. Strong normalisation of the typed terms is inferred from that of the simply-typed λ-calculus, using the notions of safe and minimal reductions as well as a simulation in Nederpelt-Klop’s λI-calculus. It is also shown that the type-free terms enjoy the preservation of strong normalisation (PSN) property with respect to β-reduction in an isomorphic image of the type-free λ-calculus.


Reduction Rule Natural Deduction Sequent Calculus Minimal Reduction Dependency Pair 
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.


  1. 1.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoret. Comput. Sci. 236(1–2), 133–178 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, Cambridge (1998)Google Scholar
  3. 3.
    Benaissa, Z., Briaud, D., Lescanne, P., Rouyer-Degli, J.: λυ, a calculus of explicit substitutions which preserves strong normalisation. J. Funct. Programming 6(5), 699–722 (1996)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bloo, R.: Preservation of Termination for Explicit Substitution. PhD thesis, Technische Universiteit Eindhoven, IPA Dissertation Series 1997-05 (1997)Google Scholar
  5. 5.
    Bloo, R., Geuvers, H.: Explicit substitution: On the edge of strong normalization. Theoret. Comput. Sci. 211(1–2), 375–395 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Church, A.: The Calculi of Lambda Conversion. Princeton University Press, Princeton (1941)Google Scholar
  7. 7.
    Curien, P.-L., Herbelin, H.: The duality of computation. In: Proc. of the 5th ACM SIGPLAN Int. Conf. on Functional Programming (ICFP 2000), pp. 233–243. ACM Press, New York (2000)CrossRefGoogle Scholar
  8. 8.
    Danos, V., Joinet, J.-B., Schellinx, H.: LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Proc. of the Work. on Advances in Linear Logic. London Math. Soc. Lecture Note Ser., vol. 222, pp. 211–224. Cambridge University Press, Cambridge (1995)Google Scholar
  9. 9.
    Danos, V., Joinet, J.-B., Schellinx, H.: A new deconstructive logic: Linear logic. J. of Symbolic Logic 62(3), 755–807 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dyckhoff, R., Pinto, L.: Permutability of proofs in intuitionistic sequent calculi. Theoret. Comput. Sci. 212(1–2), 141–155 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dyckhoff, R., Urban, C.: Strong normalization of Herbelin’s explicit substitution calculus with substitution propagation. J. Logic Comput. 13(5), 689–706 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Herbelin, H.: A lambda-calculus structure isomorphic to Gentzen-style sequent calculus structure. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 61–75. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  13. 13.
    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, London (1980) (reprint of a manuscript written 1969)Google Scholar
  14. 14.
    Kamin, S., Lévy, J.-J.: Attempts for generalizing the recursive path orderings. Handwritten paper. University of Illinois (1980)Google Scholar
  15. 15.
    Kesner, D., Lengrand, S.: Resource operators for λ-calculus. Inform. and Comput. 205(4), 419–473 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    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
  17. 17.
    Kikuchi, K.: Confluence of cut-elimination procedures for the intuitionistic sequent calculus. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) CiE 2007. LNCS, vol. 4497, pp. 398–407. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Kikuchi, K., Lengrand, S.: Strong normalisation of cut-elimination that simulates β-reduction - long version,
  19. 19.
    Klop, J.-W.: Combinatory Reduction Systems, Mathematical Centre Tracts, PhD Thesis, vol. 127, CWI, Amsterdam (1980)Google Scholar
  20. 20.
    Lengrand, S.: Call-by-value, call-by-name, and strong normalization for the classical sequent calculus. In: Gramlich, B., Lucas, S. (eds.) Post-proc. of the 3rd Int. Work. on Reduction Strategies in Rewriting and Programming (WRS 2003). ENTCS, vol. 86, Elsevier, Amsterdam (2003)Google Scholar
  21. 21.
    Lengrand, S.: Induction principles as the foundation of the theory of normalisation: concepts and techniques. Technical report, Université Paris 7 (March 2005),
  22. 22.
    Lengrand, S.: Normalisation & Equivalence in Proof Theory & Type Theory. PhD thesis, Université Paris 7 & University of St. Andrews (2006)Google Scholar
  23. 23.
    Nakazawa, K.: An isomorphism between cut-elimination procedure and proof reduction. In: Della Rocca, S.R. (ed.) TLCA 2007. LNCS, vol. 4583, pp. 336–350. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  24. 24.
    Nederpelt, R.: Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Eindhoven University of Technology (1973)Google Scholar
  25. 25.
    Pottinger, G.: Normalization as a homomorphic image of cut-elimination. Ann. of Math. Logic 12, 323–357 (1977)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Santo, J.E.: 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
  27. 27.
    Sørensen, M.H.B.: Strong normalization from weak normalization in typed lambda-calculi. Inform. and Comput. 37, 35–71 (1997)CrossRefGoogle Scholar
  28. 28.
    Sørensen, M.H.B., Urzyczyn, P.: Lectures on the Curry-Howard Isomorphism. Studies in Logic and the Foundations of Mathematics, vol. 149. Elsevier, Amsterdam (2006)Google Scholar
  29. 29.
    Sørensen, M.H.B., Urzyczyn, P.: Strong cut-elimination in sequent calculus using Klop’s ι-translation and perpetual reduction (available from the authors) (submitted for publication, 2007)Google Scholar
  30. 30.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge Tracts in Theoret. Comput. Sci., vol. 43. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  31. 31.
    Urban, C.: Classical Logic and Computation. PhD thesis, University of Cambridge (2000)Google Scholar
  32. 32.
    Urban, C., Bierman, G.M.: Strong normalisation of cut-elimination in classical logic. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 365–380. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  33. 33.
    van Raamsdonk, F., Severi, P., Sørensen, M.H.B., Xi, H.: Perpetual reductions in λ-calculus. Inform. and Comput. 149(2), 173–225 (1999)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Xi, H.: Weak and strong beta normalisations in typed lambda-calculi. In: de Groote, P., Hindley, J.R. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 390–404. Springer, Heidelberg (1997)Google Scholar
  35. 35.
    Zucker, J.: The correspondence between cut-elimination and normalization. Ann. of Math. Logic 7, 1–156 (1974)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Kentaro Kikuchi
    • 1
  • Stéphane Lengrand
    • 2
  1. 1.RIECTohoku UniversityJapan
  2. 2.CNRSLaboratoire d’Informatique de l’Ecole PolytechniqueFrance

Personalised recommendations