Skip to main content
Springer Nature Link
Log in

Strong Cut-Elimination Systems for Hudelmaier’s Depth-Bounded Sequent Calculus for Implicational Logic

  • Conference paper
Automated Reasoning (IJCAR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4130))

Included in the following conference series:

  • 1100 Accesses

Abstract

Inspired by the Curry-Howard correspondence, we study normalisation procedures in the depth-bounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cut-admissibility. We decorate proofs with terms and introduce various term-reduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cut-admissibility suggesting weakly normalising procedures for Cut-elimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  2. Dyckhoff, R., Kesner, D., Lengrand, S.: Strong cut-elimination systems for Hudelmaier’s depth-bounded sequent calculus for implicational logic(2006), Full version available at http://www.pps.jussieu.fr/~lengrand/Work/Papers.html

  3. Dyckhoff, R., Lengrand, S.: LJQ, a strongly focused calculus for intuitionistic logic. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  4. Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Communications of the ACM 22(8), 465–476 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dyckhoff, R., Negri, S.: Admissibility of structural rules for contraction-free systems of intuitionistic logic. The Journal of Symbolic Logic 65(4), 1499–1518 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dyckhoff, R.: Contraction-free sequent calculi for intuitionistic logic. The Journal of Symbolic Logic 57(3), 795–807 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hudelmaier, J.: Bounds for Cut Elimination in Intuitionistic Logic. PhD thesis, Universität Tübingen (1989)

    Google Scholar 

  8. Hudelmaier, J.: Bounds on cut-elimination in intuitionistic propositional logic. Archive for Mathematical Logic 31, 331–354 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kamin, S., Lévy, J.-J.: Attempts for generalizing the recursive path orderings. Handwritten paper, University of Illinois (1980)

    Google Scholar 

  10. Lincoln, P., Scedrov, A., Shankar, N.: Linearizing intuitionistic implication. In: Proc. of the Sixth Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, pp. 51–62 (1991)

    Google Scholar 

  11. Matthes, R.: Contraction-aware λ-calculus, Seminar at Oberwolfach (2002)

    Google Scholar 

  12. O’Donnell, M.J.: Computing in Systems Described by Equations. LNCS, vol. 58. Springer, Heidelberg (1977)

    MATH  Google Scholar 

  13. Otten, J., Raths, T., Kreitz, C.: The ILTP Library. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 333–337. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  14. Pitts, A.M.: On an interpretation of second order quantification in first-order intuitionistic propositional logic. Journal of Symbolic Logic 57, 33–52 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pitts, A.M.: Nominal logic, a first order theory of names and binding. Information and Computation 186, 165–193 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  17. Vestergaard, R.: Revisiting Kreisel: A computational anomaly in the Troelstra-Schwichtenberg g3i system (March 1999), available at http://www.cee.hw.ac.uk/homedirjrvest/

  18. Vorob’ev, N.N.: A new algorithm for derivability in the constructive propositional calculus. American Mathematical Society Translations 94(2), 37–71 (1970)

    MATH  MathSciNet  Google Scholar 

  19. van Oostrom, V., van Raamsdonk, F.: Weak orthogonality implies confluence: the higher-order case. In: Matiyasevich, Y.V., Nerode, A. (eds.) LFCS 1994. LNCS, vol. 813, pp. 379–392. Springer, Heidelberg (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, University of St Andrews, Scotland

    Roy Dyckhoff & Stéphane Lengrand

  2. PPS, CNRS and Université Paris 7, France

    Delia Kesner & Stéphane Lengrand

Authors
  1. Roy Dyckhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Delia Kesner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stéphane Lengrand
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Artificial Intelligence Research Group, University of Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz

    Ulrich Furbach

  2. SRI International, MS EL256,, 333 Ravenswood Avenue, 94025-3493, Menlo Park, CA, USA

    Natarajan Shankar

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dyckhoff, R., Kesner, D., Lengrand, S. (2006). Strong Cut-Elimination Systems for Hudelmaier’s Depth-Bounded Sequent Calculus for Implicational Logic. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_31

Download citation

  • DOI: https://doi.org/10.1007/11814771_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37187-8

  • Online ISBN: 978-3-540-37188-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation