Skip to main content
SpringerLink
Log in
SpringerLink
Log in

What Is the Problem with Proof Nets for Classical Logic?

  • Conference paper
Programs, Proofs, Processes (CiE 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6158))

Included in the following conference series:

Abstract

This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Lambek, J., Scott, P.J.: Introduction to higher order categorical logic. In: Cambridge studies in advanced mathematics, vol. 7. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  2. Girard, J.Y.: A new constructive logic: Classical logic. Math. Structures in Comp. Science 1, 255–296 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Straßburger, L.: On the axiomatisation of Boolean categories with and without medial. Theory and Applications of Categories 18(18), 536–601 (2007)

    MATH  MathSciNet  Google Scholar 

  4. Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and Types. In: Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  5. Parigot, M.: λμ-calculus: An algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS (LNAI), vol. 624, pp. 190–201. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  6. Thielecke, H.: Categorical Structure of Continuation Passing Style. PhD thesis, University of Edinburgh (1997)

    Google Scholar 

  7. Streicher, T., Reus, B.: Classical logic, continuation semantics and abstract machines. J. of Functional Programming 8(6), 543–572 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Selinger, P.: Control categories and duality: on the categorical semantics of the lambda-mu calculus. Math. Structures in Comp. Science 11, 207–260 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Laurent, O.: Polarized proof-nets and λμ-calculus. Theoretical Computer Science 290(1), 161–188 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Führmann, C., Pym, D.: Order-enriched categorical models of the classical sequent calculus. J. of Pure and Applied Algebra (2004) (to appear)

    Google Scholar 

  11. Lamarche, F., Straßburger, L.: Constructing free Boolean categories. In: LICS 2005, pp. 209–218 (2005)

    Google Scholar 

  12. McKinley, R.: Classical categories and deep inference. In: Structures and Deduction 2005 Satellite Workshop of ICALP 2005 (2005)

    Google Scholar 

  13. Lamarche, F.: Exploring the gap between linear and classical logic. Theory and Applications of Categories 18(18), 473–535 (2007)

    MATH  MathSciNet  Google Scholar 

  14. Došen, K., Petrić, Z.: Proof-Theoretical Coherence. KCL Publ., London (2004)

    MATH  Google Scholar 

  15. Gentzen, G.: Untersuchungen über das logische Schließen. I. Mathematische Zeitschrift 39, 176–210 (1934)

    Article  MathSciNet  Google Scholar 

  16. Robinson, E.P.: Proof nets for classical logic. Journal of Logic and Computation 13, 777–797 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Danos, V., Regnier, L.: The structure of multiplicatives. Annals of Mathematical Logic 28, 181–203 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  18. Buss, S.R.: The undecidability of k-provability. Annals of Pure and Applied Logic 53, 72–102 (1991)

    MathSciNet  Google Scholar 

  19. Lamarche, F., Straßburger, L.: Naming proofs in classical propositional logic. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 246–261. Springer, Heidelberg (2005)

    Google Scholar 

  20. Andrews, P.B.: Refutations by matings. IEEE Transactions on Computers C-25, 801–807 (1976)

    Article  MATH  Google Scholar 

  21. Bibel, W.: On matrices with connections. Journal of the ACM 28, 633–645 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guglielmi, A., Straßburger, L.: Non-commutativity and MELL in the calculus of structures. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 54–68. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  23. Brünnler, K., Tiu, A.F.: A local system for classical logic. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 347–361. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  24. Straßburger, L.: From deep inference to proof nets via cut elimination. Journal of Logic and Computation (2009) (To appear)

    Google Scholar 

  25. Guglielmi, A., Gundersen, T.: Normalisation control in deep inference via atomic flows. Logical Methods in Computer Science 4(1-9), 1–36 (2008)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Saclay – Île-de-France and LIX, École Polytechnique,  

    Lutz Straßburger

Authors
  1. Lutz Straßburger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matemática, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016, Lisboa, Portugal

    Fernando Ferreira

  2. Institute for Logic, Language and Computation (ILLC), Universiteit van Amsterdam, P.O. Box 94242, 1090, Amsterdam, GE, The Netherlands

    Benedikt Löwe

  3. Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, María de Luna 1, 50018, Zaragoza, Spain

    Elvira Mayordomo

  4. Universidade dos Açores, Campus de Ponta Delgada, Apartado 1422, 9501-801, Ponta Delgada, Açores, Portugal

    Luís Mendes Gomes

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Straßburger, L. (2010). What Is the Problem with Proof Nets for Classical Logic?. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_45

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-13962-8_45

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13961-1

  • Online ISBN: 978-3-642-13962-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation