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Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic

  • Lutz Straßburger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5608)

Abstract

We investigate the question of what constitutes a proof when quantifiers and multiplicative units are both present. On the technical level this paper provides two new aspects of the proof theory of MLL2 with units. First, we give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference, which allows us to observe a decomposition which is a version of Herbrand’s theorem that is not visible in the sequent calculus. Second, we show a new notion of proof nets which is independent from any deductive system. We have “sequentialisation” into the calculus of structures as well as into the sequent calculus. Since cut elimination is terminating and confluent, we have a category of MLL2 proof nets. The treatment of the units is such that this category is star-autonomous.

Keywords

Deductive System Linear Logic Proof Theory Sequent Calculus Multiplicative Unit 
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.

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References

  1. 1.
    Miller, D.: A compact representation of proofs. Studia Logica 46(4), 347–370 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    McKinley, R.: On Herbrand’s theorem and cut elimination (extended abstract) (preprint) (2008)Google Scholar
  3. 3.
    Barr, M.: *-Autonomous Categories. Lecture Notes in Mathematics, vol. 752. Springer, Heidelberg (1979)zbMATHGoogle Scholar
  4. 4.
    Lafont, Y.: Logique, Catégories et Machines. PhD thesis, Université Paris 7 (1988)Google Scholar
  5. 5.
    Lamarche, F., Straßburger, L.: From proof nets to the free *-autonomous category. Logical Methods in Computer Science 2(4:3), 1–44 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Girard, J.Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Girard, J.Y.: Quantifiers in linear logic II. Preépublication de l’Equipe de Logique, Université Paris VII, Nr. 19 (1990)Google Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Guglielmi, A.: A system of interaction and structure. ACM Transactions on Computational Logic 8(1) (2007)Google Scholar
  10. 10.
    Straßburger, L.: Linear Logic and Noncommutativity in the Calculus of Structures. PhD thesis, Technische Universität Dresden (2003)Google Scholar
  11. 11.
    Brünnler, K.: Deep Inference and Symmetry for Classical Proofs. PhD thesis, Technische Universität Dresden (2003)Google Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Retoré, C.: Réseaux et Séquents Ordonnés. Thèse de Doctorat, spécialité mathématiques, Université Paris VII (February 1993)Google Scholar
  14. 14.
    Blute, R., Cockett, R., Seely, R., Trimble, T.: Natural deduction and coherence for weakly distributive categories. Journal of Pure and Applied Algebra 113, 229–296 (1996)Google Scholar
  15. 15.
    Devarajan, H., Hughes, D., Plotkin, G., Pratt, V.R.: Full completeness of the multiplicative linear logic of Chu spaces. In: 14th IEEE Symposium on Logic in Computer Science (LICS 1999) (1999)Google Scholar
  16. 16.
    Buss, S.R.: The undecidability of k-provability. Annals of Pure and Applied Logic 53, 72–102 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guglielmi, A., Gundersen, T.: Normalisation control in deep inference via atomic flows. Logical Methods in Computer Science 4(1:9), 1–36 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Straßburger, L., Lamarche, F.: On proof nets for multiplicative linear logic with units. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 145–159. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Danos, V., Regnier, L.: The structure of multiplicatives. Annals of Mathematical Logic 28, 181–203 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hughes, D.: Simple free star-autonomous categories and full coherence (preprint) (2005)Google Scholar
  21. 21.
    Straßburger, L.: From deep inference to proof nets. In: Structures and Deduction — The Quest for the Essence of Proofs (Satellite Workshop of ICALP 2005) (2005)Google Scholar
  22. 22.
    Hughes, D.: Simple multiplicative proof nets with units (preprint) (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Lutz Straßburger
    • 1
  1. 1.INRIA Saclay – Île-de-France — Équipe-projet Parsifal[1ex] École Polytechnique — LIXPalaiseau CedexFrance

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