Naming Proofs in Classical Propositional Logic

  • François Lamarche
  • Lutz Straßburger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3461)


We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cut-elimination procedure. This gives us a “Boolean” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.


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  1. 1.
    Andrews, P.B.: Refutations by matings. IEEE Transactions on Computers C-25, 801–807 (1976)Google Scholar
  2. 2.
    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
  3. 3.
    Buss, S.R.: The undecidability of k-provability. Annals of Pure and Applied Logic 53, 72–102 (1991)MathSciNetGoogle Scholar
  4. 4.
    Carbone, A.: Interpolants, cut elimination and flow graphs for the propositional calculus. Annals of Pure and Applied Logic 83, 249–299 (1997)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44(1), 36–50 (1979)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coquand, T.: A semantics of evidence for classical arithmetic. The Journal of Symbolic Logic 60(1), 325–337 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Danos, V., Joinet, J.-B., Schellinx, H.: A new deconstructive logic: Linear logic. The Journal of Symbolic Logic 62(3), 755–807 (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Došen, K.: Identity of proofs based on normalization and generality. The Bulletin of Symbolic Logic 9, 477–503 (2003)MATHCrossRefGoogle Scholar
  9. 9.
    Došen, K., Petrić, Z.: Proof-Theoretical Coherence. KCL Publications, London (2004)MATHGoogle Scholar
  10. 10.
    Führmann, C., Pym, D.: On the geometry of interaction for classical logic (extended abstract). In: LICS 2004, pp. 211–220 (2004)Google Scholar
  11. 11.
    Führmann, C., Pym, D.: Order-enriched categorical models of the classical sequent calculus (2004)Google Scholar
  12. 12.
    Girard, J.-Y.: A new constructive logic: Classical logic. Mathematical Structures in Computer Science 1, 255–296 (1991)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. In: Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1989)Google Scholar
  14. 14.
    Guglielmi, A.: A system of interaction and structure (2002), To appear in ACM Transactions on Computational Logic. On the web at,
  15. 15.
    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
  16. 16.
    Hughes, D., van Glabbeek, R.: Proof nets for unit-free multiplicative-additive linear logic. In: LICS 2003, pp. 1–10 (2003)Google Scholar
  17. 17.
    Martin, J., Hyland, E.: Abstract interpretation of proofs: Classical propositional calculus. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 6–21. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Lamarche, F., Straßburger, L.: Constructing free Boolean categories (2005) (Submitted)Google Scholar
  19. 19.
    Laurent, O.: Etude de la Polarisation en Logique. PhD thesis, Univ. Aix-Marseille II (2002)Google Scholar
  20. 20.
    Laurent, O.: Polarized proof-nets and λμ-calculus. Theoretical Computer Science 290(1), 161–188 (2003)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lane, S.M.: Categories for the Working Mathematician. In: Graduate Texts in Mathematics, vol. 5. Springer, Heidelberg (1971)Google Scholar
  22. 22.
    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)Google Scholar
  23. 23.
    Retoré, C.: Pomset logic: A non-commutative extension of classical linear logic. In: de Groote, P., Hindley, J.R. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 300–318. Springer, Heidelberg (1997)Google Scholar
  24. 24.
    Robinson, E.P.: Proof nets for classical logic. Journal of Logic and Computation 13, 777–797 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • François Lamarche
    • 1
  • Lutz Straßburger
    • 2
  1. 1.LORIA & INRIA-Lorraine, Projet CalligrammeVillers-lès-NancyFrance
  2. 2.Informatik — ProgrammiersystemeUniversität des SaarlandesSaarbrückenGermany

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