Deep Inference and Its Normal Form of Derivations

  • Kai Brünnler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


We see a notion of normal derivation for the calculus of structures, which is based on a factorisation of derivations and which is more general than the traditional notion of cut-free proof in this formalism.


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  1. 1.
    Brünnler, K.: Atomic cut elimination for classical logic. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 86–97. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Brünnler, K.: Deep Inference and Symmetry in Classical Proofs. PhD thesis, Technische Universität Dresden (September 2003)Google Scholar
  3. 3.
    Brünnler, K.: Cut elimination inside a deep inference system for classical predicate logic. Studia Logica 82(1), 51–71 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brünnler, K., Lengrand, S.: On two forms of bureaucracy in derivations. In: Bruscoli, P., Lamarche, F., Stewart, C. (eds.) Structures and Deduction, Technische Universität Dresden, pp. 69–80 (2005)Google Scholar
  5. 5.
    Brünnler, K., Tiu, A.F.: A local system for classical logic. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS, vol. 2250, pp. 347–361. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Bruscoli, P., Guglielmi, A.: On the proof complexity of deep inference (in preparation, 2006)Google Scholar
  7. 7.
    Di Gianantonio, P.: Structures for multiplicative cyclic linear logic: Deepness vs cyclicity. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 130–144. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Fürmann, C., Pym, D.: Order-enriched categorical models of the classical sequent calculus (submitted, 2004)Google Scholar
  9. 9.
    Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland Publishing Co., Amsterdam (1969)Google Scholar
  10. 10.
    Guglielmi, A.: The calculus of structures website, Available from:
  11. 11.
    Guglielmi, A.: A system of interaction and structure. Technical Report WV-02-10, Technische Universität Dresden. ACM Transactions on Computational Logic (to appear, 2002)Google Scholar
  12. 12.
    Guglielmi, A.: The problem of bureaucracy and identity of proofs from the perspective of deep inference. In: Bruscoli, P., Lamarche, F., Stewart, C. (eds.) Structures and Deduction, Technische Universität Dresden, pp. 53–68 (2005)Google Scholar
  13. 13.
    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
  14. 14.
    Guiraud, Y.: The three dimensions of proofs. Annals of pure and applied logic (to appear, 2005)Google Scholar
  15. 15.
    Hein, R., Stewart, C.: Purity through unravelling. In: Bruscoli, P., Lamarche, F., Stewart, C. (eds.) Structures and Deduction, Technische Universität Dresden, pp. 126–143 (2005)Google Scholar
  16. 16.
    Kahramanoğullari, O.: Reducing nondeterminism in the calculus of structures (manuscript, 2005)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    McKinley, R.: Categorical models of first-order classical proofs. The degree of Doctor of Philosophy of the University of Bath (submitted, 2005)Google Scholar
  19. 19.
    Pym, D.J.: The Semantics and Proof Theory of the Logic of Bunched Implications. Applied Logic Series, vol. 26. Kluwer Academic Publishers, Dordrecht (2002)MATHGoogle Scholar
  20. 20.
    Schütte, K.: Schlussweisen-Kalküle der Prädikatenlogik. Mathematische Annalen 122, 47–65 (1950)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Schütte, K.: Proof Theory. Springer, Heidelberg (1977)CrossRefMATHGoogle Scholar
  22. 22.
    Stewart, C., Stouppa, P.: A systematic proof theory for several modal logics. In: Schmidt, R., Pratt-Hartmann, I., Reynolds, M., Wansing, H. (eds.) Advances in Modal Logic, vol. 5, pp. 309–333. King’s College Publications (2005)Google Scholar
  23. 23.
    Stouppa, P.: A deep inference system for the modal logic S5. Studia Logica (to appear, 2006)Google Scholar
  24. 24.
    Straßburger, L.: Linear Logic and Noncommutativity in the Calculus of Structures. Ph.D thesis, Technische Universität Dresden (2003)Google Scholar
  25. 25.
    Straßburger, L.: On the axiomatisation of boolean categories with and without medial (manuscript, 2005)Google Scholar
  26. 26.
    Tiu, A.F.: Properties of a Logical System in the Calculus of Structures. Master’s thesis, Technische Universität Dresden (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kai Brünnler
    • 1
  1. 1.Institut für angewandte Mathematik und InformatikBernSwitzerland

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