An Algorithmic Interpretation of a Deep Inference System

  • Kai Brünnler
  • Richard McKinley
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5330)


We set out to find something that corresponds to deep inference in the same way that the lambda-calculus corresponds to natural deduction. Starting from natural deduction for the conjunction-implication fragment of intuitionistic logic we design a corresponding deep inference system together with reduction rules on proofs that allow a fine-grained simulation of beta-reduction.


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  1. 1.
    Brünnler, K., Lengrand, S.: On two forms of bureaucracy in derivations. In: Bruscoli, P., Lamarche, F., Stewart, C. (eds.) Structures and Deduction, pp. 69–80. Technische Universität Dresden (2005)Google Scholar
  2. 2.
    Curien, P.-L.: Categorical Combinators, Sequential Algorithms and Functional Programming, 2nd edn. Research Notes in Theoretical Computer Science. Birkhäuser, Basel (1993)CrossRefMATHGoogle Scholar
  3. 3.
    de Bruijn, N.G.: Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem. Indagationes Mathematicae (Proceedings) 75(5), 381–392 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science, vol. 7. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  5. 5.
    Guglielmi, A.: A system of interaction and structure. ACM Transactions on Computational Logic 8(1), 1–64 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Guglielmi, A., Gundersen, T.: Normalisation control in deep inference via atomic flows. Logical Methods in Computer Science 4(1:9), 1–36 (2008), MathSciNetMATHGoogle Scholar
  7. 7.
    Hardin, T.: From categorical combinators to λσ-calculi, a quest for confluence. Technical report, INRIA Rocquencourt (1992),
  8. 8.
    Lambek, J., Scott, P.J.: Introduction to higher order categorical logic. Cambridge University Press, New York (1986)MATHGoogle Scholar
  9. 9.
    Mac Lane, S.: Categories for the Working Mathematician. In: Graduate Texts in Mathematics. Springer, Heidelberg (1971)Google Scholar
  10. 10.
    Parigot, M.: λμ-calculus: an algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  11. 11.
    Tiu, A.F.: A local system for intuitionistic logic. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS, vol. 4246, pp. 242–256. Springer, Heidelberg (2006), CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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