Normal Forms and Cut-Free Proofs as Natural Transformations

  • Jean-Yves Girard
  • Andre Scedrov
  • Philip J. Scott
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 21)


What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds’ relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac Lane-Mints approach to coherence problems in category theory.


Normal Form Natural Transformation Natural Deduction Sequent Calculus Lambda Calculus 
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Copyright information

© Springer-Verlag New York, Inc 1992

Authors and Affiliations

  • Jean-Yves Girard
    • 1
  • Andre Scedrov
    • 2
    • 3
  • Philip J. Scott
    • 4
  1. 1.Équipe de Logique, UA 753 du CNRS, Mathématiques, t. 45–55Univ. de Paris 7Paris Cedex 05France
  2. 2.Dept. of Computer ScienceStanford UniversityStanfordUSA
  3. 3.Dept. of MathematicsUniv. of PennsylvaniaPhiladelphiaUSA
  4. 4.Dept. of MathematicsUniv. of OttawaOttawaCanada

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