Advertisement

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)

Abstract

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.

Keywords

Normal Form Natural Transformation Natural Deduction Sequent Calculus Lambda Calculus 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Abramsky, J. C. Mitchell, A. Scedrov, P. Wadler. Relators (to appear.).Google Scholar
  2. 2.
    E.S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial Polymorphism, Theoretical Computer Science 70 (1990), pp. 35–64.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    E.S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial Polymorphism: Preliminary Report, Logical Foundations of Functional Programming, G. Huet, ed., Addison Wesley (1990), pp. 315–327. Google Scholar
  4. 4.
    M. Barr. *-Autonomous Categories. Springer LNM 752, 1979.Google Scholar
  5. 5.
    S. Eilenberg and G. M Kelly. A generalization of the functorial calculus, J. Algebra 3 (1966), pp. 366–375. MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. Felty . A Logic Program for Transforming Sequent Proofs to Natural Deduction Proofs. In: Proc. December 1989 Workshop on Extension of Logic Programming, ed. by P. Schroeder-Heister, Springer LNCS, to appear.Google Scholar
  7. 7.
    P. Freyd. Structural Polymorphism, Manuscript, Univ. of Pennsylvania (1989), to appear in Th. Comp. Science.Google Scholar
  8. 8.
    P. Freyd, J-Y. Girard, A. Scedrov, and P. J. Scott. Semantic Parametricity in Polymorphic Lambda Calculus, Proc. 3rd IEEE Symposium on Logic in Computer Science, Edinburgh, 1988. Google Scholar
  9. 9.
    J-Y. Girard . Normal functors, power series, and A-calculus, Ann. Pure and Applied Logic 37 (1986) pp. 129–177.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J-Y. Girard. The System F of Variables Types, Fifteen Years Later, Theoretical Computer Science, 45, pp. 159–192.Google Scholar
  11. 11.
    J-Y. Girard . Linear Logic, Theoretical Computer Science, 50, 1987, pp. 1–102. MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J-Y. Girard, Y.Lafont, and P. Taylor. Proofs and Types, Cambridge Tracts in Theoretical Computer Science, 7, Cambridge University Press, 1989.Google Scholar
  13. 13.
    G. M. Kelly. Many-variable functorial calculus.I, Coherence in Categories Springer LNM 281, pp. 66–105.Google Scholar
  14. 14.
    G. M. Kelly and S. Mac Lane. Coherence in closed categories, J. Pure Appl. Alg. 1 (1971), pp. 97–140.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Lambek. Deductive Systems and Categories I, J. Math. Systems Theory 2 (1968), pp. 278–318.MathSciNetGoogle Scholar
  16. 16.
    J. Lambek. Deductive Systems and Categories II, Springer LNM 86 (1969), pp. 76–122.MathSciNetGoogle Scholar
  17. 17.
    J. Lambek. Multicategories Revisited, Contemp. Math. 92 (1989), pp. 217–239.MathSciNetGoogle Scholar
  18. 18.
    J. Lambek . Logic without structural rules, Manuscript. (McGill University), 1990. Google Scholar
  19. 19.
    J. Lambek and P. J. Scott. Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press, 1986. Google Scholar
  20. 20.
    S. Mac Lane. Categories for the Working Mathematician, Springer Graduate Texts in Mathematics, Springer-Verlag, 1971.zbMATHGoogle Scholar
  21. 21.
    S. Mac Lane. Why commutative diagrams coincide with equivalent proofs, Contemp. Math. 13 (1982), pp. 387–401.MathSciNetzbMATHGoogle Scholar
  22. 22.
    G. E. Mints. Closed categories and the theory of proofs, J. Soviet Math 15 (1981), pp. 45–62.zbMATHCrossRefGoogle Scholar
  23. 23.
    J. C. Mitchell and P. J. Scott. Typed Lambda Models and Cartesian Closed Categories, Contemp. Math. 92, pp. 301–316.Google Scholar
  24. 24.
    J.C. Mitchell and R. Harper, The Essence of ML, Proc. 15th Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (POPL), San Diego, 1988, pp. 28–46.Google Scholar
  25. 25.
    A. Pitts . Polymorphism is set-theoretic, constructively, in: Category Theory and Computer Science, Springer LNCS 283 (D. H. Pitt, ed.) (1987) pp. 12–39. Google Scholar
  26. 26.
    D. Prawitz. Natural Deduction, Almquist & Wiksell, Stockhom, 1965.zbMATHGoogle Scholar
  27. 273.
    J.C.Reynolds . Types, Abstraction, and Parametric Polymorphism, in: Information Processing ’83, R. E. A. Mason, ed. North-Holland, 1983, pp. 513–523.Google Scholar
  28. 28.
    R. A. G. Seely. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. J. Symb. Logic 52(1987), pp. 969–989.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    R. A. G. Seely. Linear Logic, *-Autonomous Categories, And Cofree Coalgebras. Con-temp. Math. 92(1989), pp. 371–382.MathSciNetGoogle Scholar
  30. 30.
    R. Statman. Logical Relations and the Typed Lambda Calculus. Inf. and Control 65(1985), pp. 85–97.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    A. S. Troelstra and D. van Dalen. Constructivism in Mathematics, Vols. I and II. North-Holland, 1988.Google Scholar
  32. 32.
    P. Wadler. Theorems for Free! 4 th International Symposium on Functional Programming Languages and Computer Architecture, Assn. Comp. Machinery, London, Sept. 1989.Google Scholar
  33. 33.
    J. Zucker. The Correspondence between Cut-Elimination and Normalization, Ann. Math. Logic 7 (1974) pp. 1–112.MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations