A Semantic Formulation of ⊤ ⊤-Lifting and Logical Predicates for Computational Metalanguage

  • Shin-ya Katsumata
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

Abstract

A semantic formulation of Lindley and Stark’s ⊤ ⊤-lifting is given. We first illustrate our semantic formulation of the ⊤ ⊤-lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤ ⊤-lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures.

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References

  1. 1.
    Abadi, M.: TT-closed relations and admissibility. MSCS 10(3), 313–320 (2000)MATHMathSciNetGoogle Scholar
  2. 2.
    Amadio, R., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science, vol. 46. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  3. 3.
    G.-Larrecq, J., Lasota, S., Nowak, D.: Logical relations for monadic types. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 553–568. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Girard, J.Y.: Linear logic. Theor. Comp. Sci. 50, 1–102 (1987)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hasegawa, M.: Categorical glueing and logical predicates for models of linear logic. Technical Report RIMS-1223, Research Institute for Mathematical Sciences, Kyoto University (1999)Google Scholar
  6. 6.
    Hermida, C.: Fibrations, Logical Predicates and Indeterminants. PhD thesis, University of Edinburgh (1993)Google Scholar
  7. 7.
    Jacobs, B.: Categorical Logic and Type Theory. Elsevier, Amsterdam (1999)MATHGoogle Scholar
  8. 8.
    Johann, P.: Short cut fusion is correct. J. Funct. Program. 13(4), 797–814 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jung, A., Tiuryn, J.: A new characterization of lambda definability. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 245–257. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  10. 10.
    Kock, A.: Strong functors and monoidal monads. Archiv der Mathematik 23, 113–120 (1970)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lindley, S.: Normalisation by Evaluation in the Compilation of Typed Functional Programming Languages. PhD thesis, University of Edinburgh (2004)Google Scholar
  12. 12.
    Lindley, S., Stark, I.: Reducibility and TT-lifting for computation types. In: TLCA, pp. 262–277 (2005)Google Scholar
  13. 13.
    Ma, Q., Reynolds, J.: Types, abstractions, and parametric polymorphism, part 2. In: Schmidt, D., Main, M.G., Melton, A.C., Mislove, M.W., Brookes, S.D. (eds.) MFPS 1991. LNCS, vol. 598, pp. 1–40. Springer, Heidelberg (1992)Google Scholar
  14. 14.
    MacLane, S.: Categories for theWorking Mathematician, 2nd edn. Graduate Texts in Mathematics, vol. 5. Springer, Heidelberg (1998)Google Scholar
  15. 15.
    Melliès, P.-A., Vouillon, J.: Recursive polymorphic types and parametricity in an operational framework. In: Proc. LICS 2005 (2005) (to appear)Google Scholar
  16. 16.
    Mitchell, J.: Representation independence and data abstraction. In: Proc. POPL, pp. 263–276 (1986)Google Scholar
  17. 17.
    Mitchell, J., Scedrov, A.: Notes on sconing and relators. In: Martini, S., Börger, E., Kleine Büning, H., Jäger, G., Richter, M.M. (eds.) CSL 1992. LNCS, vol. 702, pp. 352–378. Springer, Heidelberg (1993)Google Scholar
  18. 18.
    Moggi, E.: Notions of computation and monads. Information and Computation 93(1), 55–92 (1991)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Nishimura, S.: Correctness of a higher-order removal transformation through a relational reasoning. In: Ohori, A. (ed.) APLAS 2003. LNCS, vol. 2895, pp. 358–375. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Parigot, M.: Proofs of strong normalisation for second order classical natural deduction. Journal of Symbolic Logic 62(4), 1461–1479 (1997)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pitts, A.: Parametric polymorphism and operational equivalence. Mathematical Structures in Computer Science 10(3), 321–359 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Pitts, A., Stark, I.: Operational reasoning for functions with local state. In: Gordon, A.D., Pitts, A.M. (eds.) Higher Order Operational Techniques in Semantics, Publications of the Newton Institute, pp. 227–273. Cambridge University Press, Cambridge (1998)Google Scholar
  23. 23.
    Plotkin, G.: Lambda-definability in the full type hierarchy. In: To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 367–373. Academic Press, San Diego (1980)Google Scholar
  24. 24.
    Tait, W.: Intensional interpretation of functionals of finite type I. Journal of Symbolic Logic 32 (1967)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Shin-ya Katsumata
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto University 

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