Second-Order Pre-logical Relations and Representation Independence

  • Hans Leiβ
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)


We extend the notion of pre-logical relation between models of simply typed lambda-calculus, recently introduced by F. Honsell and D. Sannella, to models of second-order lambda calculus. With pre-logical relations, we obtain characterizations of the lambda-definable elements of and the observational equivalence between second-order models. These are are simpler than those using logical relations on extended models.

We also characterize representation independence for abstract data types and abstract data type constructors by the existence of a pre-logical relation between the representations, thereby varying and generalizing results of J.C. Mitchell to languages with higher-order constants.


Logical Relation Logical Predicate Type Constructor Basic Lemma 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BMM90]
    Kim B. Bruce, Albert R. Meyer, and John C. Mitchell, The semantics of second-order lambda calculus, Information and Computation 85 (1990), 76–134.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Han99]
    Jo Erskine Hannay, Specification refinement with system F, In Proc. CSL’99, LNCS 1683, Springer Verlag, 1999, pp. 530–545.Google Scholar
  3. [HLST00]
    Furio Honsell, John Longley, Donald Sannella, and Andrzej Tarlecki, Constructive data refinement in typed lambda calculus, 3rd Intl. Conf. on Foundations of Software Science and Computation Structures. European Joint Conferences on Theory and Practice of Software (ETAPS’2000), LNCS 1784, Springer Verlag, 2000, pp. 149–164.Google Scholar
  4. [HS99]
    Furio Honsell and Donald Sannella, Pre-logical relations, Proc. Computer Science Logic, CSL’99, LNCS 1683, Springer Verlag, 1999, pp. 546–561.CrossRefGoogle Scholar
  5. [Ler95]
    Xavier Leroy, Applicative functors and fully transparent higher-order modules, Proc. of the 22nd Annual ACM Symposium on Principles of Programming Languages, ACM, 1995, pp. 142–153.Google Scholar
  6. [Mit86]
    John C. Mitchell, Representation independence and data abstraction, Proceedings of the 13th ACM Symposium on Principles of Programming Languages, January 1986, pp. 263–276.Google Scholar
  7. [Mit91]
    John C. Mitchell, On the equivalence of data representations, Artificial Intelligence and Mathematical Theory of Computation: Papers in Honour of John C. McCarthy (V. Lifschitz, ed.), Academic Press, 1991, pp. 305–330.Google Scholar
  8. [Mit96]
    John C. Mitchell, Foundations for programming languages, The MIT Press, Cambridge, Mass., 1996.Google Scholar
  9. [MM85]
    John C. Mitchell and Albert Meyer, Second-order logical relations, Proc. Logics of Programs, LNCS 193, Springer Verlag, 1985, pp. 225–236.Google Scholar
  10. [MP85]
    John C. Mitchell and Gordon D. Plotkin, Abstract types have existential type, 12-th ACM Symposium on Principles of Programming Languages, 1985, pp. 37–51.Google Scholar
  11. [MTHM97]
    Robin Milner, Mads Tofte, Robert Harper, and David MacQueen, The definition of Standard ML (revised), The MIT Press, Cambridge, MA, 1997.Google Scholar
  12. [Plo80]
    Gordon D. Plotkin, Lambda definability in the full type hierarchy, To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, 1980, pp. 363–373.Google Scholar
  13. [PPST00]
    Gordon Plotkin, John Power, Don Sannella, and Robert Tennent, Lax logical relations, ICALP 2000, Springer LNCS 1853, 2000, pp. 85–102.Google Scholar
  14. [PR00]
    John Power and Edmund Robinson, Logical relations and data abstraction, Proc. Computer Science Logic, CSL 2000, LNCS 1862, Springer-Verlag, 2000, pp. 497–511.Google Scholar
  15. [Sta85]
    R. Statman, Logical relations and the typed lambda calculus, Information and Control 65 (1985), 85–97.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Tai67]
    W.W. Tait, Intensional interpretation of functionals of finite type, Journal of Symbolic Logic 32 (1967), 198–212.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Hans Leiβ
    • 1
  1. 1.Centrum für Informations- und SprachverarbeitungUniversität MünchenMünchenGermany

Personalised recommendations