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Lax Logical Relations

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Automata, Languages and Programming (ICALP 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

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Abstract

Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambda-calculus terms.We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the pre-logical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.

This author was supported by EPSRC grants GR/J84205 and GR/M56333.

This author was supported by EPSRC grants GR/J84205 and GR/M56333, and by a grant from the British Council.

This author was supported by EPSRC grant GR/K63795.

This author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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References

  1. S. Abramsky. Abstract interpretation, logical relations and Kan extensions. J. of Logic and Computation, 1:5–40, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146:5–23, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Blackwell, H. M. Kelly, and A. J. Power. Two dimensional monad theory. J. of Pure and Applied Algebra, 59:1–41, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  4. Francis Borceux. Handbook of Categorical Algebra 2, volume 51 of Encyclopedia of Mathematics and its Applications.Cambridge University Press, 1994.

    Google Scholar 

  5. P.-L. Curien. Categorical Combinators, Sequential Algorithms, and Functional Programming. Birkhauser, Boston, 1993.

    MATH  Google Scholar 

  6. J. Flum and M. Rodriguez-Artalejo, editors. Computer Science Logic, 13th International Workshop, CSL’99, volume 1683 of Lecture Notes in Computer Science, Madrid, Spain, September 1999. Springer-Verlag, Berlin (1999).

    MATH  Google Scholar 

  7. A. Ginzburg. Algebraic Theory of Automata. Academic Press, 1968.

    Google Scholar 

  8. Claudio A. Hermida. Fibrations, logical predicates, and indeterminates. Ph.D. thesis, The University of Edinburgh, 1993. Available as Computer Science Report CST-103-93 or ECS-LFCS-93-277.

    Google Scholar 

  9. [HL+]_F. Honsell, J. Longley, D. Sannella, and A. Tarlecki. Constructive data refinement in typed lambda calculus. To appear in the Proceedings of FOSSACS 2000, Springer-Verlag Lecture Notes in Computer Science.

    Google Scholar 

  10. F. Honsell and D. Sannella. Pre-logical relations. In M. Rodriguez-Artalejo, editors. Computer Science Logic, 13th International Workshop, CSL’99, volume 1683 of Lecture Notes in Computer Science, Madrid, Spain, September 1999. Springer-Verlag, Berlin (1999). Flum and Rodriguez-Artalejo [FRA99], pages 546–561.

    Google Scholar 

  11. He Jifeng and C. A. R. Hoare. Data refinement in a categorical setting. Technical monograph PRG-90, Oxford University Computing Laboratory, Programming Research Group, Oxford, November 1990.

    Google Scholar 

  12. A. Jung and J. Tiuryn. A new characterization of lambda definability. In M. Bezen and J. F. Groote, editors, Typed Lambda Calculi and Applications, volume 664 of Lecture Notes in Computer Science, pages 245–257, Utrecht, The Netherlands, March 1993. Springer-Verlag, Berlin.

    Chapter  Google Scholar 

  13. Y. Kinoshita, P. O’Hearn, A. J. Power, M. Takeyama, and R. D. Tennent. An axiomatic approach to binary logical relations with applications to data refinement. In M. Abadi and T. Ito, editors, Theoretical Aspects of Computer Software, volume 1281 of Lecture Notes in Computer Science, pages 191–212, Sendai, Japan, 1997. Springer-Verlag, Berlin.

    Chapter  Google Scholar 

  14. Y. Kinoshita and A. J. Power. Data refinement by enrichment of algebraic structure. To appear in Acta Informatica.

    Google Scholar 

  15. G. M. Kelly and A. J. Power. Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads. Journal of Pure and Applied Algebra, 89:163–179, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Kinoshita and A. J. Power. Lax naturality through enrichment. J. Pure and Applied Algebra, 112:53–72, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Kinoshita and J. Power. Data refinement for call-by-value programming languages. In M. Rodriguez-Artalejo, editors. Computer Science Logic, 13th International Workshop, CSL’99, volume 1683 of Lecture Notes in Computer Science, Madrid, Spain, September 1999. Springer-Verlag, Berlin (1999). Flum and Rodriguez-Artalejo [FRA99], pages 562–576.

    Google Scholar 

  18. Y. Lafont. Logiques, Categories et Machines. Thése de Doctorat, Université de Paris VII, 1988.

    Google Scholar 

  19. R. Milner. An algebraic definition of simulation between programs. In Proceedings of the Second International Joint Conference on Artificial Intelligence, pages 481–489. The British Computer Society, London, 1971. Also Technical Report CS-205, Computer Science Department, Stanford University, February 1971.

    Google Scholar 

  20. J. C. Mitchell. Type systems for programming languages. InJ. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 365–458. Elsevier, Amsterdam, and The MIT Press, Cambridge, Mass., 1990.

    Google Scholar 

  21. J. C. Mitchell. On the equivalence of data representations. In V. Lifschitz, editor, Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy, pages 305–330. Academic Press, 1991.

    Google Scholar 

  22. J. C. Mitchell. Foundations for Programming Languages. The MIT Press, 1996.

    Google Scholar 

  23. Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55–92, July 1991.

    Google Scholar 

  24. QingMing Ma and J. C. Reynolds. Types, abstraction, and parametric polymorphism, part 2. In S. Brookes, M. Main, A. Melton, M. Mislove, and D. Schmidt, editors, Mathematical Foundations of Programming Semantics, Proceedings of the 7th International Conference, volume 598 of Lecture Notes in Computer Science, pages 1–40, Pittsburgh, PA, March 1991. Springer-Verlag, Berlin (1992).

    Google Scholar 

  25. R. E. Milne and C. Strachey. A Theory of Programming Language Semantics. Chapman and Hall London, and Wiley, New York, 1976.

    MATH  Google Scholar 

  26. J. C. Mitchell and A. Scedrov. Notes on sconing and relators. In E. Börger, G. Jager, H. Kleine Büning, S. Martini, and M. M. Richter, editors, Computer Science Logic: 6th Workshop, CSL’ 92: Selected Papers, volume 702 of Lecture Notes in Computer Science, pages 352–378, San Miniato, Italy, 1992. Springer-Verlag, Berlin (1993).

    Google Scholar 

  27. P. O’Hearn and J. Riecke. Kripke logical relations and PCF. Information and Computation, 120(1):107–116, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  28. P. W. O’Hearn and R. D. Tennent. Parametricity and local variables. J. ACM, 42(3):658–709, May 1995.

    Google Scholar 

  29. G. D. Plotkin. Lambda-definability and logical relations. Memorandum SAIRM-4, School of Artificial Intelligence, University of Edinburgh, October 1973.

    Google Scholar 

  30. G. D. Plotkin. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 363–373. Academic Press, 1980.

    Google Scholar 

  31. A. J. Power. Categories with algebraic structure. In M. Nielsen and W. Thomas, editors, Computer Science Logic, 11th International Workshop, CSL’99, volume 1414 of Lecture Notes in Computer Science, pages 389–405, Aarhus, Denmark, August 1997. Springer-Verlag, Berlin (1998).

    Google Scholar 

  32. J. C. Reynolds. On the relation between direct and continuation semantics. InJ. Loeckx, editor, Proc. 2nd Int. Colloq. on Automata, Languages and Programming, volume 14 of Lecture Notes in Computer Science, pages 141–156. Springer-Verlag, Berlin, 1974.

    Google Scholar 

  33. J. C. Reynolds. Types, abstraction and parametric polymorphism. In R. E. A. Mason, editor, Information Processing 83, pages 513–523, Paris, France, 1983. North-Holland, Amsterdam.

    Google Scholar 

  34. O. Schoett. Data abstraction and the correctness of modular programming. Ph.D. thesis, University of Edinburgh, February1987. Report CST-42-87.

    Google Scholar 

  35. I. Stark. Categorical models for local names. Lisp and Symbolic Computation, 9(1):77–107, February 1996.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory for Foundations of Computer Science, University of Edinburgh, Edinburgh, UK, EH9 3JZ

    Gordon Plotkin, John Power, Donald Sannella & Robert Tennent

  2. Department of Computing and Information Science, Queen’s University, Kingston, Canada, K7L 3N6

    Robert Tennent

Authors
  1. Gordon Plotkin
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  2. John Power
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  3. Donald Sannella
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  4. Robert Tennent
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

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Plotkin, G., Power, J., Sannella, D., Tennent, R. (2000). Lax Logical Relations. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_9

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  • DOI: https://doi.org/10.1007/3-540-45022-X_9

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  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

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