# Lax Logical Relations

Conference paper

First Online:

## 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.

## Keywords

Binary Relation Algebraic Structure Logical Relation Small Category Preserve Functor
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.

## References

- [Ab90]S. Abramsky. Abstract interpretation, logical relations and Kan extensions.
*J. of Logic and Computation*, 1:5–40, 1990.zbMATHCrossRefMathSciNetGoogle Scholar - [Al95]M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism.
*Theoretical Computer Science*, 146:5–23, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - [BKP89]R. Blackwell, H. M. Kelly, and A. J. Power. Two dimensional monad theory.
*J. of Pure and Applied Algebra*, 59:1–41, 1989.zbMATHCrossRefMathSciNetGoogle Scholar - [Bo94]Francis Borceux.
*Handbook of Categorical Algebra 2*, volume 51 of*Encyclopedia of Mathematics and its Applications*.Cambridge University Press, 1994.Google Scholar - [Cu93]P.-L. Curien.
*Categorical Combinators, Sequential Algorithms, and Functional Programming*. Birkhauser, Boston, 1993.zbMATHGoogle Scholar - [FRA99]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).zbMATHGoogle Scholar - [Gi68]A. Ginzburg.
*Algebraic Theory of Automata*. Academic Press, 1968.Google Scholar - [He93]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 - [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 - [HS99]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 - [JH90]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
- [JT93]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.CrossRefGoogle Scholar - 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.CrossRefGoogle Scholar - [KP]Y. Kinoshita and A. J. Power. Data refinement by enrichment of algebraic structure. To appear in
*Acta Informatica*.Google Scholar - [KP93]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.zbMATHCrossRefMathSciNetGoogle Scholar - [KP96]Y. Kinoshita and A. J. Power. Lax naturality through enrichment.
*J. Pure and Applied Algebra*, 112:53–72, 1996.zbMATHCrossRefMathSciNetGoogle Scholar - [KP99]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 - [La88]Y. Lafont.
*Logiques, Categories et Machines*. Thése de Doctorat, Université de Paris VII, 1988.Google Scholar - [Mi71]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 - [Mi90]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 - [Mi91]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 - [Mi96]J. C. Mitchell.
*Foundations for Programming Languages*. The MIT Press, 1996.Google Scholar - [Mo91]Eugenio Moggi. Notions of computation and monads.
*Information and Computation*, 93(1):55–92, July 1991.Google Scholar - [MR91]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 - [MS76]R. E. Milne and C. Strachey.
*A Theory of Programming Language Semantics*. Chapman and Hall London, and Wiley, New York, 1976.zbMATHGoogle Scholar - [MS92]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 - [OR95]P. O’Hearn and J. Riecke. Kripke logical relations and PCF.
*Information and Computation*, 120(1):107–116, 1995.zbMATHCrossRefMathSciNetGoogle Scholar - [OT95]P. W. O’Hearn and R. D. Tennent. Parametricity and local variables.
*J. ACM*, 42(3):658–709, May 1995.Google Scholar - [Pl73]G. D. Plotkin. Lambda-definability and logical relations. Memorandum SAIRM-4, School of Artificial Intelligence, University of Edinburgh, October 1973.Google Scholar
- [Pl80]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 - [Po97]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 - [Re74]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 - [Re83]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 - [Sc87]O. Schoett.
*Data abstraction and the correctness of modular programming*. Ph.D. thesis, University of Edinburgh, February1987. Report CST-42-87.Google Scholar - [St96]I. Stark. Categorical models for local names. Lisp
*and Symbolic Computation*, 9(1):77–107, February 1996.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2000