Skip to main content
SpringerLink
Log in

Lambda calculus with constrained types

Extended abstract

  • Conference paper
  • First Online:
Logics of Programs (Logic of Programs 1985)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 193))

Included in the following conference series:

Abstract

Motivated by domain equations, we consider types satisfying arbitrary equational constraints thus generalizing a range of situations with the finitely typed case at one extreme and the type-free case at the other. The abstract model theory of the β η type-free case is generalized. We investigate the relation between lambda calculus with constrained types and cartesian closed categories (cccs) at proof-theoretic and model-theoretic levels. We find an adjoint equivalence between the category of typed λ-algebras and that of cccs. The subcategories of typed λ-models and concrete cccs correspond to each other under this equivalence. All these results are parameterized by an arbitrary set of higher-order constants and an arbitrary set of higher-order equations.

This research was supported in part by NSF Grant MCS80-10707

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barendregt, Henk P. Studies in Logic. Volume 103: The Lambda Calculus: Its Syntax and Semantics. North-Holland, 1984. 2nd & revised edition.

    Google Scholar 

  2. Curien, P-L. Combinateurs Catégoriques, Algorithmes Séquentiels et Programmation Applicative. PhD thesis, Université Paris VII, 1983.

    Google Scholar 

  3. Dybjer, P. Category-Theoretic Logics and Algebras of Programs. PhD thesis, Chalmers University of Technology, Sweden, 1983.

    Google Scholar 

  4. Friedman, H. Equality between functionals. In R. Parikh (editor), LNMath. Volume 453: Logic Colloqium, 73, pages 22–37. Springer-Verlag, 1975.

    Google Scholar 

  5. Halpern, Joseph Y., Albert R. Meyer, and Boris Trakhtenbrot. The semantics of local storage, or what makes the free-list free? In 11th ACM Symp. on Principles of Programming Languages, pages 245–257., 1984.

    Google Scholar 

  6. Hindley, R., and G. Longo. Lambda-calculus models and extensionality. ZMLGM 26:289–310, 1980.

    Google Scholar 

  7. Koymans, Christiaan P.J. Models of the λ-calculus. Information and Control 52:306–332, 1982.

    Google Scholar 

  8. Koymans, Christiaan P.J. Models of the Lambda Calculus. PhD thesis, University of Utrecht, 1984.

    Google Scholar 

  9. Lambek, J. Deductive Systems and Categories III. In Lawvere, F.W. (editor), Lecture Notes in Mathematics. Volume 274: Toposes, Algebraic Geometry and Logic; Proc. 1971 Dahlhousie Conf., pages 57–82. Springer-Verlag, 1972.

    Google Scholar 

  10. Lambek, J. Functional Completeness of Cartesian Categories. Ann. Math. Logic 6:259–292, 1974.

    Google Scholar 

  11. Lambek, J. From lambda calculus to cartesian closed categories. In Seldin, J.P. and J.R. Hindley (editors), To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 375–402. Academic Press, 1980.

    Google Scholar 

  12. Lambek, J. and P. J. Scott. Aspects of Higher-Order Categorical Logic. Contemporary Mathematics 30:145–174, 1984.

    Google Scholar 

  13. Meyer, Albert R. What is a model of the lambda calculus? Information and Control 52:87–122, 1982.

    Google Scholar 

  14. Mitchell, John C. Semantic models for second-order lambda calculus. In Proc. 25 th IEEE Symp. on Foundations of Computer Science, pages 289–299., 1984.

    Google Scholar 

  15. Parsaye-Ghomi, K. Higher Order Abstract Data Types. PhD thesis, University of California, 1981.

    Google Scholar 

  16. Poigné, A. Higher-Order data Structures-Cartesian Closure Versus λ-Calculus. In LNCS. Volume 166: Symposium of Theoretical Aspects of Computer Science, Proceedings, pages 174–185. Springer-Verlag, 1984.

    Google Scholar 

  17. Poigné, A. On Specifications, Theories and Models with Higher Types. 1984. to appear Information and Control.

    Google Scholar 

  18. Scott, Dana S. Continuous Lattices. In Lawvere, F.W. (editor), Lecture Notes in Mathematics. Volume 274: Toposes, Algebraic Geometry and Logic; Proc. 1971 Dahlhousie Conf., pages 97–136. Springer-Verlag, 1972.

    Google Scholar 

  19. Scott, Dana S. Data types as lattices. SIAM J. Computing 5:522–587, 1976.

    Google Scholar 

  20. Scott, Dana S. Relating theories of the lambda calculus. In Seldin, J.P. and J.R. Hindley (editors), To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 403–450. Academic Press, 1980.

    Google Scholar 

  21. Smyth, M.B., and Gordon D. Plotkin. The category-theoretic solution of recursive domain equations. SIAM J. Computing 11:761–783, 1982.

    Google Scholar 

  22. Statman, R. λ-definable functionals and β η conversion. Archiv fur Mathematische Logik und Grundlagenforschung 22:1–6, 1982.

    Google Scholar 

  23. Statman, R. Equality between functionals, revisited. 1984. Friedman Volume, to appear.

    Google Scholar 

  24. Statman, R. Logical relations in the typed lambda-calculus. 1984. To appear, Information and Control.

    Google Scholar 

  25. Wand, Mitchell. Fixed-point Constructions in Order-enriched Categories. Theoretical Computer Science:13–30, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, USA

    Val Breazu-Tannen & Albert R. Meyer

  2. Laboratory for Computer Science, 545 Technology Sq., 02139, Cambridge, MA

    Val Breazu-Tannen & Albert R. Meyer

Authors
  1. Val Breazu-Tannen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert R. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rohit Parikh

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Breazu-Tannen, V., Meyer, A.R. (1985). Lambda calculus with constrained types. In: Parikh, R. (eds) Logics of Programs. Logic of Programs 1985. Lecture Notes in Computer Science, vol 193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15648-8_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-15648-8_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15648-2

  • Online ISBN: 978-3-540-39527-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation