Lambda calculus with constrained types

Extended abstract
  • Val Breazu-Tannen
  • Albert R. Meyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 193)


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.


Environment Model Finite Type Proof System Homomorphic Image Combinatory Model 
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  1. [Barendregt 84]
    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 83]
    Curien, P-L. Combinateurs Catégoriques, Algorithmes Séquentiels et Programmation Applicative. PhD thesis, Université Paris VII, 1983.Google Scholar
  3. [Dybjer 83]
    Dybjer, P. Category-Theoretic Logics and Algebras of Programs. PhD thesis, Chalmers University of Technology, Sweden, 1983.Google Scholar
  4. [Friedman 75]
    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, Meyer, Trakhtenbrot 84]
    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, Longo 80]
    Hindley, R., and G. Longo. Lambda-calculus models and extensionality. ZMLGM 26:289–310, 1980.Google Scholar
  7. [Koymans 82]
    Koymans, Christiaan P.J. Models of the λ-calculus. Information and Control 52:306–332, 1982.Google Scholar
  8. [Koymans 84]
    Koymans, Christiaan P.J. Models of the Lambda Calculus. PhD thesis, University of Utrecht, 1984.Google Scholar
  9. [Lambek 72]
    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 74]
    Lambek, J. Functional Completeness of Cartesian Categories. Ann. Math. Logic 6:259–292, 1974.Google Scholar
  11. [Lambek 80]
    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, Scott 84]
    Lambek, J. and P. J. Scott. Aspects of Higher-Order Categorical Logic. Contemporary Mathematics 30:145–174, 1984.Google Scholar
  13. [Meyer 82]
    Meyer, Albert R. What is a model of the lambda calculus? Information and Control 52:87–122, 1982.Google Scholar
  14. [Mitchell 84]
    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 81]
    Parsaye-Ghomi, K. Higher Order Abstract Data Types. PhD thesis, University of California, 1981.Google Scholar
  16. [Poigne 84a]
    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. [Poigne 84b]
    Poigné, A. On Specifications, Theories and Models with Higher Types. 1984. to appear Information and Control.Google Scholar
  18. [Scott 72]
    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 76]
    Scott, Dana S. Data types as lattices. SIAM J. Computing 5:522–587, 1976.Google Scholar
  20. [Scott 80]
    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, Plotkin 82]
    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 82]
    Statman, R. λ-definable functionals and β η conversion. Archiv fur Mathematische Logik und Grundlagenforschung 22:1–6, 1982.Google Scholar
  23. [Statman 84a]
    Statman, R. Equality between functionals, revisited. 1984. Friedman Volume, to appear.Google Scholar
  24. [Statman 84b]
    Statman, R. Logical relations in the typed lambda-calculus. 1984. To appear, Information and Control.Google Scholar
  25. [Wand 79]
    Wand, Mitchell. Fixed-point Constructions in Order-enriched Categories. Theoretical Computer Science:13–30, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Val Breazu-Tannen
    • 1
    • 2
  • Albert R. Meyer
    • 1
    • 2
  1. 1.Massachusetts Institute of TechnologyUSA
  2. 2.Laboratory for Computer ScienceCambridge

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