Skip to main content
SpringerLink
Log in

Infinite Intersection and Union Types for the Lazy Lambda Calculus

  • Conference paper
  • First Online:
Theoretical Aspects of Computer Software (TACS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2215))

Included in the following conference series:

  • 264 Accesses

Abstract

A type theory with infinitary intersection and union types for the lazy λ-calculus is introduced. Types are viewed as upper closed subsets of a Scott domain. Intersection and union type constructors are interpreted as the set-theoretic intersection and union, respectively, even when they are not finite. The assignment of types to λ-terms extends naturally the basic type assignment system. We prove soundness and completeness using a generalization of Abramsky’s finitary domain logic for applicative transition systems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Abramsky. The lazy lambda calculus. In Research Topics in Functional Programming, pp. 65–117, Addison Wesley, 1990.

    Google Scholar 

  2. S. Abramsky. Domain theory in logical form. Annals of Pure and Applied Logic, 51(1):1–77, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Barbanera and M. Dezani-Ciancaglini. Intersection and union types. In Proceedings of TACS’91, vol. 526 of Lecture Notes in Computer Science, pp. 651–674. Springer-Verlag, 1991.

    Google Scholar 

  4. F. Barbanera, M. Dezani-Ciancaglini, and U. dé Liguoro. Intersection and union types: syntax and semantics. Information and Computation, 119:202–230, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North Holland, 1984.

    Google Scholar 

  6. H. Barendregt, M. Coppo, and M. Dezani-Ciancaglini. A filter lambda model and the completeness of type assignment. Journal of Symbolic Logic 48(4):931–940, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  7. M.M. Bonsangue, B. Jacobs, and J.N. Kok. Duality beyond sober spaces: topological spaces and observation frames. Theoretical Computer Science, 151(1):79–124, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  8. M.M. Bonsangue. Topological Dualities in Semantics. Vol. 8 of Electronic Notes in Theoretical Computer Science. Elsevier Science, 1997.

    Google Scholar 

  9. M.M. Bonsangue and J.N. Kok. Towards an infinitary logic of domains: Abramsky logic for transition systems. Information and Computation, 155:170–201, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Coppo and M. Dezani-Ciancaglini. A new type-assignment for λ-term. In Archiv. Math. Logik, 19:139–156, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Dezani-Ciancaglini, F. Honsell and Y. Motohama. Compositional characterizations of λ-terms using intersection types. In Proceedings of MFCS 2000, vol. 1893 of Lecture Notes in Computer Science, pp. 304–313. Springer-Verlag, 2000.

    Google Scholar 

  12. K.H. Hofmann and M.W. Mislove. Local compactness and continuous lattices. In Continuous lattices-Proceedings Bremen 1979, vol. 871 of Lecture Notes in Mathematics, pp. 209–248. Springer-Verlag, 1981.

    Article  Google Scholar 

  13. P.T. Johnstone. Stone Spaces. Cambridge University Press, 1982.

    Google Scholar 

  14. D. Leivant. Discrete polymorphism. In Proceedings of 1990 ACM Conference of LISP and Functional Programming, pp. 288–297.

    Google Scholar 

  15. B.C. Pierce. Programming with intersection types, union types, and polymorphism. Technical Report CMU-CS-91-106, Carnegie Mellon University, 1991.

    Google Scholar 

  16. G. Pottinger. A type assignment for the strongly normalizable λ-term. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, pp. 561–577. Academic Press, 1980.

    Google Scholar 

  17. G. Raney. Completely distributive complete lattices. In Proceedings of the American Mathematical Society vol. 3(4), pp. 677–680, 1952.

    Article  MATH  MathSciNet  Google Scholar 

  18. D.S. Scott Domains for denotational semantics. In M. Nielson and E.M. Schmidt (eds.), International Colloquioum on Automata, Languages, and Programs, volume 140 of Lecture Notes in Computer Science, pp 577–613, Springer-Verlag, 1982.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Leiden Institute of Advanced Computer Science, Leiden University, Leiden, The Netherlands

    Marcello M. Bonsangue & Joost N. Kok

Authors
  1. Marcello M. Bonsangue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joost N. Kok
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Tokyo Institute of Technology Graduate School of Information Science and Engineering, 2-12-1 Oo-okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Naoki Kobayashi

  2. Department of Computer and Information Science, University of Pennsylvania, 200 S. 33rd St., Philadelphia, PA, 19104, USA

    Benjamin C. Pierce

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bonsangue, M.M., Kok, J.N. (2001). Infinite Intersection and Union Types for the Lazy Lambda Calculus. In: Kobayashi, N., Pierce, B.C. (eds) Theoretical Aspects of Computer Software. TACS 2001. Lecture Notes in Computer Science, vol 2215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45500-0_22

Download citation

  • DOI: https://doi.org/10.1007/3-540-45500-0_22

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42736-0

  • Online ISBN: 978-3-540-45500-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation