Skip to main content

Completeness Theorems and λ-Calculus

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3461))

Abstract

The purpose of this note is to present a variation of Hindley’s completeness theorem for simply typed λ-calculus based on Kripke model. This variation was obtained indirectly by simplifying an analysis of a fragment of polymorphic λ-calculus [2].

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

Buying options

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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aehlig, K.: On Fragments of Analysis with Strengths of Finitely Iterated Inductive Definitions. PhD thesis, Munich (2003)

    Google Scholar 

  2. Altenkirch, T., Coquand, T.: A finitary subsystem of the polymorphic λ-calculus. In: Abramsky, S. (ed.) TLCA 2001. LNCS, vol. 2044, pp. 22–28. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  3. Buchholz, W., Feferman, S., Pohlers, W., Sieg, W.: Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies. Lecture Notes in Mathematics, vol. 897. Springer, Berlin (1981)

    MATH  Google Scholar 

  4. Buchholz, W., Schütte, K.: Proof theory of impredicative subsystems of analysis. In: Studies in Proof Theory, Monographs, Bibliopolis, Naples, vol. 2 (1988)

    Google Scholar 

  5. Gödel, K.: Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse einers math. Koll., Heft 4, 34–38 (1933)

    Google Scholar 

  6. Hindley, R.: The completeness theorem for typing λ-terms. Theoret. Comput. Sci. 22(1-2), 1–17 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hindley, R.: Basic simple type theory. In: Cambridge Tracts in Theoretical Computer Science, vol. 42. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  8. Lorenzen, P.: Logical reflection and formalism. J. Symb. Logic 23, 241–249 (1958)

    Article  MathSciNet  Google Scholar 

  9. Lorenzen, P.: Algebraische und logistische Untersuchungen über freie Verbände. J. Symbolic Logic 16, 81–106 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  10. Martin-Löf, P.: Notes on constructive mathematics. In: Almquist and Wixsekk, Stockholm (1968)

    Google Scholar 

  11. Martin-Löf, P.: Infinite terms and a system of natural deduction. Compositio Math. 24, 93–103 (1972)

    MATH  MathSciNet  Google Scholar 

  12. Martin-Löf, P.: A construction of the provable wellorderings of the theory of species. In: Logic, meaning and computation. Synthese Lib., vol. 305, pp. 343–351. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  13. Novikov, P.: On the consistency of a certain logical calculus. Matematicesky sbovnik 12(3), 353–369 (1943)

    Google Scholar 

  14. Pitts, A.: On an interpretation of second-order quantification in first-order intuitionistic propositional logic. J. Symbolic Logic 57(1), 33–52 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Poincaré, H.: La logique de l’infini. In: Revue de metaphysique et de morale (1909)

    Google Scholar 

  16. Russell, B., Whitehead, A.: Principia Mathematica, Cambridge, pp. 1910–1913

    Google Scholar 

  17. Takeuti, G.: On the fundamental conjecture of GLC. I. J. Math. Soc. Japan 7, 249–275 (1955)

    Article  MathSciNet  Google Scholar 

  18. Takeuti, G.: Consistency proofs of subsystems of classical analysis. Ann. of Math. 86(2), 299–348 (1967)

    Article  MathSciNet  Google Scholar 

  19. Takeuti, I.: Proof of calculability through cut elimination. In: Proof theory and reverse mathematics, Kyoto, pp. 78–93 (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Coquand, T. (2005). Completeness Theorems and λ-Calculus. In: Urzyczyn, P. (eds) Typed Lambda Calculi and Applications. TLCA 2005. Lecture Notes in Computer Science, vol 3461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11417170_1

Download citation

  • DOI: https://doi.org/10.1007/11417170_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25593-2

  • Online ISBN: 978-3-540-32014-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics