Some normalization properties of martin-löf's type theory, and applications

  • David A. Basin
  • Douglas J. Howe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 526)


For certain kinds of applications of type theories, the faithfulness of formalization in the theory depends on intensional, or structural, properties of objects constructed in the theory. For type theories such as LF, such properties can be established via an analysis of normal forms and types. In type theories such as Nuprl or Martin-Löf's polymorphic type theory, which are much more expressive than LF, the underlying programming language is essentially untyped, and terms proved to be in types do not necessarily have normal forms. Nevertheless, it is possible to show that for Martin-Löf's type theory, and a large class of extensions of it, a sufficient kind of normalization property does in fact hold in certain well-behaved subtheories. Applications of our results include the use of the type theory as a logical framework in the manner of LF, and an extension of the proofs-as-programs paradigm to the synthesis of verified computer hardware. For the latter application we point out some advantages to be gained by working in a more expressive type theory.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. F. Allen. A non-type theoretic definition of Martin-Löf's types. In Proceedings of the Second Annual Symposium on Logic in Computer Science, pages 215–221. IEEE, 1987.Google Scholar
  2. [2]
    S. F. Allen. A Non-Type-Theoretic Semantics for Type-Theoretic Language. PhD thesis, Cornell University, 1987.Google Scholar
  3. [3]
    D. A. Basin. Extracting circuits from constructive proofs. In IFIP-IEEE International Workshop on Formal Methods in VLSI Design, Miami, USA, January 1991.Google Scholar
  4. [4]
    A. Camilleri, M. Gordon, and T. Melham. Hardware verification using higher-order logic. In D. Borrione, editor, From HDL Descriptions to Guaranteed Correct Circuit Designs, pages 43–67. Elsevier Science Publishers B. V. (North-Holland), 1987.Google Scholar
  5. [5]
    R. Constable and D. Howe. Nuprl as a general logic. In P. Odifreddi, editor, Logic in Computer Science, pages 77–90. Academic Press, 1990.Google Scholar
  6. [6]
    R. L. Constable, et al. Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Englewood Cliffs, New Jersey, 1986.Google Scholar
  7. [7]
    M. P. Fourman. Formal methods for modeling design. In Conference on Modeling the Innovation: Communications, Automation and Information Systems, Rome, Italy, 1990.Google Scholar
  8. [8]
    F. Hanna, N. Daeche, and M. Longley. VERITAS+: A specification language based on type theory. In Hardware Specification, Verification and Synthesis: Mathematical Aspects, Ithaca, New York, 1989. Springer-Verlag.Google Scholar
  9. [9]
    F. K. Hanna, M. Longley, and N. Daeche. Formal synthesis of digital systems. In IMEC-IFIP International Workshop on: Applied Formal Methods For Correct VLSI Design, volume 2, pages 532–548, Leuven, Belgium, 1989.Google Scholar
  10. [10]
    R. Harper, F. Honsell, and G. Plotkin. A framework for defining logics. In The Second Annual Symposium on Logic in Computer Science. IEEE, 1987.Google Scholar
  11. [11]
    D. Howe. Computational metatheory in Nuprl. Proc. of 9th International Conference on Automated Deduction, pages 238–257, 1988.Google Scholar
  12. [12]
    D. Howe. Equality in lazy computation systems. Proc. Fourth Annual Symposium on Logic in Computer Science, IEEE, pages 198–203, June 1989.Google Scholar
  13. [13]
    D. Howe. On computational open-endedness in Martin-Löf's type theory. To appear in Proc. Sixth Annual Symposium on Logic in Computer Science, IEEE.Google Scholar
  14. [14]
    P. Martin-Löf. Constructive mathematics and computer programming. In Sixth International Congress for Logic, Methodology, and Philosophy of Science, pages 153–175, Amsterdam, 1982. North Holland.Google Scholar
  15. [15]
    B. Nördstrom, K. Petersson, and J. M. Smith. Programming in Martin-Löf's Type Theory, volume 7 of International Series of Monographs on Computer Science. Oxford Science Publications, 1990.Google Scholar
  16. [16]
    L. C. Paulson. Natural deduction proof as higher-order resolution. Journal of Logic Programming, (3):237–258, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • David A. Basin
    • 1
  • Douglas J. Howe
    • 2
  1. 1.Department of Artificial IntelligenceUniversity of EdinburghEdinburghScotland
  2. 2.Department of Computer ScienceCornell UniversityIthacaU.S.A.

Personalised recommendations