Skip to main content
SpringerLink
Log in

Inductive definitions and type theory an introduction (preliminary version)

  • Invited Talk
  • Conference paper
  • First Online:
Book cover Foundation of Software Technology and Theoretical Computer Science (FSTTCS 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 880))

Abstract

We give a short introduction to Martin-Löf's Type Theory, seen as a theory of inductive definitions. The first part contains historical remarks that motivate this approach. The second part presents a computational semantics, which explains how proof trees can be represented using the notations of functional programming.

This research has been done within the ESPRIT Basic Research Action “Types for Proofs and Programs”. It has been paid by NUTEK, Chalmers and the University of Göteborg.

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. P. Aczel. An introduction to inductive definitions. In J. Barwise, editor, Handbook of Mathematical Logic, pages 739–782. North-Holland, 1977.

    Google Scholar 

  2. S. Allen. A Non-Type-Theoretic Semantics for Type-Theoretic Language. PhD thesis, Department of Computer Science, Cornell University, 1987.

    Google Scholar 

  3. H. P. Barendregt. The Lambda Calculus. North-Holland, 1984. Revised edition.

    Google Scholar 

  4. M. Beeson. Foundations of Constructive Mathematics. Springer-Verlag, 1985.

    Google Scholar 

  5. F. S. P. W. Buchholz, W. and S. W. Iterated Inductive Definitions and Subsystems of Analysis. Springer, Berlin, 1981.

    Google Scholar 

  6. R. M. Burstall. Proving properties of programs by structural induction. Computer Journal, (39): 135–154, 1985.

    Google Scholar 

  7. T. Coquand. Infinite objects in type theory. In Proceedings of the 1993 TYPES Workshop, Nijmegen, LNCS 806, 1993.

    Google Scholar 

  8. J. Despeyroux. Proof of translation in natural semantics. In Proceedings of the First ACM Conference on Logic in Computer Science, pages 193–205, 1986.

    Google Scholar 

  9. E. W. Dijkstra. A Discipline of Programming. Prentice-Hall, 1976.

    Google Scholar 

  10. P. Dybjer. Inductive families. Formal Aspects of Computing. To appear.

    Google Scholar 

  11. K. Gödel. Collected Works, Volumes I and II. Oxford University Press, 1986.

    Google Scholar 

  12. J. A. Goguen, J. W. Thatcher, E. G. Wagner, and J. B. Wright. An Initial Algebra Approach to the Specification, Correctness, and Implementation of Abstract Data Types. Prentice Hall, 1978.

    Google Scholar 

  13. D. Gries. The Science of Programming. Springer Texts and Monographs in Computer Science, 1981.

    Google Scholar 

  14. M. Hagiya and T. Sakurai. Foundation of logic programming based on inductive definition. New Generation Computing, 2: 59–77, 1984.

    Google Scholar 

  15. L. Hallnäs. Partial inductive definitions. Theoretical Computer Science, 53: 335–343, 1991.

    Google Scholar 

  16. J. Herbrand. On the consistency of arithmetic. In J. van Heijenoort, editor, From Frege to Gödel, pages 618–628. Harvard University Press.

    Google Scholar 

  17. C. Hoare. Recursive data structures. International Journal of Computer and Information Sciences, 4: 105–124, 1975.

    Google Scholar 

  18. W. Howard. Functional interpretation of bar induction by bar recursion. Compositio Mathematica, 20:107–124.

    Google Scholar 

  19. G. Kahn. Natural semantics. Technical Report 601, 1987.

    Google Scholar 

  20. S. Kleene. On the interpretation of intuitionistic number theory. Journal of Symbolic Logic, 10: 109–124, 1945.

    Google Scholar 

  21. G. Kreisel. Interpretation of analysis by mean of constructive functionals of finite type. In Heyting, editor, Constructivity in Mathematics, pages 101–128. North-Holland, 1959.

    Google Scholar 

  22. F. W. Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, editor, Applications of Categorical Algebra, Proceedings of Symposia in Pure Mathematics. AMS, 1970.

    Google Scholar 

  23. K. Lorenzen. Métamathématique. Edition Gauthier-Villars, 1962.

    Google Scholar 

  24. P. Martin-Löf. Notes on Constructive Mathematics. Almqvist & Wiksell, Stockholm, 1968.

    Google Scholar 

  25. P. Martin-Löf. Hauptsatz for the intuitionistic theory of iterated inductive definitions. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, pages 179–216. North-Holland, 1971.

    Google Scholar 

  26. P. Martin-Löf. An intuitionistic theory of types: Predicative part. In Logic Colloquium'73, pages 73–118. North-Holland, 1975.

    Google Scholar 

  27. P. Martin-Löf. The domain interpretation of type theory, lecture notes. In K. Karlsson and K. Petersson, editors, Workshop on Semantics of Programming Languages, Abstracts and Notes, Chalmers University of Technology and University of Göteborg, August 1983. Programming Methodology Group.

    Google Scholar 

  28. P. Martin-Löf. Intuitionistic Type Theory. Bibliopolis, 1984.

    Google Scholar 

  29. J. McCarthy and J. A. Painter. Correctness of a compiler for arithmetic expressions. In Mathematical Aspects of Computer Science, pages 33–41. AMS, 1967.

    Google Scholar 

  30. D. R. Musser. On proving inductive properties of abstract data types. POPL, pages 154–162, 1980.

    Google Scholar 

  31. C. Paulin-Mohring. Extracting fω programs from proofs in the calculus of constructions. In Sixteenth Annual ACM Symposium on Principles of Programing Languages. ACM, 1989.

    Google Scholar 

  32. G. D. Plotkin. A structural approach to operational semantics. Technical Report DAIMI FN-19, Computer Science Depratment, Aarhus University, Aarhus, Denmark, 1981.

    Google Scholar 

  33. E. Post. Absolutely unsolvable problems and relatively undecidable propositions. account of an anticipation. In M. Davis, editor, The undecidable. Raven Press, Hewlett, NY, 1965.

    Google Scholar 

  34. M. Sato. Adding proof objects and inductive definition mechanisms to frege structure. In D. H. Pitt, A. Poigné, and D. E. Rydeheard, editors, Proceeding of International Conference on Theoretical Aspects of Computer Software, pages 53–87. Springer-Verlag, LNCS 526, 1991.

    Google Scholar 

  35. D. S. Scott. Constructive validity. In Symposium on Automatic Demonstration, pages 237–275. Springer Lecture Notes in Mathematics 125, 1970.

    Google Scholar 

  36. W. W. Tait. Normal Derivability in Classical Logic. Springer, 1968.

    Google Scholar 

  37. M. Tatsuta. Realiability interpretation of coinductive definitions and program synthesis with streams. Proceedings of International Conference on Fifth Generation Computer Systems, pages 666–673, 1992.

    Google Scholar 

  38. A. S. Troelstra and D. van Dalen. Constructivism in Mathematics, an introduction. North-Holland, 1988.

    Google Scholar 

  39. J. Webb. Mechanism, Mentalism and Metamathematics. D. Reidel Publishing Company, 1980.

    Google Scholar 

  40. G. Winskel. The Formal Semantics of Programming Languages, an Introduction. MIT Press, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Programming Methodology Group. Department of Computer Sciences, Chalmers University of Technology and University of Göteborg, S-412 96, Göteborg, Sweden

    Thierry Coquand & Peter Dybjer

Authors
  1. Thierry Coquand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Dybjer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. S. Thiagarajan

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Coquand, T., Dybjer, P. (1994). Inductive definitions and type theory an introduction (preliminary version). In: Thiagarajan, P.S. (eds) Foundation of Software Technology and Theoretical Computer Science. FSTTCS 1994. Lecture Notes in Computer Science, vol 880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58715-2_114

Download citation

  • DOI: https://doi.org/10.1007/3-540-58715-2_114

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58715-6

  • Online ISBN: 978-3-540-49054-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation