Skip to main content

A Categorical Semantics for Inductive-Inductive Definitions

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 6859)

Abstract

Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A → Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-22944-2_6
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   74.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-22944-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   99.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abbott, M., Altenkirch, T., Ghani, N.: Containers: Constructing strictly positive types. Theoretical Computer Science 342(1), 3–27 (2005)

    MathSciNet  MATH  CrossRef  Google Scholar 

  2. Altenkirch, T., Morris, P.: Indexed containers. In: 24th Annual IEEE Symposium on Logic In Computer Science, LICS 2009, pp. 277–285 (2009)

    Google Scholar 

  3. Chapman, J.: Type theory should eat itself. Electronic Notes in Theoretical Computer Science 228, 21–36 (2009)

    CrossRef  Google Scholar 

  4. Danielsson, N.A.: A formalisation of a dependently typed language as an inductive-recursive family. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 93–109. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  5. Dybjer, P.: Inductive families. Formal Aspects of Computing 6(4), 440–465 (1994)

    MATH  CrossRef  Google Scholar 

  6. Dybjer, P.: Internal type theory. In: Berardi, S., Coppo, M. (eds.) TYPES 1995. LNCS, vol. 1158, pp. 120–134. Springer, Heidelberg (1996)

    Google Scholar 

  7. Dybjer, P., Setzer, A.: A finite axiomatization of inductive-recursive definitions. In: Girard, J.-Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  8. Dybjer, P., Setzer, A.: Induction–recursion and initial algebras. Annals of Pure and Applied Logic 124(1-3), 1–47 (2003)

    MathSciNet  MATH  CrossRef  Google Scholar 

  9. Ghani, N., Johann, P., Fumex, C.: Fibrational induction rules for initial algebras. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 336–350. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  10. Goguen, J., Thatcher, J., Wagner, E., Wright, J.: Initial algebra semantics and continuous algebras. Journal of the ACM 24(1), 68–95 (1977)

    MathSciNet  MATH  CrossRef  Google Scholar 

  11. Hagino, T.: A Categorical Programming Language. Ph.D. thesis, University of Edinburgh (1987)

    Google Scholar 

  12. Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Information and Computation 145(2), 107–152 (1998)

    MathSciNet  MATH  CrossRef  Google Scholar 

  13. Hofmann, M.: Syntax and semantics of dependent types. In: Semantics and Logics of Computation, pp. 79–130. Cambridge University Press, Cambridge (1997)

    CrossRef  Google Scholar 

  14. Martin-Löf, P.: Intuitionistic type theory. Bibliopolis Naples (1984)

    Google Scholar 

  15. Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s type theory: an introduction. Oxford University Press, Oxford (1990)

    MATH  Google Scholar 

  16. Nordvall Forsberg, F., Setzer, A.: Inductive-inductive definitions. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 454–468. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  17. Norell, U.: Towards a practical programming language based on dependent type theory. Ph.D. thesis, Chalmers University of Technology (2007)

    Google Scholar 

  18. Poll, E., Zwanenburg, J.: From algebras and coalgebras to dialgebras. Electronic Notes in Theoretical Computer Science 44(1), 289–307 (2001)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Altenkirch, T., Morris, P., Nordvall Forsberg, F., Setzer, A. (2011). A Categorical Semantics for Inductive-Inductive Definitions. In: Corradini, A., Klin, B., Cîrstea, C. (eds) Algebra and Coalgebra in Computer Science. CALCO 2011. Lecture Notes in Computer Science, vol 6859. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22944-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22944-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22943-5

  • Online ISBN: 978-3-642-22944-2

  • eBook Packages: Computer ScienceComputer Science (R0)