A Categorical Semantics for Inductive-Inductive Definitions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6859)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbott, M., Altenkirch, T., Ghani, N.: Containers: Constructing strictly positive types. Theoretical Computer Science 342(1), 3–27 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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. 3.
    Chapman, J.: Type theory should eat itself. Electronic Notes in Theoretical Computer Science 228, 21–36 (2009)CrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Dybjer, P.: Inductive families. Formal Aspects of Computing 6(4), 440–465 (1994)zbMATHCrossRefGoogle Scholar
  6. 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. 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)CrossRefGoogle Scholar
  8. 8.
    Dybjer, P., Setzer, A.: Induction–recursion and initial algebras. Annals of Pure and Applied Logic 124(1-3), 1–47 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Goguen, J., Thatcher, J., Wagner, E., Wright, J.: Initial algebra semantics and continuous algebras. Journal of the ACM 24(1), 68–95 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hagino, T.: A Categorical Programming Language. Ph.D. thesis, University of Edinburgh (1987)Google Scholar
  12. 12.
    Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Information and Computation 145(2), 107–152 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hofmann, M.: Syntax and semantics of dependent types. In: Semantics and Logics of Computation, pp. 79–130. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  14. 14.
    Martin-Löf, P.: Intuitionistic type theory. Bibliopolis Naples (1984)Google Scholar
  15. 15.
    Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s type theory: an introduction. Oxford University Press, Oxford (1990)zbMATHGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Norell, U.: Towards a practical programming language based on dependent type theory. Ph.D. thesis, Chalmers University of Technology (2007)Google Scholar
  18. 18.
    Poll, E., Zwanenburg, J.: From algebras and coalgebras to dialgebras. Electronic Notes in Theoretical Computer Science 44(1), 289–307 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of NottinghamUK
  2. 2.Department of Computer ScienceSwansea UniversityUK

Personalised recommendations