Skip to main content

Positive Inductive-Recursive Definitions

  • Conference paper
  • 639 Accesses

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8089)

Abstract

We introduce a new theory of data types which allows for the definition of data types as initial algebras of certain functors Fam ℂ → Fam ℂ. This theory, which we call positive inductive-recursive definitions, is a generalisation of Dybjer and Setzer’s theory of inductive-recursive definitions within which ℂ had to be discrete – our work can therefore be seen as lifting this restriction. This is a substantial endeavour as we need to not only introduce a type of codes for such data types (as in Dybjer and Setzer’s work), but also a type of morphisms between such codes (which was not needed in Dybjer and Setzer’s development). We show how these codes are interpreted as functors on Famℂ and how these morphisms of codes are interpreted as natural transformations between such functors. We then give an application of positive inductive-recursive definitions to the theory of nested data types. Finally we justify the existence of positive inductive-recursive definitions by adapting Dybjer and Setzer’s set-theoretic model to our setting.

Keywords

  • Type Theory
  • Natural Transformation
  • Discrete Category
  • Inductive Type
  • Introduction Rule

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-40206-7_3
  • 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   49.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-40206-7
  • 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   64.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abbott, M.: Category of Containers. Ph.D. thesis, University of Leicester (2003)

    Google Scholar 

  2. Abel, A., Matthes, R., Uustalu, T.: Iteration and coiteration schemes for higher-order and nested datatypes. Theoretical Computer Science 333(1-2), 3–66 (2005)

    MathSciNet  MATH  CrossRef  Google Scholar 

  3. Aczel, P.: Frege structures and the notions of proposition, truth and set. In: Barwise, J., Keisler, H.J., Kunen, K. (eds.) The Kleene Symposium. Studies in Logic and the Foundations of Mathematics, vol. 101, pp. 31–59. Elsevier (1980)

    Google Scholar 

  4. Adámek, J., Milius, S., Moss, L.: Initial algebras and terminal coalgebras: A survey (June 29, 2010), draft

    Google Scholar 

  5. Blampied, P.: Structured Recursion for Non-uniform Data-types. Ph.D. thesis, University of Nottingham (2000)

    Google Scholar 

  6. Carboni, A., Johnstone, P.: Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science 5(4), 441–459 (1995)

    MathSciNet  MATH  CrossRef  Google Scholar 

  7. Dybjer, P.: A general formulation of simultaneous inductive-recursive definitions in type theory. Journal of Symbolic Logic 65(2), 525–549 (2000)

    MathSciNet  MATH  CrossRef  Google Scholar 

  8. 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 

  9. 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 

  10. Dybjer, P., Setzer, A.: Indexed induction–recursion. Journal of Logic and Algebraic Programming 66(1), 1–49 (2006)

    MathSciNet  MATH  CrossRef  Google Scholar 

  11. Ek, L., Holmström, O., Andjelkovic, S.: Formalizing Arne Andersson trees and Left-leaning Red-Black trees in Agda. Bachelor thesis, Chalmers University of Technology (2009)

    Google Scholar 

  12. Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proc. Logic in Computer Science, pp. 193–202 (1999)

    Google Scholar 

  13. Ghani, N., Hamana, M., Uustalu, T., Vene, V.: Representing cyclic structures as nested types (2006), presented at Trends in Functional Programming

    Google Scholar 

  14. Hancock, P., McBride, C., Ghani, N., Malatesta, L., Altenkirch, T.: Small induction recursion. In: Hasegawa, M. (ed.) TLCA 2013. LNCS, vol. 7941, pp. 156–172. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  15. Ghani, N., Malatesta, L., Nordvall Forsberg, F., Setzer, A.: Fibred data types. In: LICS 2013 (2013)

    Google Scholar 

  16. Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Elsevier (1999)

    Google Scholar 

  17. Mark, W.: Familial 2-functors and parametric right adjoints. Theory and Applications of Category Theory 18(22), 665–732 (2007)

    MATH  Google Scholar 

  18. Martin-Löf, P.: An intuitionistic theory of types (1972), published in Twenty-Five Years of Constructive Type Theory

    Google Scholar 

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

    Google Scholar 

  20. McBride, C., McKinna, J.: The view from the left. Journal of Functional Programming 14(1), 69–111 (2004)

    MathSciNet  MATH  CrossRef  Google Scholar 

  21. Nordvall Forsberg, F., Setzer, A.: A finite axiomatisation of inductive-inductive definitions. In: Berger, U., Hannes, D., Schuster, P., Seisenberger, M. (eds.) Logic, Construction, Computation, Ontos Mathematical Logic, vol. 3, pp. 259–287. Ontos Verlag (2012)

    Google Scholar 

  22. Ralf, H.: Functional pearl: Perfect trees and bit-reversal permutation. Journal of Functional Programming 10(3), 305–317 (2000)

    MathSciNet  MATH  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ghani, N., Malatesta, L., Nordvall Forsberg, F. (2013). Positive Inductive-Recursive Definitions. In: Heckel, R., Milius, S. (eds) Algebra and Coalgebra in Computer Science. CALCO 2013. Lecture Notes in Computer Science, vol 8089. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40206-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-40206-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40205-0

  • Online ISBN: 978-3-642-40206-7

  • eBook Packages: Computer ScienceComputer Science (R0)