Skip to main content
Springer Nature Link
Log in

(Co-)Iteration for Higher-Order Nested Datatypes

  • Conference paper
  • First Online:
Types for Proofs and Programs (TYPES 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2646))

Included in the following conference series:

Abstract

The problem of defining iteration for higher-order nested datatypes of arbitrary (finite) rank is solved within the framework of System Fω of higher-order parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the shape of the type constructors such as positivity or even being polynomial. Its use is demonstrated for some rank-2 heterogeneous/nested datatypes such as powerlists and de Bruijn terms with explicit substitutions. An important feature is the availability of an iterative definition of the mapping operation (the functoriality)for those rank-1 type transformers (i. e., functions from types to types) arising as least fixed-points of monotone rank-2 type transformers. Strong normalization is shown by an embedding into Fω. The results dualize to greatest fixed-points, hence to coinductive constructors with coiteration.

The first author gratefully acknowledges the support by the PhD Programme Logic in Computer Science (GKLI) of the Deutsche Forschungs-Gemeinschaft.

The second author has benefitted from financial support by the EC ESPRIT project 21900 “TYPES” and the EU IST working group 29001 “TYPES”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Andreas Abel. Haskell examples for iteration and coiteration on higher-order datatypes. Haskell code accompanying this article, available on the author’s homepage, 2003.

    Google Scholar 

  2. Thorsten Altenkirch. Representations of first order function types as terminal coalgebras. In Samson Abramsky, editor, Fifth International Conference on Typed Lambda Calculi and Applications, volume 2044 of Lecture Notes in Computer Science, pages 8–21. Springer, 2001.

    Chapter  Google Scholar 

  3. Thorsten Altenkirch and Conor McBride. Generic programming within dependently typed programming. To appear in the proceedings of WCGP’02, 2002.

    Google Scholar 

  4. Thorsten Altenkirch and Bernhard Reus. Monadic presentations of lambda terms using generalized inductive types. In Jörg Flum and Mario Rodríguez-Artalejo, editors, Computer Science Logic, 13th International Workshop, CSL’ 99, Proceedings, volume 1683 of Lecture Notes in Computer Science, pages 453–468. Springer, 1999.

    Google Scholar 

  5. Henk P. Barendregt. Lambda calculi with types. In Samson Abramsky, Dov M. Gabbay, and Tom S. E. Maibaum, editors, Background: Computational Structures, volume 2 of Handbook of Logic in Computer Science, pages 117–309. 1992.

    Google Scholar 

  6. Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, volume 897 of Lecture Notes in Mathematics. Springer Verlag, 1981.

    Google Scholar 

  7. Richard Bird, Jeremy Gibbons, and Geraint Jones. Program optimisation, naturally. In Millenial Perspectives in Computer Science, Palgrave, 2000.

    Google Scholar 

  8. Richard Bird and Lambert Meertens. Nested datatypes. In Johan Jeuring, editor, Mathematics of Program Construction, MPC’98, Proceedings, volume 1422 of Lecture Notes in Computer Science, pages 52–67. Springer Verlag, 1998.

    Chapter  Google Scholar 

  9. Richard Bird and Ross Paterson. Generalised folds for nested datatypes. Formal Aspects of Computing, 11(2):200–222, 1999.

    Article  MATH  Google Scholar 

  10. Richard S. Bird and Ross Paterson. De Bruijn notation as a nested datatype. Journal of Functional Programming, 9(1):77–91, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  11. Herman Geuvers. Inductive and coinductive types with iteration and recursion. In Bengt Nordström, Kent Pettersson, and Gordon Plotkin, editors, Proceedings of the 1992 Workshop on Types for Proofs and Programs, Båstad, Sweden, June 1992, pages 193–217, 1992. Only published via ftp://ftp.cs.chalmers.se/pub/cs-reports/baastad.92/proc.dvi.Z.

  12. Jean-Yves Girard. Interprétation fonctionnelle et élimination des coupures dans l’arithmétique d’ordre supérieur. Thèse de Doctorat d’État, Université de Paris VII, 1972.

    Google Scholar 

  13. Tatsuya Hagino. A typed lambda calculus with categorical type constructors. In D. H. Pitt, A. Poigné, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 283 of Lecture Notes in Computer Science, pages 140–157. Springer Verlag, 1987.

    Google Scholar 

  14. Ralf Hinze. Efficient generalized folds. In Johan Jeuring, editor, Proceedings of the Second Workshop on Generic Programming, WGP 2000, Ponte de Lima, Portugal, July 2000.

    Google Scholar 

  15. Ralf Hinze. Polytypic values possess polykinded types. Science of Computer Programming, MPC Special Issue, 43:129–159, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. A. Howard. The formulae-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 479–490. Academic Press, 1980.

    Google Scholar 

  17. C. Barry Jay, Gianna Bellè, and Eugenio Moggi. Functorial ML. Journal of Functional Programming, 8(6):573–619, 1998.

    Article  MathSciNet  Google Scholar 

  18. Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, second edition, 1998.

    Google Scholar 

  19. Daniel Leivant. Contracting proofs to programs. In Piergiorgio Odifreddi, editor, Logic and Computer Science, volume 31 of APIC Studies in Data Processing, pages 279–327. Academic Press, 1990.

    Google Scholar 

  20. Ralph Matthes. Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. PhD thesis, Ludwig-Maximilians-University, May 1998.

    Google Scholar 

  21. Ralph Matthes. Monotone inductive and coinductive constructors of rank 2. In Laurent Fribourg, editor, Proceedings of CSL 2001, volume 2142 of Lecture Notes in Computer Science, pages 600–614. Springer Verlag, 2001.

    Google Scholar 

  22. Nax P. Mendler. Recursive types and type constraints in second-order lambda calculus. In Proceedings of the Second Annual IEEE Symposium on Logic in Computer Science, Ithaca, N.Y., pages 30–36. IEEE Computer Society Press, 1987.

    Google Scholar 

  23. Chris Okasaki. From fast exponentiation to square matrices: An adventure in types. In Proceedings of the fourth ACM SIGPLAN International Conference on Functional Programming (ICFP’ 99), Paris, France, September 27–29, 1999, volume 34 of SIGPLAN Notices, pages 28–35. ACM, 1999.

    Google Scholar 

  24. William W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium Boston 1971/72, volume 453 of Lecture Notes in Mathematics, pages 240–251. Springer Verlag, 1975.

    Google Scholar 

  25. G. C. Wraith. A note on categorical datatypes. In D. H. Pitt, A. Poigné, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 118–127. Springer Verlag, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Munich, Germany

    Andreas Abel

  2. PPS, CNRS, Université Paris VII, France

    Ralph Matthes

Authors
  1. Andreas Abel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralph Matthes
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Nijmegen, Toernooiveld 1, 6525 ED, Nijmegen, The Netherlands

    Herman Geuvers  & Freek Wiedijk  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Abel, A., Matthes, R. (2003). (Co-)Iteration for Higher-Order Nested Datatypes. In: Geuvers, H., Wiedijk, F. (eds) Types for Proofs and Programs. TYPES 2002. Lecture Notes in Computer Science, vol 2646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-39185-1_1

Download citation

  • DOI: https://doi.org/10.1007/3-540-39185-1_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-14031-3

  • Online ISBN: 978-3-540-39185-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation