Small induction recursion

Peter Hancock, Conor McBride, Neil Ghani, Lorenzo Malatesta, Thorsten Altenkirch

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

9 Citations (Scopus)


There are several different approaches to the theory of data types. At the simplest level, polynomials and containers give a theory of data types as free standing entities. At a second level of complexity, dependent polynomials and indexed containers handle more sophisticated data types in which the data have an associated indices which can be used to store important computational information. The crucial and salient feature of dependent polynomials and indexed containers is that the index types are defined in advance of the data. At the most sophisticated level, induction-recursion allows us to define data and indices simultaneously.
This work investigates the relationship between the theory of small inductive recursive definitions and the theory of dependent polynomials and indexed containers. Our central result is that the expressiveness of small inductive recursive definitions is exactly the same as that of dependent polynomials and indexed containers. A second contribution of this paper is the definition of morphisms of small inductive recursive definitions. This allows us to extend our main result to an equivalence between the category of small inductive recursive definitions and the category of dependent polynomials/indexed containers. We comment on both the theoretical and practical ramifications of this result.
Original languageEnglish
Title of host publicationTyped Lambda Calculus and Applications
Subtitle of host publication11th International Conference, TLCA 2013, Eindhoven, The Netherlands, June 26-28, 2013. Proceedings
EditorsMasahito Hasegawa
Place of PublicationBerlin
Number of pages17
ISBN (Print)9783642389450
Publication statusPublished - 6 Jun 2013
Event11th International Conference, TLCA 2013 - Eindhoven, Netherlands
Duration: 26 Jun 201328 Jun 2013

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743


Conference11th International Conference, TLCA 2013


  • data type
  • inductive-recursive definitions
  • morphisms
  • recursions
  • salient features
  • second level


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