Positive inductive-recursive definitions

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


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.
Original languageEnglish
Title of host publicationAlgebra and Coalgebra in Computer Science
Subtitle of host publication5th International Conference, CALCO 2013, Warsaw, Poland, September 3-6, 2013. Proceedings
EditorsReiko Heckel, Stefan Milius
Place of PublicationBerlin
Number of pages15
Publication statusPublished - 8 Aug 2013
Event5th Conference on Algebra and Coalgebra in Computer Science, CALCO 2013 - Warsaw, Poland
Duration: 3 Sept 20136 Sept 2013

Publication series

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


Conference5th Conference on Algebra and Coalgebra in Computer Science, CALCO 2013


  • data type
  • generalisation
  • inductive-recursive definitions
  • initial algebras
  • natural transformations
  • nested data
  • new theory
  • set-theoretic models


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