Positive Inductive-Recursive Definitions

  • Neil Ghani
  • Lorenzo Malatesta
  • Fredrik Nordvall Forsberg
Conference paper

DOI: 10.1007/978-3-642-40206-7_3

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8089)
Cite this paper as:
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

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Neil Ghani
    • 1
  • Lorenzo Malatesta
    • 1
  • Fredrik Nordvall Forsberg
    • 2
  1. 1.University of StrathclydeUK
  2. 2.Swansea UniversityUK

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