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Variations on Inductive-Recursive Definitions

Authors Neil Ghani, Conor McBride, Fredrik Nordvall Forsberg, Stephan Spahn



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Neil Ghani
Conor McBride
Fredrik Nordvall Forsberg
Stephan Spahn

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Neil Ghani, Conor McBride, Fredrik Nordvall Forsberg, and Stephan Spahn. Variations on Inductive-Recursive Definitions. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 63:1-63:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.63

Abstract

Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families - i.e. of families (U : Set, T : U -> D) such that the inductive definition of U may depend on the recursively defined T --- by defining a type DS D E of codes. Each c : DS D E defines a functor [c] : Fam D -> Fam E, and (U, T) = \mu [c] : Fam D is exhibited as the initial algebra of [c]. This paper considers the composition of DS-definable functors: Given F : Fam C -> Fam D and G : Fam D -> Fam E, is G \circ F : Fam C -> Fam E DS-definable, if F and G are? We show that this is the case if and only if powers of families are DS-definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF a subsytem of DS a subsystem of PN. Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN-functors exist by adapting Dybjer-Setzer's proof for DS.
Keywords
  • Type Theory
  • induction-recursion
  • initial-algebra semantics

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