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DOI: 10.4230/LIPIcs.MFCS.2017.63

Variations on Inductive-Recursive Definitions

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

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

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.

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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)

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@InProceedings{ghani_et_al:LIPIcs.MFCS.2017.63,
  author =	{Ghani, Neil and McBride, Conor and Nordvall Forsberg, Fredrik and Spahn, Stephan},
  title =	{{Variations on Inductive-Recursive Definitions}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{63:1--63:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.63},
  URN =		{urn:nbn:de:0030-drops-81184},
  doi =		{10.4230/LIPIcs.MFCS.2017.63},
  annote =	{Keywords: Type Theory, induction-recursion, initial-algebra semantics}
}

Document

DOI: 10.4230/LIPIcs.TLCA.2015.45

Models for Polymorphism over Physical Dimension

Authors: Robert Atkey, Neil Ghani, Fredrik Nordvall Forsberg, Timothy Revell, and Sam Staton

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract

We provide a categorical framework for models of a type theory that has special types for physical quantities. The types are indexed by the physical dimensions that they involve. Fibrations are used to organize this index structure in the models of the type theory. We develop some informative models of this type theory: firstly, a model based on group actions, which captures invariance under scaling, and secondly, a way of constructing new models using relational parametricity.

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Robert Atkey, Neil Ghani, Fredrik Nordvall Forsberg, Timothy Revell, and Sam Staton. Models for Polymorphism over Physical Dimension. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 45-59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{atkey_et_al:LIPIcs.TLCA.2015.45,
  author =	{Atkey, Robert and Ghani, Neil and Nordvall Forsberg, Fredrik and Revell, Timothy and Staton, Sam},
  title =	{{Models for Polymorphism over Physical Dimension}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{45--59},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.45},
  URN =		{urn:nbn:de:0030-drops-51547},
  doi =		{10.4230/LIPIcs.TLCA.2015.45},
  annote =	{Keywords: Category Theory, Units of Measure, Dimension Types, Type Theory}
}

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Variations on Inductive-Recursive Definitions

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Models for Polymorphism over Physical Dimension

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