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DOI: 10.4230/LIPIcs.TYPES.2014.72

Typeful Normalization by Evaluation

Authors: Olivier Danvy, Chantal Keller, and Matthias Puech

Published in: LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)

Abstract

We present the first typeful implementation of Normalization by Evaluation for the simply typed lambda-calculus with sums and control operators: we guarantee type preservation and eta-long (modulo commuting conversions), beta-normal forms using only Generalized Algebraic Data Types in a general-purpose programming language, here OCaml; and we account for sums and control operators with Continuation-Passing Style. First, we implement the standard NbE algorithm for the implicational fragment in a typeful way that is correct by construction. We then derive its call-by-value continuation-passing counterpart, that maps a lambda-term with sums and call/cc into a CPS term in normal form, which we express in a typed dedicated syntax. Beyond showcasing the expressive power of GADTs, we emphasize that type inference gives a smooth way to re-derive the encodings of the syntax and typing of normal forms in Continuation-Passing Style.

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Olivier Danvy, Chantal Keller, and Matthias Puech. Typeful Normalization by Evaluation. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 72-88, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{danvy_et_al:LIPIcs.TYPES.2014.72,
  author =	{Danvy, Olivier and Keller, Chantal and Puech, Matthias},
  title =	{{Typeful Normalization by Evaluation}},
  booktitle =	{20th International Conference on Types for Proofs and Programs (TYPES 2014)},
  pages =	{72--88},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-88-0},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{39},
  editor =	{Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.72},
  URN =		{urn:nbn:de:0030-drops-54921},
  doi =		{10.4230/LIPIcs.TYPES.2014.72},
  annote =	{Keywords: Normalization by Evaluation, Generalized Algebraic Data Types, Continuation-Passing Style, partial evaluation}
}

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DOI: 10.4230/LIPIcs.CSL.2012.381

Parametricity in an Impredicative Sort

Authors: Chantal Keller and Marc Lasson

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract

Reynold's abstraction theorem is now a well-established result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions (CIC), the underlying formal language of Coq, in which parametricity relations' codomain is the impredicative sort of propositions. To proceed, we need to refine this calculus by splitting the sort hierarchy to separate informative terms from non-informative terms. This refinement is very close to CIC, but with the property that typing judgments can distinguish informative terms. Among many applications, this natural encoding of parametricity inside CIC serves both theoretical purposes (proving the independence of propositions with respect to the logical system) as well as practical aspirations (proving properties of finite algebraic structures). We finally discuss how we can simply build, on top of our calculus, a new reflexive Coq tactic that constructs proof terms by parametricity.

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Chantal Keller and Marc Lasson. Parametricity in an Impredicative Sort. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 381-395, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{keller_et_al:LIPIcs.CSL.2012.381,
  author =	{Keller, Chantal and Lasson, Marc},
  title =	{{Parametricity in an Impredicative Sort}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{381--395},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.381},
  URN =		{urn:nbn:de:0030-drops-36851},
  doi =		{10.4230/LIPIcs.CSL.2012.381},
  annote =	{Keywords: Calculus of Inductive Constructions, parametricity, impredicativity, Coq, universes}
}

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Documents authored by Keller, Chantal

Typeful Normalization by Evaluation

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Parametricity in an Impredicative Sort

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