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Typeful Normalization by Evaluation

Authors Olivier Danvy, Chantal Keller, Matthias Puech

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Olivier Danvy
Chantal Keller
Matthias Puech

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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) https://doi.org/10.4230/LIPIcs.TYPES.2014.72

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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Keywords
  • Normalization by Evaluation
  • Generalized Algebraic Data Types
  • Continuation-Passing Style
  • partial evaluation

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