Normalization by Evaluation for Typed Weak lambda-Reduction

Author Filippo Sestini

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Filippo Sestini
  • Functional Programming Laboratory, University of Nottingham, United Kingdom


We thank Maria Emilia Maietti, Claudio Sacerdoti Coen, Thorsten Altenkirch, Andreas Abel, and Delia Kesner for helpful discussions and feedback on this work.

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Filippo Sestini. Normalization by Evaluation for Typed Weak lambda-Reduction. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Weak reduction relations in the lambda-calculus are characterized by the rejection of the so-called xi-rule, which allows arbitrary reductions under abstractions. A notable instance of weak reduction can be found in the literature under the name restricted reduction or weak lambda-reduction. In this work, we attack the problem of algorithmically computing normal forms for lambda-wk, the lambda-calculus with weak lambda-reduction. We do so by first contrasting it with other weak systems, arguing that their notion of reduction is not strong enough to compute lambda-wk-normal forms. We observe that some aspects of weak lambda-reduction prevent us from normalizing lambda-wk directly, thus devise a new, better-behaved weak calculus lambda-ex, and reduce the normalization problem for lambda-w to that of lambda-ex. We finally define type systems for both calculi and apply Normalization by Evaluation to lambda-ex, obtaining a normalization proof for lambda-wk as a corollary. We formalize all our results in Agda, a proof-assistant based on intensional Martin-Löf Type Theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Lambda calculus
  • normalization
  • lambda-calculus
  • reduction
  • term-rewriting
  • Agda


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