We show that the standard normalization-by-evaluation construction for the simply-typed ${\lambda}_{\beta \eta}$-calculus has a natural counterpart for the untyped ${\lambda}_{\beta}$-calculus, with the central type-indexed logical relation replaced by a “recursively defined” invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation.In the untyped setting, not all terms have normal forms, so the normalization function is necessarily partial. We establish its correctness in the senses of soundness (the output term, if any, is in normal form and $\beta $-equivalent to the input term); identification ($\beta $-equivalent terms are mapped to the same result); and completeness (the function is defined for all terms that do have normal forms). We also show how the semantic construction enables a simple yet formal correctness proof for the normalization algorithm, expressed as a functional program in an ML-like, call-by-value language. Finally, we generalize the construction to produce an infinitary variant of normal forms, namely Böhm trees. We show that the three-part characterization of correctness, as well as the proofs, extend naturally to this generalization.

Keywords: normalization by evaluation, untyped $\lambda $-calculus, denotational semantics, functional programming, Böhm trees, computational adequacy

@article{ITA_2005__39_3_423_0, author = {Filinski, Andrzej and Rohde, Henning Korsholm}, title = {Denotational aspects of untyped normalization by evaluation}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {423--453}, publisher = {EDP-Sciences}, volume = {39}, number = {3}, year = {2005}, doi = {10.1051/ita:2005026}, zbl = {1096.03010}, mrnumber = {2157044}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ita:2005026/} }

TY - JOUR AU - Filinski, Andrzej AU - Rohde, Henning Korsholm TI - Denotational aspects of untyped normalization by evaluation JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2005 DA - 2005/// SP - 423 EP - 453 VL - 39 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita:2005026/ UR - https://zbmath.org/?q=an%3A1096.03010 UR - https://www.ams.org/mathscinet-getitem?mr=2157044 UR - https://doi.org/10.1051/ita:2005026 DO - 10.1051/ita:2005026 LA - en ID - ITA_2005__39_3_423_0 ER -

%0 Journal Article %A Filinski, Andrzej %A Rohde, Henning Korsholm %T Denotational aspects of untyped normalization by evaluation %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2005 %P 423-453 %V 39 %N 3 %I EDP-Sciences %U https://doi.org/10.1051/ita:2005026 %R 10.1051/ita:2005026 %G en %F ITA_2005__39_3_423_0

Filinski, Andrzej; Rohde, Henning Korsholm. Denotational aspects of untyped normalization by evaluation. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 3, pp. 423-453. doi : 10.1051/ita:2005026. http://www.numdam.org/articles/10.1051/ita:2005026/

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