Denotational aspects of untyped normalization by evaluation

Andrzej Filinski; Henning Korsholm Rohde

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 39, Issue: 3, page 423-453
  • ISSN: 0988-3754

Abstract

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We show that the standard normalization-by-evaluation construction for the simply-typed λβη-calculus has a natural counterpart for the untyped λβ-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 β-equivalent to the input term); identification (β-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.

How to cite

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Filinski, Andrzej, and Rohde, Henning Korsholm. "Denotational aspects of untyped normalization by evaluation." RAIRO - Theoretical Informatics and Applications 39.3 (2010): 423-453. <http://eudml.org/doc/92775>.

@article{Filinski2010,
abstract = { We show that the standard normalization-by-evaluation construction for the simply-typed λβη-calculus has a natural counterpart for the untyped λβ-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 β-equivalent to the input term); identification (β-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. },
author = {Filinski, Andrzej, Rohde, Henning Korsholm},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Normalization by evaluation; untyped λ-calculus; denotational semantics; functional programming; Böhm trees; computational adequacy.; untyped -calculus; denotational semantics; computational adequacy; normalization-by-evaluation construction},
language = {eng},
month = {3},
number = {3},
pages = {423-453},
publisher = {EDP Sciences},
title = {Denotational aspects of untyped normalization by evaluation},
url = {http://eudml.org/doc/92775},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Filinski, Andrzej
AU - Rohde, Henning Korsholm
TI - Denotational aspects of untyped normalization by evaluation
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 3
SP - 423
EP - 453
AB - We show that the standard normalization-by-evaluation construction for the simply-typed λβη-calculus has a natural counterpart for the untyped λβ-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 β-equivalent to the input term); identification (β-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.
LA - eng
KW - Normalization by evaluation; untyped λ-calculus; denotational semantics; functional programming; Böhm trees; computational adequacy.; untyped -calculus; denotational semantics; computational adequacy; normalization-by-evaluation construction
UR - http://eudml.org/doc/92775
ER -

References

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  11. A.M. Pitts, Computational adequacy via “mixed” inductive definitions, in Mathematical Foundations of Programming Semantics. Springer-Verlag. Lect. Notes Comput. Sci.802 (1993) 72–82.  
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