Denotational aspects of untyped normalization by evaluation

Abstract

We show that the standard normalization-by-evaluation construction for the simply-typed λ βη-calculus has a natural counterpart for the untyped λ β-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.

Strong Call by Value is Reasonable for Time

“…Our starting point is the higher-order evaluator from [13] presented in Figure 1. It computes β-normal forms by following the principles of normalization by evaluation [19], where the idea is to map a λ-term to an object in the meta-language (here OCaml) from which a syntactic normal form of the input term can subsequently be read off. 1 The applicative order, i.e., leftmost-innermost reduction, where arguments of a function are directly reduced to a strong normal form, is not a conservative extension of weak CbV and therefore we do not consider it as a strong CbV strategy.…”

“…The normalization function first evaluates terms into the semantic domain represented by the recursive type sem -it is completely standard and implemented by the function eval. Then the normal form is extracted from the semantic object by the function reify that mediates between syntax and semantics in the way known from Filinski and Rohde's work [19] on NbE for the untyped λ-calculus.…”

Formal neighbourhoods, combinatory Böhm trees, and untyped normalization by evaluation

“…A Standard ML version of this program was then analysed by Filinski and Rohde [13] who instead used denotational semantics (and a proof technique due to Pitts [21,20]) in their proof that the algorithm outputs Böhm trees. They argued that in this way they could prove the correctness of the precise Standard ML program rather than of a (subtly different) operational model, as Aehlig and Joachimski.…”