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. * Extended version of an article to appear in

On Normalization by Evaluation for Object Calculi

“…This provides a neat solution to the problem of interpreting fresh name generation in the meta-language, and allows for a conceptually clear presentation of NBE and its correctness. While the overall structure of our proof closely follows that of Filinski and Rohde's [11], proofs of individual properties have a distinct flavour due to the continuation semantics. In particular, the central relation between denotations and syntax is an instance of the relational -lifting of [14,19].…”

“…While identification and completeness are fairly direct consequences of Theorem 1, the proof of soundness requires more work: as explained by Filinski and Rohde, the property is closely related to proofs of adequacy of a denotational semantics with respect to an operational one [11].…”

“…Following [11], the correctness properties we expect from norm : obj → obj ⊥ are split into three parts:…”

“…[12], and we expect that our results can be straightforwardly adapted.) Thus, we draw heavily on work about NBE of untyped lambda calculus [11,2]. In hindsight this is not surprising, since indeed it bears many similarities to the self-application model of method invocation in object calculi.…”

“…In hindsight this is not surprising, since indeed it bears many similarities to the self-application model of method invocation in object calculi. To take into account the partiality of the normalization function, the correctness criterion is weakened accordingly [11]: it comprises of soundness (results are normal forms of the respective input), identification (convertible inputs yield the same result), and completeness (the function is defined on every input that is convertible to a normal form), which we also establish for our NBE procedure.…”