A Denotational Account 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 β-equivalent to the input term); standardization (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. * Extended version of an article

From Reduction-based to Reduction-free Normalization

“…Instead of repeatedly reducing a term towards its normal form, as in the traditional reduction-based approach, one uses an extensional normalization function that does not construct any intermediate term and directly yields a normal form, if there is any [22]. Normalization by evaluation has been developed in intuitionistic type theory [14,37,44], proof theory [9,10], category theory [5,16,41], λ-definability [32], partial evaluation [18,19,26], and formal semantics [1,29,30]. The more complicated the terms and the notions of reduction, the more complicated the normalization functions.…”

Typed self-evaluation via intensional type functions

“…Our nbe is not type directed, so it does not produce η-long normal forms. This is sometimes called "untyped normalization by evaluation" [4,18], though this conflicts with our nomenclature of calling a meta-program typed or untyped to foldExp : indicate whether it operates on typed or untyped abstract syntax. We call our NbE typed, but not type-directed.…”

Kripke models for classical logic

“…The interpretation of the objectlanguage inside the meta-language, that goes via evaluation/soundness, is usually done using some form of Kripke models. So far, NBE has been used to show normalisation of various intuitionistic proof systems [5,11,2,1,28,30] as well as purely computational calculi [12]. One advantage of taking this approach to that of studying a reduction relation for a proof calculus for classical logic, explicitly as a rewrite system, is that one circumvents both difficulties of rewrite systems and validating equalities arising from η-conversion.…”