DENOTATIONAL ASPECTS OF UNTYPED NORMALIZATION BY EVALUATION

摘要

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 infini-tary variant of normal forms, namely Boehm trees. We show that the three-part characterization of correctness, as well as the proofs, extend naturally to this generalization.