Document Type

Article

Publication Title

Logical Methods in Computer Science

Abstract

We study the equivalence between eval-readback and eval-apply big-step evaluators in the general setting of the pure lambda calculus. We study ‘one-step’ equivalence (same strategy) and also discuss ‘big-step’ equivalence (same final result). One-step equivalence extends for free to evaluators in other settings (calculi, programming languages, proof assistants, etc.) by restricting the terms (closed, convergent) while maintaining the strategy. We present a proof that one-step equivalence holds when the ‘readback’ stage satisfies straightforward well-formedness provisos, the ‘eval’ stage implements a ‘uniform’ strategy, and the eval-apply evaluator implements a ‘balanced hybrid’ strategy that has ‘eval’ as a subsidiary strategy. The proof proceeds by the ‘lightweight fusion by fixed-point promotion’ program transformation on evaluator implementations to fuse readback and eval into the balanced hybrid. The proof can be followed with no previous knowledge of the transformation. We use Haskell 2010 as the implementation language, with all evaluators written in monadic style to guarantee semantics (strategy) preservation, but the choice of implementation language is immaterial. To illustrate the large scope of the equivalence, we provide an extensive survey of the strategy space using canonical eval-apply evaluators in code and big-step ‘natural’ operational semantics. We discuss the strategies’ properties, some of their uses, and their abstract machines. We improve the formal definition of uniform and hybrid strategy, use it to structure the strategy space, and to obtain generic higher-order evaluators which are used in the equivalence proof. We introduce a systematic notation for both evaluator styles and use it to summarise strategy and evaluator equivalences, including (non-)equivalences within a style and (non-)equivalences between styles not proven by the transformation.

DOI

https://doi.org/10.46298/lmcs-20(4:24)2024

Publication Date

12-2024

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