Abstract machines for Open Call-by-Value

Abstract

The theory of the call-by-value λ -calculus relies on weak evaluation and closed terms, that are natural hypotheses in the study of programming languages. To model proof assistants, however, strong evaluation and open terms are required. Open call-by-value is the intermediate setting of weak evaluation with (possibly) open terms, on top of which Grégoire and Leroy designed one of the abstract machines of Coq. This paper provides a theory of abstract machines for the fireball calculus , the simplest presentation of open call-by-value. The literature contains machines that are either simple but inefficient, as they have an exponential overhead, or efficient but heavy, as they rely on a labeling of environments and a technical optimization. We introduce a machine that is simple and efficient : it does not use labels and it implements the fireball calculus within a bilinear overhead. Moreover, we provide a new fine understanding of how different optimizations impact on the complexity of the overhead, and evidence that the time cost model we work with is minimal. • Study of the implementation of Open Call-by-Value via simple, reasonable and efficient abstract machines. • Comparison with abstract machines in the literature closely related to the ones we introduce. • Examples of machine executions and of size-explosion problems, together with refined explanations and insights. • Focus on the key property leading to reasonable machines for Open CbV: avoiding the substitution of inert terms. • Fine understanding of how different optimizations impact on the implementation of Open CbV.