Types of Fireballs (Extended Version)

TL;DR

This paper develops a denotational semantics for open call-by-value lambda calculus using multi types, providing a quantitative analysis of evaluation costs and normalisation properties.

Contribution

It introduces a type system for open call-by-value that characterises normalisation and quantifies evaluation costs, extending prior operational studies.

Findings

Type derivations bound evaluation steps and normal form size.

The type system characterises normalisation and provides exact cost bounds.

A refined fireball calculus facilitates denotational analysis.

Abstract

The good properties of Plotkin's call-by-value lambda-calculus crucially rely on the restriction to weak evaluation and closed terms. Open call-by-value is the more general setting where evaluation is weak but terms may be open. Such an extension is delicate, and the literature contains a number of proposals. Recently, Accattoli and Guerrieri provided detailed operational and implementative studies of these proposals, showing that they are equivalent from the point of view of termination, and also at the level of time cost models. This paper explores the denotational semantics of open call-by-value, adapting de Carvalho's analysis of call-by-name via multi types (aka non-idempotent intersection types). Our type system characterises normalisation and thus provides an adequate relational semantics. Moreover, type derivations carry quantitative information about the cost of evaluation: their size bounds the number of evaluation steps and the size of the normal form, and we also characterise derivations giving exact bounds. The study crucially relies on a new, refined presentation of the fireball calculus, the simplest proposal for open call-by-value, that is more apt to denotational investigations.