Types by Need (Extended Version)

TL;DR

This paper introduces a novel multi type system for call-by-need evaluation in lambda calculus, providing exact bounds on evaluation lengths and establishing a precise denotational model for call-by-need, advancing the quantitative understanding of this evaluation strategy.

Contribution

It develops the first multi type system for call-by-need that yields exact evaluation bounds and offers a tight quantitative denotational semantics.

Findings

Provides exact bounds for call-by-need evaluation lengths

Induces a denotational model of call-by-need

Advances quantitative semantics of lambda calculus

Abstract

A cornerstone of the theory of lambda-calculus is that intersection types characterise termination properties. They are a flexible tool that can be adapted to various notions of termination, and that also induces adequate denotational models. Since the seminal work of de Carvalho in 2007, it is known that multi types (i.e. non-idempotent intersection types) refine intersection types with quantitative information and a strong connection to linear logic. Typically, type derivations provide bounds for evaluation lengths, and minimal type derivations provide exact bounds. De Carvalho studied call-by-name evaluation, and Kesner used his system to show the termination equivalence of call-by-need and call-by-name. De Carvalho's system, however, cannot provide exact bounds on call-by-need evaluation lengths. In this paper we develop a new multi type system for call-by-need. Our system produces exact bounds and induces a denotational model of call-by-need, providing the first tight quantitative semantics of call-by-need.