A strong call-by-need calculus

TL;DR

This paper introduces a novel call-by-need lambda calculus that supports strong reduction, ensuring arguments are evaluated only when necessary and at most once, with formal proofs of normalization and minimal reduction sequences.

Contribution

It presents a new strong call-by-need calculus with explicit substitutions, enabling more reduction sequences and shorter ones while preserving neededness, and formalizes part of it using the Abella proof assistant.

Findings

The calculus guarantees normalization to normal forms.

A restriction with the diamond property performs minimal-length reductions.

Formalization in Abella influenced the calculus design.

Abstract

We present a call-by-need $\lambda$-calculus that enables strong reduction (that is, reduction inside the body of abstractions) and guarantees that arguments are only evaluated if needed and at most once. This calculus uses explicit substitutions and subsumes the existing strong-call-by-need strategy, but allows for more reduction sequences, and often shorter ones, while preserving the neededness. The calculus is shown to be normalizing in a strong sense: Whenever a $\lambda$-term t admits a normal form n in the $\lambda$-calculus, then any reduction sequence from t in the calculus eventually reaches a representative of the normal form n. We also exhibit a restriction of this calculus that has the diamond property and that only performs reduction sequences of minimal length, which makes it systematically better than the existing strategy. We have used the Abella proof assistant to formalize part of this calculus, and discuss how this experiment affected its design. In particular, it led us to derive a new description of call-by-need reduction based on inductive rules.