A System F accounting for scalars

TL;DR

This paper introduces a System F-like type system for the linear-algebraic lambda-calculus that tracks scalar quantities, ensuring properties like subject reduction and strong normalization, and guarantees barycentric normal forms.

Contribution

It presents a novel scalar type system for the linear-algebraic lambda-calculus that maintains key properties and tracks scalar amounts in terms, simplifying the calculus.

Findings

The scalar type system satisfies subject reduction.

It ensures strong normalization of terms.

It guarantees that normal forms are barycentric.

Abstract

The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this "scalar" type system enjoys both the subject-reduction property and the strong-normalisation property, our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules. But the more important, original feature of this scalar type system is that it keeps track of 'the amount of a type' that is present in each term. As an example of its use, we shown that it can serve as a guarantee that the normal form of a term is barycentric, i.e that its scalars are summing to one.