No solvable lambda-value term left behind

TL;DR

This paper introduces a new definition of solvability in the lambda-value calculus that aligns with operational relevance and allows for a consistent proof-theory where unsolvables are equated based on their argument order.

Contribution

It provides a novel solvability concept for lambda-value calculus that captures operational relevance and supports a consistent equational theory for unsolvables.

Findings

A new definition of solvability for lambda-value calculus is proposed.

Proof of a version of the Genericity Lemma for unsolvable terms.

Operational relevance is preserved in the new solvability framework.

Abstract

In the lambda calculus a term is solvable iff it is operationally relevant. Solvable terms are a superset of the terms that convert to a final result called normal form. Unsolvable terms are operationally irrelevant and can be equated without loss of consistency. There is a definition of solvability for the lambda-value calculus, called v-solvability, but it is not synonymous with operational relevance, some lambda-value normal forms are unsolvable, and unsolvables cannot be consistently equated. We provide a definition of solvability for the lambda-value calculus that does capture operational relevance and such that a consistent proof-theory can be constructed where unsolvables are equated attending to the number of arguments they take (their "order" in the jargon). The intuition is that in lambda-value the different sequentialisations of a computation can be distinguished operationally. We prove a version of the Genericity Lemma stating that unsolvable terms are generic and can be replaced by arbitrary terms of equal or greater order.