Genericity Through Stratification

TL;DR

This paper introduces a new notion called potential valuability to better capture meaningful terms in call-by-value lambda calculus, proving key properties like genericity and consistency using stratified reduction techniques.

Contribution

It validates potential valuability as an appropriate notion of meaningfulness in CbV, establishing its genericity and consistency through novel stratified reduction methods.

Findings

Potential valuability accurately models meaningfulness in CbV.

Stratified reduction provides a new approach to prove genericity.

The congruence of meaningless terms aligns with observational equivalence.

Abstract

A fundamental issue in the $\lambda$-calculus is to find appropriate notions for meaningfulness. It is well-known that in the call-by-name $\lambda$-calculus (CbN) the meaningful terms can be identified with the solvable ones, and that this notion is not appropriate in the call-by-value $\lambda$-calculus (CbV). This paper validates the challenging claim that yet another notion, previously introduced in the literature as potential valuability, appropriately represents meaningfulness in CbV. Akin to CbN, this claim is corroborated by proving two essential properties. The first one is genericity, stating that meaningless subterms have no bearing on evaluating normalizing terms. To prove this (which was an open problem), we use a novel approach based on stratified reduction, indifferently applicable to CbN and CbV, and in a quantitative way. The second property concerns consistency of the smallest congruence relation resulting from equating all meaningless terms. While the consistency result is not new, we provide the first direct operational proof of it. We also show that such a congruence has a unique consistent and maximal extension, which coincides with a well-known notion of observational equivalence. Our results thus supply the formal concepts and tools that validate the informal notion of meaningfulness underlying CbN and CbV.