Genericity Through Stratification
CoRR(2024)
摘要
A fundamental issue in the λ-calculus is to find appropriate notions
for meaningfulness. Inspired by well-known results for the call-by-name
λ-calculus (CbN), where meaningful terms are identified to the solvable
ones, this paper validates the challenging claim that the notion of potential
valuability (aka scrutability), previously introduced in the literature,
adequately represents meaningfulness in the call-by-value λ-calculus
(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, we use a novel
approach based on stratified reduction, indifferently applicable to CbN and
CbV. The second property concerns consistency of the smallest congruence
relation resulting from equating all meaningless terms (without equating all
terms). We also show that such a congruence has a unique consistent and maximal
extension, which coincides with a natural notion of observational equivalence.
Our results thus supply the formal concepts and tools that validate the
informal notion of meaningfulness underlying CbN and CbV.
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