Stability Property for the Call-by-Value $λ$-calculus through Taylor Expansion

TL;DR

The Stability Property for the call-by-value $\lambda$-calculus (CbV in the following) is proved via the tool of Taylor-resource approximation, whose power has been shown in several recent papers.

Abstract

We prove the Stability Property for the call-by-value $λ$-calculus (CbV in the following). This result states necessary conditions under which the contexts of the CbV $λ$-calculus commute with intersections of approximants. This is an important non-trivial result, which implies the sequentiality of the calculus. We prove it via the tool of Taylor-resource approximation, whose power has been shown in several recent papers. This technique is usually conceived for the ordinary $λ$-calculus, but it can be easily defined for the CbV setting. Our proof is the adaptation of the one for the ordinary calculus using the same technique, with some minimal technical modification due to the fact that in the CbV setting one linearises terms in a slightly different way than usual (cfr. $!(A\multimap B)$ vs $!A\multimap B$). The content of this article is taken from the PhD thesis of the author.

Theorems & Definitions (25)

• Definition 1.1: CbV $\lambda$-calculus
• Definition 1.2: Resource CbV $\lambda$-calculus
• Proposition 1.3
• Definition 1.4: Qualitative CbV-Taylor expansion
• Remark 1.5
• Theorem 1.6: Monotonicity Property
• Proposition 1.7: Simulation Property
• Corollary 1.8: Taylor normal form $\lambda$-theory
• Remark 1.9
• Remark 1.10