How to play the Accordion. Uniformity and the (non-)conservativity of the linear approximation of the λ-calculus (2305.02785v3)

Abstract: Twenty years after its introduction by Ehrhard and Regnier, differentiation in {\lambda}-calculus and in linear logic is now a celebrated tool. In particular, it allows to write the Taylor formula in various {\lambda}-calculi, hence providing a theory of linear approximations for these calculi. In the standard {\lambda}-calculus, this linear approximation is expressed by results stating that the (possibly) infinitary {\beta}-reduction of {\lambda}-terms is simulated by the reduction of their Taylor expansion: in terms of rewriting systems, the resource reduction (operating on Taylor approximants) is an extension of the {\beta}-reduction. In this paper, we address the converse property, conservativity: are there reductions of the Taylor approximants that do not arise from an actual {\beta}-reduction of the approximated term? We show that if we restrict the setting to finite terms and {\beta}-reduction sequences, then the linear approximation is conservative. However, as soon as one allows infinitary reduction sequences this property is broken. We design a counter-example, the Accordion. Then we show how restricting the reduction of the Taylor approximants allows to build a conservative extension of the {\beta}-reduction preserving good simulation properties. This restriction relies on uniformity, a property that was already at the core of Ehrhard and Regnier's pioneering work.