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

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.