Taylor subsumes Scott, Berry, Kahn and Plotkin

Abstract

The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential λ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in λ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity.

Strategies as Resource Terms, and their Categorical Semantics

“…This was expected -as in [21] -but some consequences deserve discussion. Firstly, x : F → G is not a valid term as it is not η-expanded: it hides some infinitary copycat behaviour that must be written explicitely in our typed resource calculus, requiring an infinite sum as in (2). This makes our calculus finitary in a stronger sense than usual: each normal resource term describes a simple, finite behaviour, and one can prove that it corresponds to a single point of the relational model of [8].…”

“…Since its inception [14], Taylor expansion was intended as a quantitative alternative to order based approximation techniques, such as Scott continuity and Böhm trees. For instance, Barbarossa and Manzonetto leveraged it to get simpler proofs of known results in pure λ-calculus [2].…”

Normalization and Taylor expansion of lambda-terms

“…Resource terms have gained special interest thanks to the definition by Ehrhard and Regnier of the Taylor expansion for λ-terms [5]. From this perspective, the resource calculus is a theory of approximation of programs and has been successfully exploited to study the computational properties of λ-terms [1,27,21,23]. Our syntax is very close to the one of polyadic calculi or rigid resource calculi [21,26].…”

Coherent differentiation

“…As shown by the recent [BMP20,MP21] formal transformations of programs related to the differential λ-calculus can be used for efficiently implementing gradient back-propagation in a purely functional framework. The differential λ-calculus and the differential linear logic are also useful as the foundation for an approach to finite approximations of programs based on the Taylor expansion [ER08,BM20] which provides a precise analysis of the use of resources during the execution of a functional program deeply related with implementations of the λ-calculus in abstract machines such as the Krivine Machine [ER06].…”