# Geometry of Resource Interaction and Taylor-Ehrhard-Regnier Expansion — A Minimalist Approach

## Abstract

The resource λ-calculus is a variation of the λ-calculus where arguments are superpositions of terms and must be linearly used, hence it is a model for linear and non-deterministic programming languages. Moreover, it is the target language of Taylor-Ehrhard-Regnier expansion of λ-terms, a linearisation of the λ-calculus which develops ordinary terms into infinite series of resource terms. In a strictly typed restriction of the resource λ-calculus, we study the notion of path persistence, and define a remarkably simple geometry of resource interaction (GoRI) that characterises it. In addition, GoRI is invariant under reduction and counts addends in normal forms. We also analyse expansion on paths in ordinary terms, showing that reduction commutes with expansion and, consequently, that persistence can be transferred back and forth between a path and its expansion. Lastly, we also provide an expanded counterpart of the execution formula, which computes paths as series of objects of GoRI, thus exchanging determinism and conciseness for linearity and simplicity.