# Krivine Machine and Taylor Expansion in a Non-uniform Setting

## TL;DR

This paper extends the Krivine machine to the algebraic lambda-calculus, enabling resource annotations and probabilistic interpretations, with coefficients derived from Taylor expansions.

## Contribution

It generalizes the resource-sensitive Krivine machine to algebraic lambda-calculus, linking resource annotations to Taylor expansion coefficients and probabilistic semantics.

## Key findings

- Coefficients in the machine correspond to resource usage probabilities.
- Taylor expansion coefficients can recover resource annotations.
- The approach applies to probabilistic lambda-calculus models.

## Abstract

The Krivine machine is an abstract machine implementing the linear head reduction of lambda-calculus. Ehrhard and Regnier gave a resource sensitive version returning the annotated form of a lambda-term accounting for the resources used by the linear head reduction. These annotations take the form of terms in the resource lambda-calculus. We generalize this resource-driven Krivine machine to the case of the algebraic lambda-calculus. The latter is an extension of the pure lambda-calculus allowing for the linear combination of lambda-terms with coefficients taken from a semiring. Our machine associates a lambda-term M and a resource annotation t with a scalar k in the semiring describing some quantitative properties of the linear head reduction of M. In the particular case of non-negative real numbers and of algebraic terms M representing probability distributions, the coefficient k gives the probability that the linear head reduction actually uses exactly the resources annotated by t. In the general case, we prove that the coefficient k can be recovered from the coefficient of t in the Taylor expansion of M and from the normal form of t.