Krivine Machine and Taylor Expansion in a Non-uniform Setting

The Krivine machine is an abstract machine implementing the linear head reduction of λ -calculus. Ehrhard and Regnier gave a resource sensitive version returning the annotated form of a λ -term accounting for the resources used by the linear head reduction. These annotations take the form of terms in the resource λ -calculus. We generalize this resource-driven Krivine machine to the case of the algebraic λ -calculus. The latter is an extension of the pure λ -calculus allowing for the linear combination of λ -terms with coefficients taken from a semiring. Our machine associates a λ -term M and a resource annotation t with a scalar α 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 α 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 α can be recovered from the coefficient of t in the Taylor expansion of M and from the normal form of t.