  A famous result by Milner is that the lambda-calculus can be simulated inside
the pi-calculus. This simulation, however, holds only modulo strong
bisimilarity on processes, i.e. there is a slight mismatch between
beta-reduction and how it is simulated in the pi-calculus. The idea is that
evaluating a lambda-term in the pi-calculus is like running an
environment-based abstract machine, rather than applying ordinary
beta-reduction. In this paper we show that such an abstract-machine evaluation
corresponds to linear weak head reduction, a strategy arising from the
representation of lambda-terms as linear logic proof nets, and that the
relation between the two is as tight as it can be. The study is also smoothly
rephrased in the call-by-value case, introducing a call-by-value analogous of
linear weak head reduction.
