We introduce a calculus which is a direct extension of both the and the π calculi. We give a simple type system for it, that encompasses both Curry's type inference for the -calculus, and Milner's sorting for the π-calculus as particular cases of typing. We observe that the various continuation passing style transformations for -terms, written in our calculus, actually correspond to encodings already given by Milner and others for evaluation strategies of -terms into the π-calculus. Furthermore, the associated sortings correspond to well-known double negation translations on types. Finally we provide an adequate cps transform from our calculus to the π-calculus. This shows that the latter may be regarded as an "assembly language", while our calculus seems to provide a better programming notation for higher-order concurrency.

Sangiorgi has shown that the semantics induced by Milner's encoding of the call-by-name -calculus in the ß-calculus is the equality of L#vy-Longo trees. Later it was realized that Milner's encodings are actually variations on well-known continuation passing style transforms. Then a question is: is the discriminating ability due to ß-calculus features, or is it already ooeered by the cps transform? We show that the latter is true: the semantics induced by the call-by-name cps transform on -terms is L#vy-Longo trees equality. Keywords continuation-passing-style transforms, -calculus, L#vy-Longo trees, B#hm-out technique. 1. Introduction The -Calculus In this note we study the semantics induced by Plotkin's call-by-name cps transform [13], which has for both source and target languages the -calculus. To start with, we øx some notations regarding this calculus (we sometimes deviate from the notations of Barendregt's book [1]), which we assume the reader is familiar with. We denote by ...