Differential Linear Logic and Processes

Corresponding reduction (cut-elimination) rules are added, which express operationally this new !/? symmetry. These reduction rules are semantically justified, when interpreting the new logical rules for “!” as standard operations on functions (in particular, codereliction corresponds to differentiation of a function at point 0 of a vector space). This extended linear logic is called Differential Linear Logic (DLL). This new symmetry adds expressive power to linear logic. In particular, we show how a fragment of the π-calculus can be translated into differential interaction nets (a system if interaction nets where cells correspond to rules of DLL) and how the dereliction/codereliction reductions of this differential interaction net simulate the reductions of the process. Last, we present a simple denotational model of differential interaction nets, in a category of sets and relations. This model, which is also a natural model of the pure lambda-calculus (β and η), becomes therefore a model of the considered fragment of the π-calculus, and we explore some of its properties.