The Geometry of Interaction of Differential Interaction Nets

The geometry of interaction purpose is to give a semantic of proofs or programs accounting for their dynamics. The initial presentation, translated as an algebraic weighting of paths in proofnets, led to a better characterization of the lambda-lambda-calculus optimal reduction. Recently Ehrhard and Regnier have introduced an extension of the multiplicative exponential fragment of linear logic (MELL) that is able to express non-deterministic behaviour of programs and a proofnet-like calculus: differential interaction nets. This paper constructs a proper geometry of interaction (GoI) for this extension. We consider it both as an algebraic theory and as a concrete reversible computation. We draw links between this GoI and the one of MELL. As a by-product we give for the first time an equational theory suitable for the GoI of the multiplicative additive fragment of linear logic.