Light types for polynomial time computation in lambda-calculus

We propose a new type system for lambda-calculus ensuring that well-typed programs can be executed in polynomial time: dual light affine logic (DIAL). DIAL has a simple type language with a linear and an intuitionistic type arrow, and one modality. It corresponds to a fragment of light affine logic (LAL). We show that contrarily to LAL, DIAL ensures good properties on lambda-terms: subject reduction is satisfied and a well-typed term admits a polynomial bound on the reduction by any strategy. Finally we establish that as LAL, DIAL allows to represent all polytime functions.