Linear logic and polynomial time

Light and Elementary Linear Logic, which form key components of the interface between logic and implicit computational complexity, were originally introduced by Girard as ‘stand-alone’ logical systems with a (somewhat awkward) sequent calculus of their own. The latter was later reformulated by Danos and Joinet as a proper subsystem of linear logic, whose proofs satisfy a certain structural condition. We extend this approach to polytime computation, finding two solutions: the first is obtained by a simple extension of Danos and Joinet's condition, closely resembles Asperti's Light Affine Logic and enjoys polystep strong normalisation (the polynomial bound does not depend on the reduction strategy); the second, which needs more complex conditions, exactly corresponds to Girard's Light Linear Logic.