Turing Machines and Differential Linear Logic

We describe the vector space semantics for linear logic in which the exponential modality is modelled by cofree coalgebras. There is a natural differentiable structure in this semantics arising from the primitive elements. We provide an investigation of this structure, through many examples of proofs and their derivatives. In particular, an encoding of Turing machines in linear logic is given in detail, based on work by Girard, but modified in order to be compatible with derivatives. It is shown that for proofs of the appropriate form, the derivatives obtained via the coalgebraic structure agree with those from elementary calculus, allowing one to write Taylor expansions of proofs. We provide an interpretation of the derivative of the Turing encoding via a simple example.