JAX Autodiff refers to the core of the automatic differentiation (AD) systems developed in projects like JAX and Dex. JAX Autodiff has recently been formalised in a linear typed calculus by Radul et al in POPL 2023. Although this formalisation suffices to express the main program transformations of AD, the calculus is very specific to this task, and it is not clear whether the type system yields a substructural logic that has interest on its own. We propose an encoding of JAX Autodiff into a linear λ- calculus that enjoys a Curry-Howard correspondence with Girard’s linear logic. We prove that the encoding is sound both qualitatively (the encoded terms are extensionally equivalent to the original ones) and quantitatively (the encoding preserves the original work cost as described by Radul et al. As a byproduct, we show that unzipping, one of the transformations used to implement backpropagation in JAX Autodiff, is, in fact, optional.

A system of session types is introduced as induced by a Curry Howard correspondence applied to Bounded Linear Logic, and then extending the thus obtained type system with probabilistic choices and ground types. The obtained system satisfies the expected properties, like subject reduction and progress, but also unexpected ones, like a polynomial bound on the time needed to reduce processes. This makes the system suitable for modelling experiments and proofs from the so-called computational model of cryptography.