ωPAP Spaces: Reasoning Denotationally About Higher-Order, Recursive Probabilistic and Differentiable Programs

We introduce a new setting, the category of.PAP spaces, for reasoning denotationally about expressive differentiable and probabilistic programming languages. Our semantics is general enough to assign meanings to most practical probabilistic and differentiable programs, including those that use general recursion, higher-order functions, discontinuous primitives, and discrete and continuous sampling. But crucially, it is also specific enough to exclude many pathological denotations, enabling us to establish new results about differentiable and probabilistic programs. In the differentiable setting, we prove general correctness theorems for automatic differentiation and its use within gradient descent. In the probabilistic setting, we establish the almost-everywhere differentiability of probabilistic programs' trace density functions, and the existence of convenient base measures for density computation in Monte Carlo inference. In some cases these results were previously known, but required detailed proofs of an operational flavor; by contrast, all our proofs work directly with programs' denotations.