A unifying framework for continuity and complexity in higher types

Abstract

We set up a parametrised monadic translation for a class of call-by-value functional languages, and prove a corresponding soundness theorem. We then present a series of concrete instantiations of our translation, demonstrating that a number of fundamental notions concerning higher-order computation, including termination, continuity and complexity, can all be subsumed into our framework. Our main goal is to provide a unifying scheme which brings together several concepts which are often treated separately in the literature. However, as a by-product, we obtain in particular (i) a method for extracting moduli of continuity for closed functionals of type \((\mathbb{N}\to\mathbb{N})\to\mathbb{N}\) in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion.