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 (Nat → Nat) → Nat in (extensions of) System T, and (ii) a characterisation of the time complexity of bar recursion.

A Gentzen-style monadic translation of Gödel's System T

“…There is also a Kuroda-style translation of System T studied [10,11], where each type ρ is translated to J[ρ] with [ρ] defined by Similarly to the Kolmogorov-style translation, we also need a nonstandard notion of application g * x : Jρ for g : J(δ → Jρ) and x : Jδ defined by g * x := (λh δ→Jρ .h κ x) κ (g) in the case of function application in the term translation (t : ρ) → ([t] : J[ρ]).…”

“…We refer to the papers such as [10,11,13] and our Agda implementation [15] for further information of these monadic translations.…”