Representing definable functions of HA by neighbourhood functions

A unifying framework for continuity and complexity in higher types

“…Moreover, explicit moduli of continuity can be defined within System T itself. In recent years there has been a renewed interest in syntactic approaches to continuity which provide explicit moduli of continuity, and several new proofs of the continuity of System T definable functionals have been given, including [7,11,13] and most recently [16,28,39]. We provide another proof of this fact via our abstract framework, which applies not just to System T but to arbitrary languages which satisfy the relevant conditions.…”

“…The approach presented here is more closely connected to recent work, where continuity and the extraction of corresponding moduli are established by appealing to monads and effects (e.g. [7,11,13,16,28,39], though this list is not exhaustive). A related monadic translation in all finite types due to the present author, but using the state monad and working in an equational setting, is also the basis for [27].…”