The Gandy-Hyland functional and a hitherto unknown computational aspect of Nonstandard Analysis

Abstract

In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving nonstandard numbers inside Nelson's internal set theory. From this classical and ineffective proof in Nonstandard Analysis, a term from Goedel's system T is extracted which computes the Gandy-Hyland functional in terms of a modulus-of-continuity functional and a special case of the fan functional. We obtain several similar relative computability results not involving Nonstandard Analysis from their associated nonstandard theorems, in particular involving the weak continuity functional. By way of reversal, we show that certain relative computability results, called Herbrandisations, also imply the nonstandard theorem from whence they were obtained. Thus, we establish a direct two-way connection between the field Computability (in particular theoretical computer science) and the field Nonstandard Analysis.