The Gandy–Hyland functional and a 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 Gödel's system T can be 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. 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.

To be or not to be constructive, that is not the question

“…First of all, since any individual real can be given a binary representation (See [52]), the equivalence between STP and STP R is immediate. Next, STP is easily seen to be equivalent to 15) and this equivalence may also be found in [94,Theorem 3.2]. For completeness, we first prove the equivalence STP ↔ (A.15).…”

“…Note however that Wattenberg (explicitly and implicitly) makes use of Transfer in [122], which is also highly non-constructive, as established in Section 4.3 and [97, §4]. In 15 In particular, the special fan functional Θ from [94] is not computable (in the sense of Kleene's S1-S9) from the Suslin functional, the functional version of Π 1 1 -CA 0 , as proved in [79].…”

“…Furthermore, (S ′′ ) is translated to the special fan functional (First introduced in [94]; see Section 4.4) during term extraction, an object which is extremely hard 15 to compute, as studied in [79].…”

“…As it turns out, the latter is extremely hard to compute in classical mathematics, while easy to compute in intuitionistic mathematics. All results on the special fan functional are from [79,94].…”

On the Herbrand functional interpretation

“…Nelson's framework has been formulated for higher-order Peano and Heyting arithmetic, and given a computational interpretation via Herbrand realizability and the associated nonstandard Dialectica interpretation [8]-we refer to the later here as the Herbrand functional interpretation. More recently, this Herbrand functional interpretation has been successfully applied in the characterisation of the computational content of nonstandard analysis [5][6][7].…”