The computational content of Nonstandard Analysis

Kohlenbach's proof mining program deals with the extraction of effective information from typically ineffective proofs. Proof mining has its roots in Kreisel's pioneering work on the so-called unwinding of proofs. The proof mining of classical mathematics is rather restricted in scope due to the existence of sentences without computational content which are provable from the law of excluded middle and which involve only two quantifier alternations. By contrast, we show that the proof mining of classical Nonstandard Analysis has a very large scope. In particular, we will observe that this scope includes any theorem of pure Nonstandard Analysis, where `pure' means that only nonstandard definitions (and not the epsilon-delta kind) are used. In this note, we survey results in analysis, computability theory, and Reverse Mathematics.Comment: In Proceedings CL&C 2016, arXiv:1606.0582

Reverse Mathematics and parameter-free Transfer

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi

From Nonstandard Analysis to various flavours of Computability Theory

As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and higher-order computability theory, where the former (resp. the latter) sub-discipline deals with objects of type zero and one (resp. of all types). The aforementioned results regarding NSA deal exclusively with the higher-order case; we show in this paper that theorems of NSA also give rise to theorems in classical computability theory by considering so-called textbook proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/

Grilliot's trick in Nonstandard Analysis

The technique known as Grilliot's trick constitutes a template for explicitly defining the Turing jump functional $(\exists^2)$ in terms of a given effectively discontinuous type two functional. In this paper, we discuss the standard extensionality trick: a technique similar to Grilliot's trick in Nonstandard Analysis. This nonstandard trick proceeds by deriving from the existence of certain nonstandard discontinuous functionals, the Transfer principle from Nonstandard analysis limited to $\Pi_1^0$-formulas; from this (generally ineffective) implication, we obtain an effective implication expressing the Turing jump functional in terms of a discontinuous functional (and no longer involving Nonstandard Analysis). The advantage of our nonstandard approach is that one obtains effective content without paying attention to effective content. We also discuss a new class of functionals which all seem to fall outside the established categories. These functionals directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page

A functional interpretation for nonstandard arithmetic

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of extensional Heyting and Peano arithmetic in all finite types, strengthening earlier results by Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To conclude the paper, we will point out some open problems and directions for future research and mention some initial results on saturation principles

On algorithm and robustness in a non-standard sense

In this paper, we investigate the invariance properties, i.e. robust- ness, of phenomena related to the notions of algorithm, finite procedure and explicit construction. First of all, we provide two examples of objects for which small changes completely change their (non)computational behavior. We then isolate robust phenomena in two disciplines related to computability