3,120 research outputs found
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 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 -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
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