A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK

@article{Westrick2014ALA,
  title={A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK},
  author={Linda Brown Westrick},
  journal={The Journal of Symbolic Logic},
  year={2014},
  volume={79},
  pages={240 - 265}
}
  • L. Westrick
  • Published 2014
  • Mathematics, Computer Science
  • The Journal of Symbolic Logic
Abstract We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal $\alpha > 0$ , the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at most α is ${{\rm{\Pi }}_{2\alpha + 1}}$ -complete. This result is expressed in… Expand
4 Citations
Computable classifications of continuous, transducer, and regular functions.
We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces and apply it to attack classification problems concerning the BanachExpand
An Effective Analysis of the Denjoy Rank
  • L. Westrick
  • Computer Science, Mathematics
  • Notre Dame J. Formal Log.
  • 2020
TLDR
The descriptive complexity of several $\Pi^1_1$ ranks from classical analysis which are associated to Denjoy integration are analyzed, answering questions of Walsh on the limsup rank on well-founded trees. Expand
Cheap Non-standard Analysis and Computability
TLDR
It is proved that many concepts from computable analysis as well as several concepts from Computability can be very elegantly and alternatively presented in this framework. Expand
Computable Analysis and Classification Problems
TLDR
A collection of mathematical objects such as the class of connected compact Polish groups or the set of all real numbers which are normal to some base is given. Expand

References

SHOWING 1-10 OF 32 REFERENCES
Ranks of differentiable functions
The purpose of this paper is to define and study a natural rank function which associates to each differentiable function (say on the interval [0, 1]) a countable ordinal number, which measures theExpand
Hyperarithmetical index sets in recursion theory
We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a II1-complete index set. We alsoExpand
Three ordinal ranks for the set of differentiable functions
Abstract The complexity of a differentiable function can be measured according to its differentiability properties, the integrability properties of its derivative, or the convergence properties ofExpand
Computable Structures and the Hyperarithmetical Hierarchy . Studies in Logic and the Foundations of Mathematics
Chris Ash started Computable Structures and the Hyperarithmetical Hierarchy to bring together, in a consistent and compact form, results from computable (recursive, effective) model theory. Much ofExpand
The Slaman-Wehner theorem in higher recursion theory
Slaman and Wehner have independently shown that there is a countable structure whose degree spectrum consists of the nonzero Turing degrees. We show that the analogue fails in the degrees ofExpand
Computable structures and the hyperarithmetical hierarchy
Preface. Computability. The arithmetical hierarchy. Languages and structures. Ordinals. The hyperarithmetical hierarchy. Infinitary formulas. Computable infinitary formulas. The Barwise-KreiselExpand
Higher recursion theory
Hyperarithmetic theory is the first step beyond classical recursion theory. It is the primary source of ideas and examples in higher recursion theory. It is also a crossroad for several areas ofExpand
On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank
We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal α differentiable functions f and g such that the Zalcwasser rank and the Kechris-WoodinExpand
Index sets for computable differential equations
TLDR
The new notion of a locally computable real function is introduced and several examples of Σ04 complete sets are provided. Expand
Recursively enumerable sets and degrees
TABLE OF CONTENTS Introduction Chapter I. The relation of the structure of an r.e. set to its degree. 1. Post's program and simple sets. 2. Dominating functions and quotient lattices. 3. Maximal setsExpand
...
1
2
3
4
...