# 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} }

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

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#### References

SHOWING 1-10 OF 32 REFERENCES

Ranks of differentiable functions

- Mathematics
- 1986

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 the… Expand

Hyperarithmetical index sets in recursion theory

- Mathematics
- 1987

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 also… Expand

Three ordinal ranks for the set of differentiable functions

- Mathematics
- 1991

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 of… Expand

Computable Structures and the Hyperarithmetical Hierarchy . Studies in Logic and the Foundations of Mathematics

- 2001

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 of… Expand

The Slaman-Wehner theorem in higher recursion theory

- Mathematics
- 2011

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 of… Expand

Computable structures and the hyperarithmetical hierarchy

- Mathematics
- 2000

Preface. Computability. The arithmetical hierarchy. Languages and structures. Ordinals. The hyperarithmetical hierarchy. Infinitary formulas. Computable infinitary formulas. The Barwise-Kreisel… Expand

Higher recursion theory

- Mathematics
- 1990

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 of… Expand

On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

- Mathematics
- 1997

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-Woodin… Expand

Index sets for computable differential equations

- Mathematics, Computer Science
- Math. Log. Q.
- 2004

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

- Mathematics
- 1987

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 sets… Expand