A lightface analysis of the differentiability rank

TL;DR

This paper analyzes the complexity of the differentiability hierarchy for functions on [0,1], showing that the set of functions with a given differentiability rank is highly complex in the arithmetical hierarchy.

Contribution

It provides a detailed classification of the complexity of differentiability ranks within the computable functions, connecting it to the arithmetical hierarchy and extending previous work.

Findings

The set of functions with differentiability rank at most alpha is Pi_{2 alpha + 1}-complete.

The analysis applies to the computable part of the differentiability hierarchy.

The results clarify the complexity of verifying differentiability for functions with various ranks.

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 alpha is Pi_{2 alpha + 1}-complete. This result is expressed in the notation of Ash and Knight.