Feedback computability on Cantor space

TL;DR

This paper introduces feedback computable functions on Cantor space, showing they are exactly the effectively Borel functions, and explores their properties and applications in computability theory.

Contribution

It extends feedback Turing computation to functions on Cantor space and characterizes feedback computable functions as effectively Borel functions.

Findings

Feedback computable functions are exactly the effectively Borel functions.

The notion of feedback computability on structures is absolute.

Characterization of functions computable from Gandy ordinals with finite subsets.

Abstract

We introduce the notion of feedback computable functions from $2^\omega$ to $2^\omega$, extending feedback Turing computation in analogy with the standard notion of computability for functions from $2^\omega$ to $2^\omega$. We then show that the feedback computable functions are precisely the effectively Borel functions. With this as motivation we define the notion of a feedback computable function on a structure, independent of any coding of the structure as a real. We show that this notion is absolute, and as an example characterize those functions that are computable from a Gandy ordinal with some finite subset distinguished.