Feedback computability on Cantor space

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.