A syntactic approach to Borel functions: Some extensions of Louveau's theorem

TL;DR

This paper extends Louveau's theorem from Borel sets to Borel functions, showing that the codes for certain classes of Borel functions can be effectively derived from their Borel codes using hyperarithmetical methods.

Contribution

It introduces extensions of Louveau's theorem to Borel functions, providing effective methods to obtain their codes in various Borel classes.

Findings

Effective hyperarithmetical coding of Borel functions in specific classes

Extension, domination, and decomposition variants of Louveau's theorem

Generalization from Borel sets to Borel functions

Abstract

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $\Gamma$, then its $\Gamma$-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If a Borel function on a Polish space happens to be a $\Sigma_t$-function, then one can effectively find its $\Sigma_t$-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions.