Notions of robust information coding

TL;DR

This paper introduces new notions of computability reducibility that are robust to partial information, motivated by reverse mathematics and encompassing existing concepts like generic and coarse reducibilities.

Contribution

It defines and analyzes several robust reducibility notions, extending the framework of computability theory with applications to reverse mathematics.

Findings

New notions of reducibility are formally introduced.

These notions generalize existing concepts like generic and coarse reducibilities.

The framework connects computability theory with reverse mathematics principles.

Abstract

We introduce and study several notions of computability-theoretic reducibility between subsets of $\omega$ that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be recovered about the output. These are motivated by reductions between $\Pi^1_2$ principles in the context of reverse mathematics, and also encompasses generic and coarse reducibilities, previously studied by Jockusch and Schupp (2012).