Comparing Notions of Dense Computability on $\omega^\omega$ and $2^\omega$

TL;DR

This paper compares two notions of robust computability, effective dense reducibility and coarse reducibility, revealing fundamental differences in their degree structures and complexity properties.

Contribution

It demonstrates that the degrees under these notions differ significantly, with effective dense degrees being more complex and introducing new forcing methods for their analysis.

Findings

Every uniform coarse degree contains a set.

Non-uniform effective dense degrees do not contain sets.

Coarse reducibility is an arithmetic property, effective dense reducibility is $$-complete.

Abstract

A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about the notions of reducibility involved still persist. In this paper, we examine two notions of robust information coding, effective dense reducibility and coarse reducibility and answer the question posed in [1]: whether the degrees of functions under these reductions are the same as the degrees of sets. Despite the surface similarity of these two reducibilities we show that every uniform coarse degree contains a set but that this fails even for the non-uniform effective dense degrees. We then further distinguish these two notions by showing that whether g is coarsely reducible to f is an arithmetic property of f and g while for non-uniform effective dense reducibility it is a $\Pi^1_1$ complete property. To prove these results we introduce notions of forcing that allow us to build generic effective dense and coarse descriptions which may be of use in further exploration of these topics - including the open questions we pose in the final section.