Failure Modes for Structural Highness Notions
Wesley Calvert, Johanna N. Y. Franklin, Dan Turetsky
TL;DR
This work maps out the landscape of structural highness by introducing three failure modes—reticent, loquacious, and collegiate highness—and systematically analyzes how these modes manifest across classes of computable structures. It establishes deep connections between these notions and classical notions in descriptive set theory, such as $\Sigma^1_1$ and $\Pi^0_1$ classes, and uses constructions like rank-saturated trees and computable functors to derive transfer results and separations. The results reveal that the reticent and loquacious variants often collapse to broad behavioral patterns (e.g., enumeration of $\Sigma^1_1$ sets or $\mathcal{O}$), while collegiate notions yield a nuanced hierarchy that, in some contexts, collapses but remains proper within restricted structure classes. The paper also demonstrates that non-lowness for isomorphism trivially propagates across these failure modes and addresses several open questions about the boundary between different highness notions. Overall, it clarifies when degrees that are high for isomorphism provide meaningful, robust computational information versus when they fail to yield outputs unless certain structural features exist, and it highlights the role of computable functorial reductions in transferring these properties between classes.
Abstract
In a previous paper, entitled "Structural Highness Notions," we defined several classes of degrees that are high in senses related to computable structure theory. Each class of degrees is characterized by a structural feature (e.g., an isomorphism) that it can compute if such a feature exists. In this paper, we examine each of these classes and characterize them based on what they do if no such object exists. We describe, in particular, reticent, loquacious, and collegiate senses of being high. These, respectively, reflect the case where a computation from the degree can give output only if the desired feature exists, the case where it will give output of some kind whether or not the feature exists, and the case where the degree will either compute the feature or the best available approximation to it.