Structural vs. computational complexity

Abstract

We consider highness in the context of computable structure theory and, particularly, the Scott rank of a structure. We define highness for Scott rank and highness for computably defined Scott rank $\hat{I}\pm$ and characterize them in terms of the ability to compute $\hat{I}{”}_{\hat{I}}^2$ sets for appropriate ${\hat{I}}^2$. We close with a discussion of the index sets of structures with a given Scott rank or computably defined Scott rank and a few words about highness for noncomputable Scott ranks.