Generically computable linear orderings

Abstract

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the ${\Sigma }_{\beta }$ hierarchy. We focus on linear orderings. We show that at the ${\Sigma }_1$ level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the ${\Sigma }_{\alpha +2}$ level for any $\alpha \in {\omega }_1^{CK}$ the set of linear orderings with generically or coarsely computable copies is ${\Sigma }_1^1$-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.