Infinite Belligerent Jump Inversion and Computable Scott Analysis

Abstract

Scott analysis provides two fundamental tools for studying countable structures: Scott sentences, which characterize structures up to isomorphism, and back-and-forth relations, which measure structural similarity. A recurring phenomenon in computable structure theory is that many notions naturally associated with level $\alpha$ of Scott analysis have effective complexity at approximately $2\alpha$ jumps. This discrepancy appears both in the complexity of the back-and-forth relations and in the passage from arbitrary infinitary formulas to computable infinitary formulas. We develop two new coding tools, the Belligerent Pairs Theorem and Belligerent Jump Inversion Theorem, which allow information at complexity level $2\alpha$ to be reflected in computable structures whose distinguishing features already appear at level $\alpha$. These results extend Harrison-Trainor's finite unfriendly jump inversion uniformly throughout the computable ordinals. As applications, we determine the optimal interaction between syntactic complexity and oracle complexity for computable Scott sentences and for formulas distinguishing computable structures. For every computable infinite ordinal $\alpha$, we determine the oracle needed to compute a $\Pi_\alpha$ Scott sentence for a computable structure which has a $\Pi_\alpha$ Scott sentence. Any computable structure with a $\Pi_\alpha$ Scott sentence has a computable $\Pi_{2\alpha}$ Scott sentence. We show that both of these bounds are sharp. We prove analogous optimal results for formulas witnessing failure of the $\alpha$-back-and-forth relation. We also obtain further applications, including a resolution of a question of Chen, Gonzalez, and Harrison-Trainor concerning the complexity of back-and-forth classes.