2025/05/01 by Ruiyuan Chen, Chen, Ruiyuan, David Gonzalez +3 · 2 citations
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2505.00893
openalex publication_date 2025/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The back-and-forth relations M≤αN are central to computable structure theory and countable model theory. It is well-known that the relation \(M,N) : M ≤αN\ is (lightface) Π02α. We show that this is optimal as the set is \mathbfΠ02α-complete. We are also interested in the one-sided relations \ N : M ≤αN\ and \ N : M ≥αN\ for a fixed M, measuring the Πα and Σα types of M. We show that these sets are always \mathbfΠ0α+ 2 and \mathbfΠ0α+3 respectively, and that for most α there are structures M for which these relations are complete at that level. In particular, there are structures M such that there is no Πα (or even Πα+1) sentence φ such that N \models φ\Longleftrightarrow M ≤αN. This is unfortunate as not all Πα+2 sentences are preserved under ≤α. We define a new hierarchy of syntactic complexity closely related to the back-and-forth game, which can both define the back-and-forth types as well as be preserved by them. These hierarchies of formulas have already been useful in certain Henkin constructions, one of which we give in this paper, and another previously used by Gonzalez and Harrison-Trainor to show that every Πα theory of linear orders has a model with Scott rank at most α+3.