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Optimal Syntactic Definitions of Back-and-Forth Types

Ruiyuan Chen, David Gonzalez, Matthew Harrison-Trainor

TL;DR

The paper precisely maps the descriptive-set-theoretic complexity of back-and-forth relations across countable structures, proving the natural Π^0_{2α}-complete barrier for the symmetric definition and establishing tight Π^0 bounds for one-sided α-types. To address non-uniformities, it introduces the AE (E_α/A_α) hierarchy of formulas, which captures back-and-forth types while enjoying preservation properties under these relations. A jump-inversion machinery then lifts lower bounds to higher α, yielding a cohesive picture of definability across the ordinal scale and enabling refined Henkin-type constructions and type-omitting results. The work further applies these ideas to linear orders, Scott sentences, and Omitting Types, revealing new tools for infinitary logic and model theory beyond classical Σ/Π analyses.

Abstract

The back-and-forth relations $M\leq_αN$ are central to computable structure theory and countable model theory. It is well-known that the relation $\{(M,N) : M \leq_αN\}$ is (lightface) $Π^0_{2α}$. We show that this is optimal as the set is $\mathbfΠ^0_{2α}$-complete. We are also interested in the one-sided relations $\{ N : M \leq_αN\}$ and $\{ N : M \geq_αN\}$ for a fixed $M$, measuring the $Π_α$ and $Σ_α$ types of $M$. We show that these sets are always $\mathbfΠ^0_{α+ 2}$ and $\mathbfΠ^0_{α+3}$ 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 $Π_α$ (or even $Π_{α+1})$ sentence $\varphi$ such that $N \models \varphi \Longleftrightarrow M \leq_αN$. This is unfortunate as not all $Π_{α+2}$ sentences are preserved under $\leq_α$. 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 $Π_α$ theory of linear orders has a model with Scott rank at most $α+3$.

Optimal Syntactic Definitions of Back-and-Forth Types

TL;DR

The paper precisely maps the descriptive-set-theoretic complexity of back-and-forth relations across countable structures, proving the natural Π^0_{2α}-complete barrier for the symmetric definition and establishing tight Π^0 bounds for one-sided α-types. To address non-uniformities, it introduces the AE (E_α/A_α) hierarchy of formulas, which captures back-and-forth types while enjoying preservation properties under these relations. A jump-inversion machinery then lifts lower bounds to higher α, yielding a cohesive picture of definability across the ordinal scale and enabling refined Henkin-type constructions and type-omitting results. The work further applies these ideas to linear orders, Scott sentences, and Omitting Types, revealing new tools for infinitary logic and model theory beyond classical Σ/Π analyses.

Abstract

The back-and-forth relations are central to computable structure theory and countable model theory. It is well-known that the relation is (lightface) . We show that this is optimal as the set is -complete. We are also interested in the one-sided relations and for a fixed , measuring the and types of . We show that these sets are always and respectively, and that for most there are structures for which these relations are complete at that level. In particular, there are structures such that there is no (or even sentence such that . This is unfortunate as not all sentences are preserved under . 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 theory of linear orders has a model with Scott rank at most .
Paper Structure (14 sections, 42 theorems, 85 equations, 1 figure)

This paper contains 14 sections, 42 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The definability of the back-and-forth relations in the $\Sigma$ and $\Pi$ hierarchies
  3. Upper bounds for the definability of the $\alpha$-types of structures
  4. General bounds
  5. Exceptional bounds
  6. Lower bounds for the definability of the $\alpha$-types of structures
  7. Lower bounds for small finite cases
  8. Critical finite cases
  9. Exceptional lower bounds at limit levels
  10. Jump inversion
  11. $\mathfrak{E}_\alpha$ and $\mathfrak{A}_\alpha$ formulas
  12. Henkin constructions
  13. Scott ranks of models of a theory of linear orders
  14. Omitting types

Key Result

theorem 1.2

For any non-zero ordinal $\alpha$, structures $\mathcal{M}$ and $\mathcal{N}$ and tuples $\bar{a}\in\mathcal{M}$ and $\bar{b}\in\mathcal{N}$, the following are equivalent:

Figures (1)

  • Figure 6.1: Hierarchies of $\Sigma_\alpha/\Pi_\alpha$ and ${\mathfrak{E}}_\alpha/{\mathfrak{A}}_\alpha$ formulas. Solid arrows denote inclusions. The dotted arrows show how syntactic operations affect the complexity of formulas.

Theorems & Definitions (93)

  • definition 1.1
  • theorem 1.2: Karp
  • theorem 1.3
  • proposition 1.4
  • proposition 1.4
  • theorem 1.5
  • corollary 1.5
  • lemma 2.1
  • proof
  • proof
  • ...and 83 more