arXiv subjects

Matthew Harrison-Trainor

Publications and source records attributed to Matthew Harrison-Trainor.

At least 19 recordsLinked to original sources

arXiv2026

Characterizing relative decidability in terms of model completeness

A theory $T$ is said to be relatively decidable if for every model of $T$, one can compute the elementary diagram of that model from its atomic diagram together with $T$. We verify a conjecture of Chubb, Miller, and Solomon by showing that for complete theories $T$, $T$ is relatively decidable if and only if $T$ has a conservative model complete extension of the form $T \cup \{\varphi(\bar{c})\}$ where $T \models \exists \bar{x} \; \varphi(\bar{x})$. We also show that no such characterization works for incomplete theories.

math.LO↗

arXiv2026

Scott spectral gaps for trees are bounded

Given a Borel class of trees, we show that there is a tree in that class whose Scott sentence is not too much more complicated than the definition of the class. In particular, if the class is definable by a $\Pi_\alpha$ sentence, then there is a model of Scott rank at most $\alpha + 2$. This gives another proof-and one that does not require first proving Vaught's conjecture for trees-of the fact that trees are not faithfully Borel complete.

math.LO↗

arXiv2025

Dichotomy results for classes of countable graphs

We study classes of countable graphs where every member does not contain a given finite graph as an induced subgraph -- denoted by $\mathsf{Free}(\mathcal{G})$ for a given finite graph $\mathcal{G}$. Our main results establish a structural dichotomy for such classes: If $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is on top under effective bi-interpretability, implying that the members of $\mathsf{Free}(\mathcal{G})$ exhibit the full range of structural and computational behaviors. In contrast, if $\mathcal{G}$ is an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is structurally simple, as witnessed by the fact that every member satisfies the computable embeddability condition. This dichotomy is mirrored in the finite setting when one considers combinatorial and complexity-theoretic properties. Specifically, it is known that $\mathsf{Free}(\mathcal{G})^{fin}$ is complete for graph isomorphism and not a well-quasi-order under embeddability whenever $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, while in all other cases $\mathsf{Free}(\mathcal{G})^{fin}$ forms a well-quasi-order and the isomorphism problem for $\mathsf{Free}(\mathcal{G})^{fin}$ is solvable in polynomial time.

math.LO↗

arXiv2025

Relative to any non-arithmetic set

Given a countable structure $\mathcal{A}$, the degree spectrum of $\mathcal{A}$ is the set of all Turing degrees which can compute an isomorphic copy of $\mathcal{A}$. One of the major programs in computable structure theory is to determine which (upwards closed, Borel) classes of degrees form a degree spectrum. We resolve one of the major open problems in this area by showing that the non-arithmetic degrees are a degree spectrum. Our main new tool is a new form of unfriendly jump inversions where the back-and-forth types are maximally complicated. This new tool has several other applications.

math.LO↗

arXiv2025

Optimal Syntactic Definitions of Back-and-Forth Types

The back-and-forth relations $M\leq_\alpha N$ are central to computable structure theory and countable model theory. It is well-known that the relation $\{(M,N) : M \leq_\alpha N\}$ is (lightface) $\Pi^0_{2\alpha}$. We show that this is optimal as the set is $\mathbf{\Pi}^0_{2\alpha}$-complete. We are also interested in the one-sided relations $\{ N : M \leq_\alpha N\}$ and $\{ N : M \geq_\alpha N\}$ for a fixed $M$, measuring the $\Pi_\alpha$ and $\Sigma_\alpha$ types of $M$. We show that these sets are always $\mathbf{\Pi}^0_{\alpha + 2}$ and $\mathbf{\Pi}^0_{\alpha+3}$ respectively, and that for most $\alpha$ there are structures $M$ for which these relations are complete at that level. In particular, there are structures $M$ such that there is no $\Pi_\alpha$ (or even $\Pi_{\alpha+1})$ sentence $\varphi$ such that $N \models \varphi \Longleftrightarrow M \leq_\alpha N$. This is unfortunate as not all $\Pi_{\alpha+2}$ sentences are preserved under $\leq_\alpha$. 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 $\Pi_\alpha$ theory of linear orders has a model with Scott rank at most $\alpha+3$.

math.LO↗

arXiv2025

On the computability of optimal Scott sentences

Given a countable mathematical structure, its Scott sentence is a sentence of the infinitary logic $\mathcal{L}_{\omega_1 \omega}$ that characterizes it among all countable structures. We can measure the complexity of a structure by the least complexity of a Scott sentence for that structure. It is known that there can be a difference between the least complexity of a Scott sentence and the least complexity of a computable Scott sentence; for example, Alvir, Knight, and McCoy showed that there is a computable structure with a $\Pi_2$ Scott sentence but no computable $\Pi_2$ Scott sentence. It is well known that a structure with a $\Pi_2$ Scott sentence must have a computable $\Pi_4$ Scott sentence. We show that this is best possible: there is a computable structure with a $\Pi_2$ Scott sentence but no computable $\Sigma_4$ Scott sentence. We also show that there is no reasonable characterization of the computable structures with a computable $\Pi_n$ Scott sentence by showing that the index set of such structures is $\Pi^1_1$-$m$-complete.

math.LO↗

arXiv2024

A Relation on ${(\omega, <)}$ of Intermediate Degree Spectrum on a Cone

We examine the degree spectra of relations on ${(\omega, <)}$. Given an additional relation $R$ on ${(\omega,<)}$, such as the successor relation, the degree spectrum of $R$ is the set of Turing degrees of $R$ in computable copies of ${(\omega,<)}$. It is known that all degree spectra of relations on ${(\omega,<)}$ fall into one of four categories: the computable degree, all of the c.e. degrees, all of the $\Delta^0_2$ degrees, or intermediate between the c.e. degrees and the $\Delta^0_2$ degrees. Examples of the first three degree spectra are easy to construct and well-known, but until recently it was open whether there is a relation with intermediate degree spectrum on a cone. Bazhenov, Kaloci\'{n}ski, and Wroclawski constructed an example of an intermediate degree spectrum, but their example is unnatural in the sense that it is constructed by diagonalization and thus not canonical, that is, which relation you obtain from their construction depends on which G\"odel encoding (and hence order of enumeration) of the partial computable functions / programs you choose. In this paper, we use the ''on-a-cone'' paradigm to restrict our attention to "natural" relations $R$. Our main result is a construction of a natural relation on ${(\omega,<)}$ which has intermediate degree spectrum. This relation has intermediate degree spectrum because of structural reasons.

math.LO↗

arXiv2024

Scott Spectral Gaps are Bounded for Linear Orderings

We demonstrate that any $\Pi_\alpha$ sentence of the infinitary logic $L_{\omega_1 \omega}$ extending the theory of linear orderings has a model with a $\Pi_{\alpha+4}$ Scott sentence and hence of Scott rank at most $\alpha+3$. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by $4$. This contrasts the situation with general structures where for any $\alpha$ there is a $\Pi_2$ sentence all of whose models have Scott rank $\alpha$. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) $\alpha$, we construct a $\Pi_\alpha$ sentence extending the theory of linear orderings such that no models have a $\Sigma_{\alpha+2}$ Scott sentence and hence no models have Scott rank less than or equal to $\alpha$.

math.LO↗

arXiv2024

Measuring the complexity of characterizing $[0, 1]$, $S^1$, and $\mathbb{R}$ up to homeomorphism

In analogy to the study of Scott rank/complexity of countable structures, we initiate the study of the Wadge degrees of the set of homeomorphic copies of topological spaces. One can view our results as saying that the classical characterizations of $[0,1]$ (e.g., as the unique continuum with exactly two non-cut points, and other similar characterizations), appropriated expressed, are the simplest possible characterizations of $[0,1]$. Formally, we show that the set of homeomorphic copies of $[0,1]$ is $\mathbf{\Pi}^0_4$-Wadge-complete. We also show that the set of homeomorphic copies of $S^1$ is $\mathbf{\Pi}^0_4$-Wadge-complete. On the other hand, we show that the set of homeomorphic copies of $\mathbb{R}$ is $\mathbf{\Pi}^1_1$-Wadge-complete. It is the local compactness that cannot be expressed in a Borel way; the set of homeomorphic copies of $\mathbb{R}$ is $\mathbf{\Pi}^0_4$-Wadge-complete within the locally compact spaces.

math.LO↗

arXiv2024

Computable learning of natural hypothesis classes

This paper is about the recent notion of computably probably approximately correct learning, which lies between the statistical learning theory where there is no computational requirement on the learner and efficient PAC where the learner must be polynomially bounded. Examples have recently been given of hypothesis classes which are PAC learnable but not computably PAC learnable, but these hypothesis classes are unnatural or non-canonical in the sense that they depend on a numbering of proofs, formulas, or programs. We use the on-a-cone machinery from computability theory to prove that, under mild assumptions such as that the hypothesis class can be computably listable, any natural hypothesis class which is learnable must be computably learnable. Thus the counterexamples given previously are necessarily unnatural.

cs.LG↗

arXiv2024

Scott analysis, linear orders and almost periodic functions

For any limit ordinal $\lambda$, we construct a linear order $L_\lambda$ whose Scott complexity is $\Sigma_{\lambda+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $\Sigma_{\lambda+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity. Moreover, we can construct the linear orders $L_\lambda$ so that not only does $L_\lambda$ have Scott complexity $\Sigma_{\lambda+1}$, but there are continuum-many structures $M \equiv_\lambda L_\lambda$ and all such structures also have Scott complexity $\Sigma_{\lambda+1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity $\Pi_{\lambda+1}$ that is only $\lambda$-equivalent to structures with Scott complexity $\Pi_{\lambda+1}$. Our construction is based on functions $f \colon \mathbb{Z}\to \mathbb{N}\cup \{\infty\}$ which are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations.

math.LO↗

arXiv2024

Back-and-forth equivalent group von Neumann algebras

We prove that if $G$ and $H$ are $\alpha$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $\alpha \geq \omega$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $\alpha$-back-and-forth equivalent. In particular, if $G$ and $H$ are $\omega$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups.

math.LO↗

arXiv2024

Two results on complexities of decision problems of groups

We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same for the isomorphism problem for recursively presented groups. Second, the fact that every Turing degree appears as the degree of the word problem of a finitely presented group is shown independently by multiple people in the 1960s. We answer the analogous question for degrees of ceers instead of Turing degrees. We show that the set of ceers which are computably equivalent to a finitely presented group is $\Sigma^0_3$-complete, which is the maximal possible complexity.

math.LO↗

arXiv2024

Computable Gelfand Duality

We establish a computable version of Gelfand Duality. Under this computable duality, computably compact presentations of metrizable spaces uniformly effectively correspond to computable presentations of unital commutative $C^*$ algebras.

math.LO↗

arXiv2023

Coding information into all infinite subsets of a dense set

Suppose you have an uncomputable set $X$ and you want to find a set $A$, all of whose infinite subsets compute $X$. There are several ways to do this, but all of them seem to produce a set $A$ which is fairly sparse. We show that this is necessary in the following technical sense: if $X$ is uncomputable and $A$ is a set of positive lower density then $A$ has an infinite subset which does not compute $X$. We also prove an analogous result for PA degree: if $X$ is uncomputable and $A$ is a set of positive lower density then $A$ has an infinite subset which is not of PA degree. We will show that these theorems are sharp in certain senses and also prove a quantitative version formulated in terms of Kolmogorov complexity. Our results use a modified version of Mathias forcing and build on work by Seetapun, Liu, and others on the reverse math of Ramsey's theorem for pairs.

math.LO↗

arXiv2022

An Effective Classification of Borel Wadge Classes

We give a new and effective classification of all Borel Wadge classes of subsets of Baire space. This relies on the true stage machinery originally developed by Montalb\'an. We use this machinery to give a new proof of Louveau and Saint-Raymond's separation theorem for Borel Wadge classes. This gives a proof of Borel Wadge determinacy in the subsystem $\text{ATR}_0+\Pi^1_1$-I of second-order arithmetic.

math.LO↗

arXiv2022

Iterated Priority Arguments in Descriptive Set Theory

We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff-Kuratowski and Wadge theorems on the structure of ${\mathbf \Delta}^0_\xi$, Louveau and Saint-Raymond's separation theorem, and Louveau's separation theorem.

math.LO↗

arXiv2022

The Logic of Cardinality Comparison Without the Axiom of Choice

We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$m$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.

math.LO↗