arXiv subjects

Dino Rossegger

Publications and source records attributed to Dino Rossegger.

At least 19 recordsLinked to original sources

arXiv2026

Relations enumerable from positive information

We study countable structures from the viewpoint of enumeration reducibility. Since enumeration reducibility is based on only positive information, in this setting it is natural to consider structures given by their positive atomic diagram, the computable join of all relations of the structure. Fixing a structure $\mathcal{A}$, a natural class of relations in this setting are the relations $R$ such that $R^{\hat{\mathcal{A}}}$ is enumeration reducible to the positive atomic diagram of $\hat{\mathcal{A}}$ for every $\hat{\mathcal{A}}\cong \mathcal{A}$, the relatively intrinsically positively enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are exactly the relations that are definable by $Σ^p_1$ formulas, a subclass of the infinitary $Σ^0_1$ formulas. We then introduce a new natural notion of the jump of a structure and study its interaction with other notions of jumps.

math.LO↗

arXiv2026

Scott Analysis below the Vaught Ordinal

We develop new tools for determining the existence of models of specific Scott ranks under countability conditions. Using these, we improve a result of Sacks by showing that any counterexample to Vaught's conjecture must have at least two models of every parameterized Scott rank -- a result that contrasts with the unparameterized case, where minimal counterexamples have only one model at many ranks. We further prove that theories with fewer than continuum many models have trivial Scott spectra and provide a general, systematic classification of low Scott rank models when only countably many $Σ_α$-types are realized. Additionally, we classify the Scott complexity spectra for many Ehrenfeucht theories, and prove the $ω$-Vaught's conjecture in this setting, answering an infinitary strengthening of a question of Pillay and Tanović. We demonstrate that the Scott complexity of prime models for $ω$-stable first-order theories is commensurate with the complexity of the theory itself. Along the way, we apply our methods to concrete theories like p-groups, trees, and Boolean algebras, answering questions of Harris--Montalbán and Alvir--Csima--MacLean regarding specific structures.

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arXiv2026

Structural vs. computational complexity

We consider highness in the context of computable structure theory and, particularly, the Scott rank of a structure. We define highness for Scott rank and highness for computably defined Scott rank $α$ and characterize them in terms of the ability to compute $Δ^0_β$ sets for appropriate $β$. We close with a discussion of the index sets of structures with a given Scott rank or computably defined Scott rank and a few words about highness for noncomputable Scott ranks.

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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.

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arXiv2025

Uniformity in learning structures

The standard framework for studying learning problems on algebraic structures assumes that the structures in the target family are pairwise nonisomorphic. Under this assumption, the most widely investigated learning criterion--Ex-learning--becomes inherently equivalent to the well-known paradigm of Bc-learning. This paper explores what happens when the nonisomorphism requirement is removed and analyzes the extent to which these two learning criteria remain uniformly equivalent.

math.LO↗

arXiv2025

Classifying the complexity of models of arithmetic

We classify the possible Scott complexities for models of Peano arithmetic. We construct models of particular complexities by first giving a complete Scott analysis of colored linear orderings and constructing models of Peano arithmetic from these colored orderings. We also provide tight connections of certain Scott complexities with notions from the classical theory of models of Peano arithmetic, such as prime, finitely generated, and recursively saturated. This effort provides a powerful set of tools to understand the models of Peano arithmetic.

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arXiv2025

The Borel complexity of the class of models of first-order theories

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmbΠ_ω^0$-complete set of models. In particular, any sequential theory (a class of foundational theories isolated by Pudlák) has a $\pmbΠ_ω^0$-complete set of models. We also give sharp conditions for theories to have a $\pmbΠ^0_n$-complete set of models.

math.LO↗

arXiv2024

Hausdorff dimension and countable Borel equivalence relations

We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally countable Borel quasi-order on $2^ω$ and $g$ is any gauge function of lower order than the identity, then there is a closed set $A$ so that $A$ is an antichain in $\leq_Q$ and $H^g(A) > 0$.

math.LO↗

arXiv2024

Feferman's completeness theorem

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of full uniform reflection of $\mathsf{PA}$. This result is commonly known as Feferman's completeness theorem. The purpose of this paper is twofold. On the one hand this is an expository paper, giving two new proofs of Feferman's completeness theorem that, we hope, shed light on this mysterious and often overlooked result. On the other hand, we combine one of our proofs with results from computable structure theory due to Ash and Knight to give sharp bounds on the order types of well-orders necessary to attain the completeness for levels of the arithmetical hierarchy.

math.LO↗

arXiv2024

Algorithmic aspects of left-orderings of solvable Baumslag--Solitar groups via its dynamical realization

We answer a question of Calderoni and Clay by showing that the conjugation equivalence relation of left orderings of the Baumslag-Solitar groups $\mathrm{BS}(1,n)$ is hyperfinite for any $n$. Our proof relies on a classification of $\mathrm{BS}(1,n)$'s left-orderings via its one-dimensional dynamical realizations. We furthermore use the effectiveness of the dynamical realizations of $\mathrm{BS}(1,n)$ to study algorithmic properties of the left-orderings on $\mathrm{BS}(1,n)$.

math.LO↗

arXiv2024

Learning Equivalence Relations on Polish Spaces

We investigate natural variations of behaviourally correct learning and explanatory learning -- two learning paradigms studied in algorithmic learning theory -- that allow us to ``learn'' equivalence relations on Polish spaces. We give a characterization of the learnable equivalence relations in terms of their Borel complexity and show that the behaviorally correct and explanatory learnable equivalence relations coincide both in uniform and non-uniform versions of learnability and provide a characterization of the learnable equivalence relations in terms of their Borel complexity. We also show that the set of uniformly learnable equivalence relations is $\pmbΠ^1_1$-complete in the codes and study the learnability of several equivalence relations arising naturally in logic as a case study.

math.LO↗

arXiv2024

Learning Families of Algebraic Structures from Text

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $Σ_2$ sentences.

math.LO↗

arXiv2024

A Lopez-Escobar Theorem for Continuous Domains

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(τ)$ be the set of countable structures with universe $ω$ in vocabulary $τ$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(τ)$ is $Π^0_α$ in the effective Borel hierarchy of this topology if and only if it is definable by a $Π^p_α$ - formula, a positive $Π^0_α$ formula in the infinitary logic $L_{ω_1,ω}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $Φ$, then for every $Π^p_α$ formula $ξ$ in the vocabulary of $K'$ there is a $Π^p_α$ formula $ξ^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models ξ^\star$ if and only if $Φ(A) \models ξ$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

math.LO↗

arXiv2023

Degree spectra of analytic complete equivalence relations

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by this relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a analytic complete equivalence relation. We then investigate the algorithmic properties of this reduction to show that every bi-embeddability spectrum of a graph is the jump spectrum of an elementary bi-embeddability spectrum of a graph.

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arXiv2023

Scott sentence complexities of linear orderings

We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman-Stanley embedding on Scott sentence complexity and show that it only preserves $Π^{\mathrm{in}}_α$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott complexities except $Σ^{\mathrm{in}}_{3}$ and $Σ^{\mathrm{in}}_{λ+1}$ for $λ$ a limit ordinal. We show that the former can not be the Scott sentence complexity of a linear ordering. In the process we develop new techniques which appear to be helpful to calculate the Scott sentence complexities of structures.

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arXiv2023

Degrees of categoricity and treeable degrees

We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $\mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $\mathbf d$ with $\mathbf 0^{(α)}\leq \mathbf d\leq \mathbf 0^{(α+1)}$ for $α$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $\mathbf d$ with $\mathbf 0'\leq \mathbf d\leq \mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $\mathbf d$ with $\mathbf 0'< \mathbf d< \mathbf 0''$ that is not the degree of categoricity of a rigid structure.

math.LO↗

arXiv2022

The structural complexity of models of arithmetic

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $ω$ and that non-standard models of true arithmetic must have Scott rank greater than $ω$. Other than that there are no restrictions. By giving a reduction via $Δ^{\mathrm{in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $ω$-jump of models of an arbitrary completion $T$ of $\mathrm{PA}$ we show that every countable ordinal $α>ω$ is realized as the Scott rank of a model of $T$.

math.LO↗

arXiv2021

Positive enumerable functors

We study reductions well suited to compare structures and classes of structures with respect to properties based on enumeration reducibility. We introduce the notion of a positive enumerable functor and study the relationship with established reductions based on functors and alternative definitions.

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