Searcharxiv⌕ Search

arXiv subjects

Dan Turetsky

Publications and source records attributed to Dan Turetsky.

15 recordsLinked to original sources
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 $\alpha$ and characterize them in terms of the ability to compute $\Delta^0_\beta$ sets for appropriate $\beta$. 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.

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 $\Sigma_\alpha$-types are realized. Additionally, we classify the Scott complexity spectra for many Ehrenfeucht theories, and prove the $\omega$-Vaught's conjecture in this setting, answering an infinitary strengthening of a question of Pillay and Tanovi\'c. We demonstrate that the Scott complexity of prime models for $\omega$-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--Montalb\'an and Alvir--Csima--MacLean regarding specific structures.

math.LO↗
arXiv2026

Topology, forcing, and graph colourings

We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.

math.GN↗
arXiv2026

Characterising SJT reducibility

SJT reducibility between sets $A,B \subseteq \mathbb N$ is defined by $A \le_{SJT} B$ if for each computable function $h$ that is unbounded and nondecreasing, there is an $h$-bounded uniformly $B$-c.e.\ trace $(T_n)_{n \in \mathbb N} $ such that for each $n$, the value $J^A(n)$ of the jump is in $T_n$, if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT reducibility, and as a main result give several characterisations of it on the $K$-trivial sets. This is the first case of extending the three lowness paradigms, weak as an oracle, computed by many, and inert, to the setting of weak reducibilities.

math.LO↗
arXiv2025

Failure Modes for Structural Highness Notions

In a previous paper, entitled "Structural Highness Notions," we defined several classes of degrees that are high in senses related to computable structure theory. Each class of degrees is characterized by a structural feature (e.g., an isomorphism) that it can compute if such a feature exists. In this paper, we examine each of these classes and characterize them based on what they do if no such object exists. We describe, in particular, reticent, loquacious, and collegiate senses of being high. These, respectively, reflect the case where a computation from the degree can give output only if the desired feature exists, the case where it will give output of some kind whether or not the feature exists, and the case where the degree will either compute the feature or the best available approximation to it.

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↗
arXiv2021

Structural Highness Notions

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

math.LO↗
arXiv2020

Computing sets from all infinite subsets

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $\Pi^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.

math.LO↗
arXiv2019

Taking the path computably travelled

We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $\Pi^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lowness for isomorphism.

math.LO↗
arXiv2019

Coding in the automorphism group of a computably categorical structure

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimulin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank, and a structure of computable dimension 2 such that there is no hyperarithmetic isomorphism between the two copies.

math.LO↗
arXiv2019

Relationships between computability-theoretic properties of problems

A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of $\omega$ cones, of 1 hyperimmunity or of 1 non-$\Sigma^0_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non-$\Sigma^0_1$ definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.

math.LO↗
arXiv2017

Finding bases of uncountable free abelian groups is usually difficult

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis.

math.LO↗