Computably enumerable equivalence relations received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility \. This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \ case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by \ on the \ equivalence relations. (...) A special focus of our work is on the existence of infima and suprema of c-degrees. (shrink)

Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal κ. We show the consistency of Eλ-clubλ++,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλλ++ in the space λ++, being continuously reducible to Eλ+-club2,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλ+λ++ in the space 2λ++. Then we show that for κ ineffable Ereg2,κ, the relation of equivalence modulo (...) the nonstationary ideal restricted to regular cardinals in the space 2κ is Σ11-complete. We finish by showing that, for Π21-indescribable κ, the isomorphism relation between dense linear orders of cardinality κ is Σ11-complete. (shrink)

The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this (...) jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$. (shrink)

We investigate the computational complexity the class of Γ-categorical computable structures. We show that hyperarithmetic categoricity is Π11-complete, while computable categoricity is Π04-hard.

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at (...) most n can be naturally identified with the set S of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class Sof rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ of x is defined to be the function. (shrink)

The set of complete groups is a complete co-analytic subset of the standard Borel space of countably infinite groups.

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman—Neumann—Neumann Embedding Theorem.

Let T be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that T has continuum-many countable models. The proof is purely first order, but it raises the question of Borel completeness of T.

We work in the context of ω-stable theories. We obtain a natural, algebraic equivalent of ENI-NDOP and discuss recent joint proofs with Shelah that if an ω-stable theory has either ENI-DOP or is ENI-NDOP and is ENI-deep, then the set of models of T with universe ω is Borel complete.

Let E be an analytic equivalence relation on a Polish space. We introduce a framework for studying the possible “reasonable” complete classifications and the complexity of possible classifying invariants for E, such that: (1) the standard results and intuitions regarding classifications by countable structures are preserved in this framework; (2) this framework respects Borel reducibility; and (3) this framework allows for a precise study of the possible invariants of certain equivalence relations which are not classifiable by countable structures, such as (...) E1. In this framework we show that E1 can be classified, with classifying invariants which are κ-sequences of E0-classes where κ=b, and it cannot be classified in such a manner if κshrink)

The singularity space consists of all germs , with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a complete, separable space for the metric given by the order to which jets agree after base change. In the terminology of descriptive set-theory, the classification of singularities up to analytic extensions of scalars is a smooth problem. Over , the following two classification problems (...) up to isomorphism are then also smooth: analytic germs; and polarized schemes. (shrink)

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we (...) describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work. (shrink)

We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Th}$$\end{document} up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite, then the theory is ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}-categorical or Borel complete; this (...) generalizes a theorem due to Schirmann. (shrink)

We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward density (...) holds and downward density fails. (shrink)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory T is not $\omega $ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct.

In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and (...) Stanley and construct, in particular, a sequence of complete first-order ω-stable theories $(T_\alpha)_{\alpha < \omega_1}$ with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility. (shrink)

In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation ΔR+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\mathbb {R}}^+$$\end{document} on RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{{\mathbb {N}}}$$\end{document} defined by xΔR+y⇔{xi:i∈N}={yi:i∈N}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x (...) \varDelta _{\mathbb {R}}^+y \Leftrightarrow \{x_i{:}\,i \in {\mathbb {N}}\}=\{y_i{:}\,i \in {\mathbb {N}}\}$$\end{document}. Moreover, we show that ΔR+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta _{\mathbb {R}}^+$$\end{document} is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications. (shrink)

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if, then many of them are ‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is ‐complete (it is, if, but can be forced not to be).

We study the Borel reducibility of isomorphism relations in the generalized Baire space. In the main result we show for inaccessible κ, that if T is a classifiable theory and is stable with the orthogonal chain property (OCP), then the isomorphism of models of T is Borel reducible to the isomorphism of models of.

Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.

We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π03 iff the invariants are countable sets of reals, it is potentially Π04 iff the invariants are countable sets of countable sets of reals, and so on. The proofs (...) use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

A computable structure [Formula: see text] is decidable if, given a formula [Formula: see text] of elementary first-order logic, and a tuple [Formula: see text], we have a decision procedure to decide whether [Formula: see text] holds of [Formula: see text]. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is [Formula: see text]-complete. We also show that for each [Formula: see text] (...) the index set of the computable structures with [Formula: see text]-decidable presentations is [Formula: see text]-complete. (shrink)

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As (...) a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. (shrink)

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 $\Pi ^{\mathrm {in}}_{\alpha }$ complexities. We then take a more direct approach and exhibit linear orderings of all Scott sentence complexities except $\Sigma ^{\mathrm {in}}_{3}$ and $\Sigma ^{\mathrm {in}}_{\lambda +1}$ for $\lambda $ a limit ordinal. We show that the former cannot 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. (shrink)

We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L ω ₁ ω ) is far from complete (see [5, 2]). In (...) this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of ω ₁ ω , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids. (shrink)

We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω.

We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...) equivalence relations and show that existence of infinitely many properly Σ11 equivalence classes that are complete as Σ11 sets is necessary but not sufficient for a relation to be complete in the context of Σ11 equivalence relations. (shrink)

We use modified Tsirelson's spaces to prove that there is no finite basis for turbulent Polish group actions. This answers a question of Hjorth and Kechris 329–346; Hjorth, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Section 3.4.3).

We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

We prove that the conjugacy problem for the automorphism group of the random graph is Borel complete, and discuss the analogous problem for some other countably categorical structures.

We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the (...) countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely $\Delta^1_2$ reductions, and hence to the infinite time computable reductions. (shrink)

We study the complexity of the classification problem for countable models of set theory (). We prove that the classification of arbitrary countable models of is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of.

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group [Formula: see text] we introduce the [Formula: see text]-jump. We study the elementary properties of the [Formula: see text]-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups [Formula: see text], the [Formula: see text]-jump is proper in the sense that for any Borel equivalence relation [Formula: see text] the [Formula: see (...) text]-jump of [Formula: see text] is strictly higher than [Formula: see text] in the Borel reducibility hierarchy. On the other hand, there are examples of groups [Formula: see text] for which the [Formula: see text]-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the full [Formula: see text]-tree, and relate these to iterates of the [Formula: see text]-jump. We also produce several new examples of equivalence relations that arise from applying the [Formula: see text]-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank. (shrink)

We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation (...) for certain classes of homogeneous and ultrahomogeneous metric spaces. (shrink)

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We study Polish spaces for which a set of possible distances $A \subseteq R^+$ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with distances in that set belong to a given class, (...) such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of A, of the relations of isometry and isometric embeddability between these Polish spaces. (shrink)

Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic164(12) (2013) 1454–1492], we show that whenever κ is a cardinal satisfying κ<κ=κ>ω, then the embeddability relation between κ-sized structures is strongly invariantly universal, and hence complete for (κ-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or (...) groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc.357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc.146 (2018) 1765–1780]. (shrink)

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable members is very complex become classifiable when we consider only computable members. We follow recent work (...) by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. For real closed fields we show that the isomorphism problem is Δ1 1 complete (the maximum possible), and for others we show that it is of relatively low complexity. We show that the isomorphism problem for algebraically closed fields, Archimedean real closed fields, or vector spaces is Π0 3 complete. (shrink)

Computable structures of Scott rank ${\omega_1^{CK}}$ are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of ${\mathcal{L}_{\omega_1 \omega}}$ , this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank ${\omega_1^{CK}}$ whose computable infinitary theories are each ${\aleph_0}$ -categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank ${\omega_1^{CK}}$ , which guarantee that the (...) resulting structure is a model of an ${\aleph_0}$ -categorical computable infinitary theory. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.