Related papers: Essential countability of treeable equivalence rel…
The main question here is the possible generalization of the following theorem on ``simple'' equivalence relation on 2^omega to higher cardinals. Theorem: (1) Assume that: (a) E is a Borel 2-place relation on 2^omega, (b) E is an…
The celebrated Trakhtenbrot's theorem states that the set of finitely valid sentences of first-order logic is not computably enumerable. In this note we will extend this theorem by proving that the finite satisfiability problem of any…
We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str$_0$-trees (the reduct of str-trees that forgets…
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous…
Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological…
We define E-theory for separable C*-algebras over second countable topological spaces and establish its basic properties. This includes an approximation theorem that relates the E-theory over a general space to the E-theories over finite…
Decision trees have long been recognized as models of choice in sensitive applications where interpretability is of paramount importance. In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing,…
The structures $\langle M,\subseteq^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle…
We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular,…
Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the…
Friedman and Stanley developed the notion of Borel reducibility and illustrated its use in comparing classification problems for some familiar classes of countable structures. For many embeddings, the fact that the embedding is $1-1$ on…
We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $\Phi$ as a class of structures in a related language. From this, we show that $\Phi$ has a Borel complete expansion if and only…
A result of Kaufmann shows that if $L_\alpha$ is countable, admissible and satisfies $\Pi_n\textsf{-Collection}$, then $\langle L_\alpha, \in \rangle$ has a proper $\Sigma_{n+1}$-elementary end extension. This paper investigates to what…
Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\mu$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group…
We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{M}^{[0,1)}$ has a…
We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations from the viewpoint of descriptive set theory. It turns out that such equivalence relations are…
In this paper, we show that, for each $p>1$, there are continuum many Borel equivalence relations between $\Bbb R^\omega/\ell_1$ and $\Bbb R^\omega/\ell_p$ ordered by $\le_B$ which are pairwise Borel incomparable.
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…
A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponentially separability. We start by studying these properties in GO spaces, where we…
The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms…