Related papers: Computable presentations of randomizations

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…

We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if $p \geq 1$ is a computable real, and if $\Omega$ is a nonzero, non-atomic, and separable measure space, then every…

A computable structure $\mathcal{A}$ is decidable if, given a formula $\varphi(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $\varphi$ holds of $\bar{a}$. We…

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function…

We investigate (2,1):1 structures, which consist of a countable set $A$ together with a function $f: A \to A$ such that for every element $x$ in $A$, $f$ maps either exactly one element or exactly two elements of $A$ to $x$. These…

Adapting a result of Bazhenov, Kalimullin, and Yamaleev, we show that if a Turing degree $\textbf{d}$ is the degree of categoricity of a computable structure $\mathcal{M}$ and is not the strong degree of categoricity of any computable…

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…

While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of…

We show that $z\in\R^n$ is computably random if and only if every computable monotone function on $\R^n$ is differentiable at $z$.

As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe ever more complex probabilistic models and inference algorithms. It is natural to ask whether there is a…

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…

In this paper, a computably definable predicate is defined and characterized. Then, it is proved that every separable infinite-dimensional Hilbert structure in an effectively presented language is computable. Moreover, every definable…

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show…

Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has…

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…

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of…

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…

This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with…

A computably presented algebraic field $F$ has a \emph{splitting algorithm} if it is decidable which polynomials in $F[X]$ are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of…

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and…