Related papers: Bi-Isolated d.c.e. Degrees and $\Sigma_1$ Inductio…
A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree $\mathbf{d}$ is isolated by a c.e. degree $\mathbf{a}<\mathbf{d}$ if all c.e. degrees that are below $\mathbf{d}$ are also below…
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $\Pi^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c},…
We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural…
We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e.…
A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a…
The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}\Sigma_1$, with lowness…
Let $C$ be a curve defined over a number field $k$. We say a closed point $x\in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parametrized by the projective line or a positive rank abelian…
Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We…
Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $a\mid 2d$, then $(a, a+d, a+2d)\in\mdeg(\Tame(\mathbb{C}^3))$. (2) If $a\nmid 2d$, then, except for arithmetic progressions…
We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $\mathcal A$ as the family of Turing degrees that compute…
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…
We initiate the computability-theoretic study of ringed spaces and schemes. In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures…
A computable structure $\mathcal{A}$ has degree of categoricity $\mathbf{d}$ if $\mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $\mathcal{A}$. Fokina, Kalimullin, and Miller…
An isolated point of degree $d$ is a closed point on an algebraic curve which does not belong to an infinite family of degree $d$ points that can be parameterized by some geometric object. We provide an algorithm to test whether a rational,…
In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \textit{``Strong Reducibilities''} (1981) and \textit{``Reducibilities''} (1999). The problem asks whether every computably…
The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists. Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its…
We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table…
A computable graph $\mathcal{G}$ is computably categorical relative to a degree $\mathbf{d}$ if and only if for all $\mathbf{d}$-computable copies $\mathcal{B}$ of $\mathcal{G}$, there is a $\mathbf{d}$-computable isomorphism…
Most databases can be configured to operate under isolation levels weaker than serializability. These enforce fewer restrictions on the concurrent access to data and consequently allow for more performant implementations. While formal…
We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove…