Related papers: Inside the Muchnik Degrees II: The Degree Structur…
Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify…
We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability…
We survey the theory of Muchnik (weak) and Medvedev (strong) degrees of subsets of ${}^\omega\omega$ with particular attention to the degrees of $\Pi^0_1$ subsets of ${}^\omega2$. Later sections present proofs, some more complete than…
We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum…
Richter, Stephan, and Zhang asked whether every nonrecursive many-one degree contains a least finite-one degree. We solve this question in the negative, already within the class of computably enumerable many-one degrees. Positive answers…
The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be…
We prove that there exists a countable infinite sequence of non-empty special $\Pi^0_1$ classes $\{\mathcal{P}_i\}_{i\in\omega}$ such that no infinite union of elements of any $\mathcal{P}_i$ computes the halting set. We then give a…
The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $\omega$-power of the Sierpi\'nski space).…
Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this…
We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of non-uniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses…
We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products,…
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 give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that…
The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a…
The article is devoted to one infinite parametric class of continuous functions with complicated local structure. In the article differential, integral, self-affine and other properties of functions, that their argument is represented by…
Intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate…
The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a…
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.…
We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of…
For $p \in [0,1]$ let $\mathcal D(p)$ be the mass problem of infinite bit sequences~$y$ (i.e., $\{0,1\}$-valued functions) such that for each computable bit sequence $x$, the bit sequence $ x \leftrightarrow y$ has asymptotic lower density…