Related papers: Measuring the complexity of reductions between equ…
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,…
The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with…
A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce…
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…
The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$ is computably…
We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.
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 study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by $ R\le S \iff \ex f \, \forall x, y \, [xRy…
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 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…
This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete "names", possibly…
Although there is a somewhat standard formalization of computability on countable sets given by Turing machines, the same cannot be said about uncountable sets. Among the approaches to define computability in these sets, order-theoretic…
We examine the degree structure $\mathbf{ER}$ of equivalence relations on $\omega$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not…
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…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in…
We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…
The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic…
The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration…