Related papers: There is No Composition in the Computable Reducibi…
We study the degrees of selector functions related to the degrees in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such degrees can be represented as the unions of upper cones…
The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).
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…
The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group…
We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence…
We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.
In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems:…
Let $C^{pr}_m$ be the upper semilattice of degrees of computable sets with respect to primitive recursive $m$-reducibility. We prove that the first-order theory of $C^{pr}_m$ is hereditarily undecidable.
In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically,…
We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
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…
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 consider equivalence relations and preorders complete for various levels of the arithmetical hierarchy under computable, component-wise reducibility. We show that implication in first order logic is a complete preorder for $\SI 1$, the…
In the last two decades there has been a wealth of results determining the precise value of the catenary degree and the tame degree. Mostly, however, only for very special classes of monoids and domains. In the present work we now show that…
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…
In this paper, we apply the machinery developed in arXiv:2401.06641(2) to study the behavior of computable categoricity relativized to non-c.e. degrees. In particular, we show that we can build a computable structure which is not computably…
An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a \cup c = b$. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary…
This abstract presents (without proofs) some new results on commutativity degree of finite groups.
The Weihrauch degrees and strong Weihrauch degrees are partially ordered structures representing degrees of unsolvability of various mathematical problems. Their study has been widely applied in computable analysis, complexity theory, and…