14 citations · 33 across the 9 of their papers we have counts for
12 papers
Primitive recursive reverse mathematics
Nikolay Bazhenov, Marta Fiori-Carones, Lu Liu +1
We use a second-order analogy of to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics.…
Computable topological groups
Heer Tern Koh, Alexander Melnikov, Keng Meng Ng
We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions in the literature. We relate these not…
Computably totally disconnected locally compact groups
Alexander Melnikov, Andre Nies
We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c.\ group…
Computable Stone spaces
Nikolay Bazhenov, Matthew Harrison-Trainor, Alexander Melnikov
We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a…
Computable topological abelian groups
Martino Lupini, Alexander Melnikov, Andre Nies
We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the dualization is computable in the important cases of compact and locally compact totally disco…
Computable classifications of continuous, transducer, and regular functions
Johanna N. Y. Franklin, Rupert Hölzl, Alexander Melnikov +2
We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary…
Foundations of Online Structure Theory II: The Operator Approach
Rod Downey, Alexander Melnikov, Keng Meng Ng
We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We sugg…
On the complexity of classifying Lebesgue spaces
Tyler Brown, Alexander G. Melnikov, Timothy H. McNicholl
Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.
Random Subgroups of Rationals
Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov +4
This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup of rationals, two main…
Computable functors and effective interpretability
Matthew Harrison-Trainor, Alexander Melnikov, Russell Miller +1
Our main result is the equivalence of two notions of reducibility between structures. One is a syntactical notion which is an effective version of interpretability as in model theo…
On computable field embeddings and difference closed fields
Matthew Harrison-Trainor, Russell Miller, Alexander Melnikov
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply ou…
Independence in computable algebra
Matthew Harrison-Trainor, Alexander Melnikov, Antonio Montalbán
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…