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Nikolay Bazhenov

Publications and source records attributed to Nikolay Bazhenov.

At least 19 recordsLinked to original sources
arXiv2026

Punctually Standard and Nonstandard Models of Natural Numbers

Abstract models of computation often treat the successor function $S$ on $\mathbb{N}$ as a primitive operation, even though its low-level implementations correspond to non-trivial programs operating on specific numerical representations. This behaviour can be analyzed without referring to notations by replacing the standard interpretation $(\mathbb{N}, S)$ with an isomorphic copy ${\mathcal A} = (\mathbb{N}, S^{\mathcal A})$, in which $S^{\mathcal A}$ is no longer computable by a single instruction. While the class of computable functions on $\mathcal{A}$ is standard if $S^{\mathcal{A}}$ is computable, existing results indicate that this invariance fails at the level of primitive recursion. We investigate which sets of operations have the property that if they are primitive recursive on $\mathcal A$ then the class of primitive recursive functions on $\mathcal A$ remains standard. We call such sets of operations \emph{bases for punctual standardness}. We exhibit a series of non-basis results which show how the induced class of primitive recursive functions on $\mathcal A$ can deviate substantially from the standard one. In particular, we demonstrate that a wide range of natural operations, including large subclasses of primitive recursive functions studied by Skolem and Levitz, fail to form such bases. On the positive side, we exhibit natural finite bases for punctual standardness. Our results answer a question recently posed by Grabmayr and establish punctual categoricity for certain natural finitely generated structures.

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arXiv2026

Primitive recursive categoricity spectra

We study the primitive recursive analogue of computable categoricity spectra for various natural classes of structures. We show that these notions coincide for all relatively $Δ_{2}^{0}$-categorical equivalence structures and linear orders, relatively $Δ_{3}^{0}$-categorical Boolean algebras, and computably categorical tree as partial orders.

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arXiv2026

Primitive recursive categoricity spectra of functional structures

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non-$Δ_{1}^{0}$-categorical injection structures, and construct an example of a $Δ_{1}^{0}$-categorical injection structure for which these notions differ. Additionally, we also show that in every non-zero c.e.~Turing degree, there exists a PR-degree that is low for punctual isomorphism (to be defined), and also a PR-degree that is a degree of punctual categoricity.

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arXiv2025

A non-computable c.e. closed subset of $[0,1]$

We prove that there exists a $Σ^0_1$ closed subset of $[0,1]$ that is not homeomorphic to any computably compact space. We show that the index set of c.e. subspaces of $[0,1]$ that admit a computably compact presentation is not arithmetical, as witnessed by subsets of $[0,1]$. The index set result is new for computable Polish spaces in general, not only for those realised as c.e. closed subsets of $[0,1]$.

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arXiv2025

Open Problems in Computability Theory and Descriptive Set Theory

These open problems were presented in the Problem Sessions held during the Tianyuan Workshop on Computability Theory and Descriptive Set Theory, June 16-20, 2025. The problems are organized into sections named after their contributors, in the order of their presentations during the workshop. Notes were taken and compiled by Wei Dai, Feng Li, Ruiwen Li, Ming Xiao, Xu Wang, Víctor Hugo Yañez Salazar, and Yang Zheng.

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arXiv2025

Online and feasible presentability: from trees to modal algebras

We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.

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arXiv2024

Classifying different criteria for learning algebraic structures

In the last years there has been a growing interest in the study of learning problems associated with algebraic structures. The framework we use models the scenario in which a learner is given larger and larger fragments of a structure from a given target family and is required to output an hypothesis about the structure's isomorphism type. So far researchers focused on $\mathbf{Ex}$-learning, in which the learner is asked to eventually stabilize to the correct hypothesis, and on restrictions where the learner is allowed to change the hypothesis a fixed number of times. Yet, other learning paradigms coming from classical algorithmic learning theory remained unexplored. We study the "learning power" of such criteria, comparing them via descriptive-set-theoretic tools thanks to the novel notion of $E$-learnability. The main outcome of this paper is that such criteria admit natural syntactic characterizations in terms of infinitary formulas analogous to the one given for $\mathbf{Ex}$-learning in [6]. Such characterizations give a powerful method to understand whether a family of structure is learnable with respect to the desired criterion.

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arXiv2024

Learning Families of Algebraic Structures from Text

We adapt the classical notion of learning from text to computable structure theory. Our main result is a model-theoretic characterization of the learnability from text for classes of structures. We show that a family of structures is learnable from text if and only if the structures can be distinguished in terms of their theories restricted to positive infinitary $Σ_2$ sentences.

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arXiv2024

A Lopez-Escobar Theorem for Continuous Domains

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(τ)$ be the set of countable structures with universe $ω$ in vocabulary $τ$ topologized by the Scott topology. We show that an invariant set $X \subseteq Mod(τ)$ is $Π^0_α$ in the effective Borel hierarchy of this topology if and only if it is definable by a $Π^p_α$ - formula, a positive $Π^0_α$ formula in the infinitary logic $L_{ω_1,ω}$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $K$ be positively computably embeddable in $K'$ by $Φ$, then for every $Π^p_α$ formula $ξ$ in the vocabulary of $K'$ there is a $Π^p_α$ formula $ξ^\star$ in the vocabulary of $K$ such that for all $A \in K$, $A \models ξ^\star$ if and only if $Φ(A) \models ξ$. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

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arXiv2023

Computable Stone spaces

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 Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space $X$, the Banach space $C(X;\mathbb{R})$ has a computable presentation if, and only if, $X$ is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.

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arXiv2022

Primitive recursive reverse mathematics

We use a second-order analogy $\mathsf{PRA}^2$ of $\mathsf{PRA}$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (\lq punctual\rq) algebra and analysis, and with results from \lq online\rq\ combinatorics. We argue that $\mathsf{PRA}^2$ is sufficiently robust to serve as an alternative base system below $\mathsf{RCA}_0$ to study the proof-theoretic content of theorems in ordinary mathematics. (The most popular alternative is perhaps $\mathsf{RCA}_0^*$.) We discover that many theorems that are known to be true in $\mathsf{RCA}_0$ either hold in $\mathsf{PRA}^2$ or are equivalent to $\mathsf{RCA}_0$ or its weaker (but natural) analogy $2^N-\mathsf{RCA}_0$ over $\mathsf{PRA}^2$. However, we also discover that some standard mathematical and combinatorial facts are incomparable with these natural subsystems.

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arXiv2022

Relatively acceptable notation

Shapiro's notations for natural numbers, and the associated desideratum of acceptability - the property of a notation that all recursive functions are computable in it - is well-known in philosophy of computing. Computable structure theory, however, although capable of fully reconstructing Shapiro's approach, seems to be off philosophers' radar. Based on the case study of natural numbers with standard order, we make initial steps to reconcile these two perspectives. First, we lay the elementary conceptual groundwork for the reconstruction of Shapiro's approach in terms of computable structures and show, on a few examples, how results pertinent to the former can inform our understanding of the latter. Secondly, we prove a new result, inspired by Shapiro's notion of acceptability, but also relevant for computable structure theory. The result explores the relationship between the classical notion of degree spectrum of a computable function on the structure in question - specifically, having all c.e. degrees as a spectrum - and our ability to compute the (image of the) successor from the (image of the) function in any computable copy of the structure. The latter property may be otherwise seen as relativized acceptability of every notation for the structure.

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arXiv2021

Learning algebraic structures with the help of Borel equivalence relations

We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct guess. We prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on reals. This motivates a novel research program, that is, using descriptive set theoretic tools to calibrate the (learning) complexity of nonlearnable families. Here, we focus on the learning power of well-known benchmark Borel equivalence relations (i.e., $E_1$, $E_2$, $E_3$, $Z_0$, and $E_{set}$).

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arXiv2021

Intrinsic complexity of recursive functions on natural numbers with standard order

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 intrinsic complexity of unary total recursive functions on nonnegative integers with standard order. According to existing results, possible spectra of such functions include three sets consisting of precisely: the computable degree, all c.e. degrees and all $Δ_2$ degrees. These results, however, fall far short of the full classification. In this paper, we obtain a more complete picture by giving a few criteria for a function to have intrinsic complexity equal to one of the three candidate sets of degrees. Our investigations are based on the notion of block functions and a broader class of quasi-block functions beyond which all functions of interest have intrinsic complexity equal to the c.e. degrees. We also answer the questions raised by Wright and Harrison-Trainor by showing that the division between computable, c.e. and $Δ_2$ degrees is insufficient in this context as there is a unary total recursive function whose spectrum contains all c.e. degrees but is strictly contained in the $Δ_2$ degrees.

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arXiv2021

Punctual equivalence relations and their (punctual) complexity

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 reducible to $S$ if there is a computable function $f \colon ω\to ω$ that induces an injective map from $R$-equivalence classes to $S$-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore $\mathbf{Peq}$, the degree structure generated by primitive recursive reducibility on punctual equivalence relations (i.e., primitive recursive equivalence relations with domain $ω$). In contrast with all other known degree structures on equivalence relations, we show that $\mathbf{Peq}$ has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of $\mathbf{Peq}$, proving, e.g., that the structure is neither rigid nor homogeneous.

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arXiv2021

Approximating approximate reasoning: Fuzzy sets and the Ershov hierarchy

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies $Δ^0_2$ sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., the real interval $[0; 1]_\mathbb{R}$). In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy $n$-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is $n$-c.e. if its membership function can be approximated by changing monotonicity at most $n-1$ times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy $n$-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all $Δ^0_2$ fuzzy sets.

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arXiv2021

On the Turing complexity of learning finite families of algebraic structures

In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structures is both finite and learnable, then any oracle which computes the Halting set is able to achieve such a learning. On the other hand, we construct a pair of structures which is learnable but no computable learner can learn it.

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