arXiv subjects

Stefan Vatev

Publications and source records attributed to Stefan Vatev.

6 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.

math.LO↗

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.

math.LO↗

arXiv2021

Computable embeddings for pairs of linear orders

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that $\{ω\cdot k,ω^\star \cdot k\}$ is computably embeddable in $\{ω\cdot t, ω^\star \cdot t\}$ iff $k$ divides $t$.

math.LO↗

arXiv2020

A Note on Computable Embeddings for Ordinals and Their Reverses

We continue the study of computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that although $\{ω\cdot 2, ω^\star \cdot 2\}$ is computably embeddable in $\{ω^2, {(ω^2)}^\star\}$, the class $\{ω\cdot k,ω^\star \cdot k\}$ is \emph{not} computably embeddable in $\{ω^2, {(ω^2)}^\star\}$ for any natural number $k \geq 3$.

math.LO↗

arXiv2020

Coding in graphs and linear orderings

There is a Turing computable embedding $Φ$ of directed graphs $A$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform interpretation; i.e., for all directed graphs $A$, these formulas interpret $A$ in $Φ(G)$. It follows that A is Medvedev reducible to $Φ(A)$ uniformly; i.e., there is a fixed Turing operator that serves for all $A$. We observe that there is a graph $G$ that is not Medvedev reducible to any linear ordering. Hence, $G$ is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $Σ_2$ formulas. Any graph can be interpreted in a linear ordering using computable $Σ_3$ formulas. Friedman and Stanley gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{ω_1,ω}$ formulas that, for all $G$, interpret the input graph $G$ in the output linear ordering $L(G)$. Harrison-Trainor and Montalbán have also shown this, by a quite different proof.

math.LO↗

arXiv2019

Cohesive Powers of Linear Orders

Cohesive powers of computable structures can be viewed as effective ultraproducts over effectively indecomposable sets called cohesive sets. We investigate the isomorphism types of cohesive powers $Π_{C}% \mathcal{L}$ for familiar computable linear orders $\mathcal{L}$. If $% \mathcal{L}$ is isomorphic to the ordered set of natural numbers $\mathbb{N}$ and has a computable successor function, then $Π_{C}\mathcal{L}$ is isomorphic to $\mathbb{N}+\mathbb{Q}\times \mathbb{Z}.$ Here, $+$ stands for the sum and $\times $ for the lexicographical product of two orders. We construct computable linear orders $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ isomorphic to $\mathbb{N},$ both with noncomputable successor functions, such that $Π_{C}\mathcal{L}_{1}\mathbb{\ }$is isomorphic to $\mathbb{N}+% \mathbb{Q}\times \mathbb{Z}$, while $Π_{C}\mathcal{L}_{2}$ is not$.$ While cohesive powers preserve all $Π_{2}^{0}$ and $Σ_{2}^{0}$ sentences, we provide new examples of $Π_{3}^{0}$ sentences $Φ$ and computable structures $% \mathcal{M}$ such that $\mathcal{M}\vDash Φ$ while $Π_{C}\mathcal{M}% \vDash \urcorner Φ.$

math.LO↗