arXiv subjects
Publications and source records attributed to Nikolay Bazhenov.
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 $t$.
A numbering of a countable family $S$ is a surjective map from the set of natural numbers $ω$ onto $S$. The paper studies Rogers semilattices, i.e. upper semilattices induced by the reducibility between numberings, for families $S\subset P(ω)$. Working in set theory ZF+DC+PD, we obtain the following results on families from various levels of the analytical hierarchy. For a non-zero number $n$, by $E^1_n$ we denote $Π^1_n$ if $n$ is odd, and $Σ^1_n$ if $n$ is even. We show that for a finite family $S$ of $E^1_n$ sets, its Rogers $E^1_n$-semilattice has the greatest element if and only if $S$ contains the least element under set-theoretic inclusion. Furthermore, if $S$ does not have the $\subseteq$-least element, then the corresponding Rogers $E^1_n$-semilattice is upwards dense.
We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $Σ^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $\mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every bi-embeddable copy of it there is an embedding computable in $2n-1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $\mathcal L$ be a computable linear order of Hausdorff rank $n\geq 1$, then $\mathbf 0^{(2n-2)}$ does not compute embeddings between it and all its computable bi-embeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable bi-embeddable copies. On the other hand, if a computable Boolean algebra is superatomic, then there is a least computable ordinal $α$ such that $\mathbf 0^{(α)}$ computes embeddings between all its computable bi-embeddable copies. The main technique used in this proof is a new variation of Ash and Knight's pairs of structures theorem.
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 $k \geq 3$.
A general theme of computable structure theory is to investigate when structures have copies of a given complexity $Γ$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $Π^0_1$ equivalence structure with no $Σ^0_1$ copy, and in fact that the isomorphism types realized by the $Π^0_1$ equivalence structures coincide with those realized by the $Δ^0_2$ equivalence structures. We also construct a $Σ^0_1$ preorder with no $Π^0_1$ copy.
A numbering of a countable family $S$ is a surjective map from the set of natural numbers $ω$ onto $S$. A numbering $ν$ is reducible to a numbering $μ$ if there is an effective procedure which given a $ν$-index of an object from $S$, computes a $μ$-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families $S \subset P(ω)$ belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers $m\neq n$, any non-trivial Rogers semilattice of a $Π^1_m$-computable family cannot be isomorphic to a Rogers semilattice of a $Π^1_n$-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.
A standard tool for classifying the complexity of equivalence relations on $ω$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $Γ$ be one of the following classes: $Σ^0_α$, $Π^0_α$, $Σ^1_n$, or $Π^1_n$, where $α\geq 2$ is a computable ordinal and $n$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $Γ$.
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 embeddings between any computable bi-embeddable copies of $\mathcal A$; the degree of bi-embeddable categoricity of $\mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $\mathbf{0}^{(α)}$ for $α$ a computable successor ordinal and $\mathbf{0}^{(λ)}$ for $λ$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.
Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $Δ^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $\leq_c$ on the $Σ^{-1}_{a}\smallsetminus Π^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $c$-degrees.
We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $Δ^0_α$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $Δ^0_α$ bi-embeddable categoricity and relative $Δ^0_α$ bi-embeddable categoricity coincide for equivalence structures for $α=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $\mathbf{0},\mathbf{0}'$, or $\mathbf{0}''$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability.