arXiv subjects

Dino Rossegger

Publications and source records attributed to Dino Rossegger.

25 records · Page 2Linked to original sources

arXiv2020

On bi-embeddable categoricity of algebraic structures

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.

math.LO↗

arXiv2019

The complexity of Scott sentences of scattered linear orders

Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.

math.LO↗

arXiv2019

Degrees of bi-embeddable categoricity

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.

math.LO↗

arXiv2019

Bi-embeddability spectra and bases of spectra

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.

math.LO↗

arXiv2018

Measuring the complexity of reductions between equivalence relations

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.

math.LO↗

arXiv2017

Degrees of bi-embeddable categoricity of equivalence structures

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.

math.LO↗

arXiv2017

On functors enumerating structures

We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the equivalence relation computable are equivalent. We also obtain results on the relation between enumerable and computable functors.

math.LO↗