# Equivalence Relations on Classes of Computable Structures

@inproceedings{Fokina2009EquivalenceRO, title={Equivalence Relations on Classes of Computable Structures}, author={Ekaterina B. Fokina and Sy-David Friedman}, booktitle={CiE}, year={2009} }

If $\mathcal{L}$ is a finite relational language then all computable $\mathcal{L}$-structures can be effectively enumerated in a sequence in such a way that for every computable $\mathcal{L}$-structure $\mathcal{B}$ an index n of its isomorphic copy can be found effectively and uniformly. Having such a universal computable numbering, we can identify computable structures with their indices in this numbering. If K is a class of $\mathcal{L}$-structures closed under isomorphism we denote by K c…

## 23 Citations

On Σ11 equivalence relations over the natural numbers

- MathematicsMath. Log. Q.
- 2012

It is shown that existence of infinitely many properlyΣ11 equivalence classes that are complete as Σ11 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ 11 equivalence relations.

FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

- Computer Science, MathematicsThe Journal of Symbolic Logic
- 2016

It is shown that, for every n, there does exist a natural equivalence relation which is ${\rm{\Pi }}_{n + 2}^0$ -complete under finitary reducibility.

Isomorphism and Bi-Embeddability Relations on Computable Structures ∗

- Mathematics
- 2010

We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and bi-embeddability. We use the notion of tc-reducibility to show completeness of the…

On the Degree Structure of Equivalence Relations Under Computable Reducibility

- Mathematics, Computer ScienceNotre Dame J. Formal Log.
- 2019

It is proved that for all the degree classes considered, upward density holds and downward density fails and the existence of the greatest element for the ω- c.e. and n-c.

The Hierarchy of Equivalence Relations on the Natural Numbers Under Computable Reducibility

- Computer Science, MathematicsComput.
- 2012

The computable reducibility hierarchy is investigated, comparing and contrasting it with the Borel reduCibility hierarchy from descriptive set theory, and the exposition extends earlier work in the literature concerning the classification of computable structures.

Computable model theory

- MathematicsTuring's Legacy
- 2014

In the last few decades there has been increasing interest in computable model theory and the work of Turing, Gödel, Kleene, Church, Post, and others in the mid-1930s established the rigorous mathematical foundations for the computability theory.

Computability theoretic classifications for classes of structures

- Mathematics
- 2014

In this paper, we survey recent work in the study of classes of structures from the viewpoint of computability theory. We consider different ways of classifying classes of structures in terms of…

Reducibilities among equivalence relations induced by recursively enumerable structures

- MathematicsTheor. Comput. Sci.
- 2016

Isomorphism relations on computable structures

- MathematicsThe Journal of Symbolic Logic
- 2012

The notion of FF-reducibility introduced in [9] is used to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω.

## References

SHOWING 1-10 OF 17 REFERENCES

Computable Structure and Non-Structure Theorems

- Mathematics, Computer Science
- 2002

The goal of the present article is to consider some possible answers to Shore's question about what would be a convincing negative result in the classification of computable members of various classes of structures of some computable language.

Analytic equivalence relations and bi-embeddability

- MathematicsThe Journal of Symbolic Logic
- 2011

This article strengthens the results of Louveau and Rosendal by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalences relation is Borel equivalent to such a relation.

Degree spectra and computable dimensions in algebraic structures

- MathematicsAnn. Pure Appl. Log.
- 2002

The isomorphism problem for classes of computable fields

- Mathematics, Computer ScienceArch. Math. Log.
- 2004

This paper follows recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable, and shows that the isomorphic problem for real closed fields is Δ11 complete, and for others it is of relatively low complexity.

The isomorphism problem for computable Abelian p-groups of bounded length

- Mathematics, Computer ScienceJournal of Symbolic Logic
- 2005

The degree of the isomorphism problem for Abelian p-groups of bounded Ulm length is calculated and a sequence of classes whose isomorphic problems are cofinal in the hyperarithmetical hierarchy is explored.

Turing computable embeddings

- Computer ScienceJournal of Symbolic Logic
- 2007

A “Pull-back Theorem” is given, saying that if Ф is a Turing computable embedding of K into K′, then for any computable infinitary sentence φ in the language of K, it can be found that for all A ∈ K A ⊨ φ* iff A has the same “complexity” as φ.

New Directions in Descriptive Set Theory

- MathematicsBulletin of Symbolic Logic
- 1999

Desc descriptive set theory is the study of the structure of definable sets and functions in separable completely metrizable spaces, usually called Polish spaces.

Comparing Classes of Finite Structures

- Mathematics
- 2004

We compare classes of structures using the notion of a computable embedding, which is a partial order on the classes of structures. Our attention is mainly, but not exclusively, focused on classes of…