Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

Bazhenov, Nikolay; Mustafa, Manat; Mauro, Luca San; Yamaleev, M. M.

doi:10.1134/s199508022002002x

Cited by 4 publications

(2 citation statements)

References 15 publications

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“…It concentrates on two main focuses: first, to calculate the complexity of natural equivalence relations on , proving, e.g., that provable equivalence in Peano Arithmetic is -complete [9], Turing equivalence on c.e. sets is -complete [18], and the isomorphism relations on several familiar classes of computable structures (e.g., trees, torsion abelian groups, and fields of characteristic or p ) are -complete [14]; secondly, to understand the structure of the collection of equivalence relations of a certain complexity (e.g., lying at some level of the arithmetical [10], analytical [8], or Ershov hierarchy [7, 21]).…”

Section: Introductionmentioning

confidence: 99%

On the Structure of Computable Reducibility on Equivalence Relations of Natural Numbers

Andrews¹,

Belin²,

Mauro³

2022

J. symb. log.

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We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\operatorname {\mathrm {\mathbf {ER}}}$ . We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$ . Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic.

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Section: Introductionmentioning

confidence: 99%

On the Structure of Computable Reducibility on Equivalence Relations of Natural Numbers

Andrews¹,

Belin²,

Mauro³

2022

J. symb. log.

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“…It concentrates on two main focuses: first, to calculate the complexity of natural equivalence relations on ω, proving, e.g., that provable equivalence in Peano Arithmetic is Σ 0 1 -complete [9], Turing equivalence on c.e. sets is Σ 0 4 -complete [18], and the isomorphism relations on several familiar classes of computable structures (e.g., trees, torsion abelian groups, fields of characteristic 0 or p) are Σ 1 1 -complete [13]; secondly, to investigate the poset of degrees generated by computable reducibility on the collection of equivalence relations of a certain complexity Γ, e.g., lying at some level of the arithmetical [10], analytical [7], or Ershov hierarchy [8,21].…”

Section: Introductionmentioning

confidence: 99%

On the structure of computable reducibility on equivalence relations of natural numbers

Andrews,

Belin,

Mauro

2021

Preprint

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We examine the degree structure ER of equivalence relations on ω under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in ER. We show that every equivalence relation has continuum many self-full strong minimal covers, and that d ' Id1 needn't be a strong minimal cover of a self-full degree d. Finally, we show that the theory of the degree structure ER as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second order arithmetic.2000 Mathematics Subject Classification. 03D55.

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Computably enumerable equivalence relations via primitive recursive reductions

Kalmurzayev,

Bazhenov,

Iskakov

2024

Journal of Logic and Computation

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The complexity classification of computably enumerable equivalence relations (or ceers, for short) has received much attention in the recent literature. A measure of complexity is typically provided by an appropriate notion of a reduction. Given binary relations $R$ and $S$ on natural numbers, a total function $f$ is a reduction from $R$ to $S$ if for arbitrary $x$ and $y$, the conditions $x~R~y$ and $f(x)~S~f(y)$ are always equivalent. If the function $f$ can be chosen primitive recursive, then we say that $R$ is primitively recursively reducible to $S$, denoted by $R \leq _{pr} S$. We investigate the degree structure $(\textbf {Ceers},\leq _{pr})$ of $\leq _{pr}$-degrees of ceers. We examine when pairs of incomparable degrees have an infimum and a supremum. In particular, we show that $(\textbf {Ceers},\leq _{pr})$ is neither an upper semilattice nor a lower semilattice. We also study first-order definable subclasses of $(\textbf {Ceers},\leq _{pr})$. In particular, we prove that the set of equivalences that have only finitely many classes is definable in $(\textbf {Ceers},\leq _{pr})$. Finally, we show that the structure of $\leq _{pr}$-degrees of computably enumerable preorders has a hereditarily undecidable theory.

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Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

Cited by 4 publications

References 15 publications

On the Structure of Computable Reducibility on Equivalence Relations of Natural Numbers

On the Structure of Computable Reducibility on Equivalence Relations of Natural Numbers

On the structure of computable reducibility on equivalence relations of natural numbers

Computably enumerable equivalence relations via primitive recursive reductions

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