Isomorphism relations on computable structures

Fokina, Ekaterina B.; Friedman, Sy-David; Harizanov, Valentina S.; Knight, Julia F.; McCoy, Charles F. D.; Montalbán, Antonio

doi:10.2178/jsl/1327068695

“…Let now E be a universal ceer such that there is no decidable set X = ∅, which is E-closed (i.e., satisfying that x ∈ X and y E x imply y ∈ X : see again Ershov [9, Proposition 8.2] for an example of such a ceer, or, more generally, this property holds of all the universal ceers given by Corollary 3. 16), and let f be a computable function reducing E ≤ R ⊕ S. If E R and E S, then the set X = {x : f(x) even} is a nontrivial decidable set that is E-closed. §3.…”

Section: Introductionmentioning

confidence: 99%

Universal Computably Enumerable Equivalence Relations

Andrews

¹

,

Lempp

²

,

Miller

³

et al. 2014

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We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

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“…Furthermore, [9] showes that the isomorphism relation on many natural classes of computable structures is FF-complete among Σ 1 1 equivalence relations. By the above results, there exist the h-degrees formed by Δ 1 1 equivalence relations with exactly n equivalence classes, for n ≤ ω, and a greatest h-degree of Σ 1 1 equivalence relations, namely, that of a complete Σ 1 1 equivalence relation.…”

Section: A Complete σ 1 1 Equivalence Relationmentioning

confidence: 99%

“…In [8,9] the notion of hyperarithmetical and computable reducibility of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω was used to study the question of completeness of natural equivalence relations on hyperarithmetical classes of computable structures as a special class of Σ 1 1 equivalence relations on ω. In this paper we use this approach to study the structure of Σ 1 1 equivalence relations on ω as a whole.…”

Section: Introductionmentioning

confidence: 99%

On Σ¹₁ equivalence relations over the natural numbers

Fokina

¹

,

Friedman

²

2011

Mathematical Logic Qtrly

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We study the structure of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly Σ 1 1 i.e., Σ 1 1 but not Δ 1 1 equivalence classes. We also show the existence of incomparable Σ 1 1 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ 1 1 equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ 1 1 equivalence classes that are complete as Σ 1 1 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ 1 1 equivalence relations.

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“…The natural effectivization to computable models of the Friedman-Stanley reducibility would be to consider hyperarithmetic reductions instead of computable reductions. We say that a class K is on top under hyperarithmetic reducibility if every Σ 1 1 equivalence relation on hyperarithmetically reduces to the isomorphism problem among computable models of K. Another unexpected empirical observation from the results in [8] is that every theory which we could prove was on top under hyperarithmetic reducibility, was already on top under effective reducibility. We show here that this should always be the case, at least among nice theories T where relativization should not be an issue.…”

mentioning

confidence: 99%

Classes of Structures With No Intermediate Isomorphism Problems

Montalbán¹

2016

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We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone. §1. Introduction. We show a connection between Vaught's conjecture and an intriguing open question about computable structures. The question we are referring to asks whether every nice theory T (given by a computably infinitary sentence) satisfies what we call the no-intermediate-extension property, which essentially means that for every nice extensionT of T (i.e.,T = T ∧ ϕ, where ϕ is a computable infinitary sentence), the isomorphism problem among the computable models ofT is either "simple," or as complicated as possible, but is never intermediate. (Throughout this paper, "theory" means "L 1 , -sentence".) By "simple" here we mean hyperarithmetic, and by "as complicated as possible" we mean universal among all Σ 1 1 -equivalences relations on under effective reducibility. See Definition 1.3. It is already known that if T has this property when relativized to all oracles, then Vaught's conjecture holds among the extensions of T (Becker [3]). The main result of this paper is a partial reversal, showing that the no-intermediate-extension property follows from a strengthening of Vaught's conjecture, which we call the uniform-effective-density property.As a bit of evidence that this strengthening is not too strong, we show that the theory of linear orderings has the uniform-effective-density property. It thus follows that the isomorphism problem among the computable models of any given theoryT extending that of linear orderings, is either hyperarithmetic or as complicated as possible, but never intermediate, at least relative to every oracle in a cone (Theorem 1.4).

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Isomorphism relations on computable structures

Cited by 71 publications

References 23 publications

Universal Computably Enumerable Equivalence Relations

Universal Computably Enumerable Equivalence Relations

On Σ¹₁ equivalence relations over the natural numbers

Classes of Structures With No Intermediate Isomorphism Problems

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Isomorphism relations on computable structures

Cited by 71 publications

References 23 publications

Universal Computably Enumerable Equivalence Relations

Universal Computably Enumerable Equivalence Relations

On Σ11 equivalence relations over the natural numbers

Classes of Structures With No Intermediate Isomorphism Problems

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On Σ¹₁ equivalence relations over the natural numbers