Every Low Boolean Algebra is Isomorphic to a Recursive One

Downey, Rodney G.; Jockusch, Carl G.

doi:10.2307/2160766

Cited by 21 publications

(34 citation statements)

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“…If A is a computable ordering, then A with the predicate adj is a ∆ 0 2 structure. In [1], a converse of the following sort was established. If (A, adj) is in the class ∆ 0 2 , then there exists a computable linear ordering A such that A and A generate isomorphic Boolean algebras (the case in point here is a standard approach used in treating Boolean algebras generated by half-open intervals of the ordering).…”

Section: Introductionmentioning

confidence: 99%

“…In [1], it was proved that every Boolean algebra with a low presentation also has a computable presentation. A low presentation is one that is computable with an oracle X such that X ≡ T ∅ .…”

Section: Introductionmentioning

confidence: 99%

“…Proofs for theorems in [1,2] were based on constructing certain linear orderings, while constructions in [3] worked directly with Boolean algebras. Our reasoning is underpinned by ideas from [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Our reasoning is underpinned by ideas from [1,2]. We will generally drop the word "linear," for all the orderings discussed are linear orderings.…”

Section: Introductionmentioning

confidence: 99%

“…In view of results of [1], we readily derive the following: if A is a 2-low ordering, then it has a computable presentation. Unfortunately, the list P 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Computability on linear orderings enriched with predicates

2009

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Our reasoning is underpinned by ideas from [1,2]. We will generally drop the word "linear," for all the orderings discussed are linear orderings.…”

Section: Introductionmentioning

confidence: 99%

“…In view of results of [1], we readily derive the following: if A is a 2-low ordering, then it has a computable presentation. Unfortunately, the list P 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Computability on linear orderings enriched with predicates

2009

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Degree spectra of real closed fields

Miller¹,

González²

2018

Arch. Math. Logic

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Several researchers have recently established that for every Turing degree c, the real closed field of all c-computable real numbers has spectrum {d : d ′ ≥ c ′′ }. We investigate the spectra of real closed fields further, focusing first on subfields of the field R0 of computable real numbers, then on archimedean real closed fields more generally, and finally on nonarchimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of R0 with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation. * This is a pre-print of an article to be published in the Archive for Mathematical Logic. The final authenticated version will be available online at: https://doi.org/10.1007/s00153-018-0638-z.

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Strongly constructive Boolean algebras

Alaev¹

2005

Algebr Logic

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A computable structure is said to be n-constructive if there exists an algorithm which, given a finite Σ n -formula and a tuple of elements, determines whether that tuple satisfies this formula. A structure is strongly constructive if such an algorithm exists for all formulas of the predicate calculus, and is decidable if it has a strongly constructive isomorphic copy. We give a complete description of relations between n-constructibility and decidability for Boolean algebras of a fixed elementary characteristic.A computable structure (in particular, a computable Boolean algebra) is said to be n-constructive if there exists an algorithm which, given a finite Σ n -formula and a tuple of elements, determines whether that tuple satisfies this formula. A strongly constructive structure is one for which such an algorithm exists for all formulas of the predicate calculus. Each strongly constructive structure is clearly n-constructive for every n ∈ ω. In [1], an example of an n-constructive Boolean algebra is furnished which has no (n+1)-constructive isomorphic copy, for any n ∈ ω. This means that the transition from n-constructibility to decidability (the existence of a strongly constructive isomorphic copy) in the class of all Boolean algebras is impossible in general. However, if we fix an elementary characteristic of a Boolean algebra, the situation changes. In [2, p. 1343], the following problem is formulated: for every elementary characteristic (m, k, ε), find a minimal n such that the property of being n-constructible implies being decidable. (Its particular cases are stated in [3][4][5] as open questions.) Here, we work to do away with this problem.The present paper naturally generalizes [6] and is essentially based on its results. For some characteristics (m, k, ε), the answer has been found earlier, in a large body of research. Below we have a brief look at the history of some studies. These are based on a natural algebraic description for n-constructibility, given in [5]. For each n ∈ ω, there is a finite set of one-place predicates, definable by first-order formulas, such that a Boolean algebra is n-constructive iff it is computable and the predicates define computable subsets in it.In [1], an example of a non-decidable Boolean algebra of characteristic (∞, 0, 0) was constructed which is n-constructive for all n ∈ ω (certainly, without uniformity in n). A non-decidable and 0-constructive (i.e., computable) algebra of characteristic (0, ∞, 0) was exemplified in [7]. And some results of [8] hold that in this case 1-constructibility implies not only decidability, but strong constructibility as well. In fact, this gives an answer for the characteristic (0, ∞, 0). For (0, k, 0) and (0, k, 1), k ∈ ω, the answer follows immediately from [8]: computability implies strong constructibility.

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Every Low Boolean Algebra is Isomorphic to a Recursive One

Cited by 21 publications

References 6 publications

Computability on linear orderings enriched with predicates

Computability on linear orderings enriched with predicates

Degree spectra of real closed fields

Strongly constructive Boolean algebras

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