Rice sequences of relations

Montalbán, Antonio

doi:10.1098/rsta.2011.0333

Cited by 12 publications

(10 citation statements)

References 33 publications

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“…In the spring of 2012, Russian, Bulgarian, and U.S. researchers gathered in Chicago for further discussions of the notions of jump, at the workshop "Definability in computable structures", funded mainly by the Packard Foundation. Later, Montalbán proved that the different-looking definitions are equivalent (see [22]).…”

Section: Sketch Of Proofmentioning

confidence: 99%

Strong jump inversion

Calvert

Frolov

Harizanov

et al. 2018

Journal of Logic and Computation

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We say that a structure A admits strong jump inversion provided that for every oracle X, if X ′ computes D(C) ′ for some C ∼ = A, then X computes D(B) for some B ∼ = A. Jockusch and Soare [13] showed that there are low linear orderings without computable copies, but Downey and Jockusch [7] showed that every Boolean algebra admits strong jump inversion. More recently, D. Marker and R. Miller [19] have shown that all countable models of DCF0 (the theory of differentially closed fields of characteristic 0) admit strong jump inversion. We establish a general result with sufficient conditions for a structure A to admit strong jump inversion. Our conditions involve an enumeration of B1-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no 1-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of DCF0. This also yields the fact that the saturated model of DCF0 has a decidable copy.

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Section: Sketch Of Proofmentioning

confidence: 99%

Strong jump inversion

Calvert

Frolov

Harizanov

et al. 2018

Journal of Logic and Computation

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“…pregeometries. As we have already mentioned in the introduction, it follows from [AKMS89] and [Chi90] that there is a tuple d in M such that these relations are uniformly defined by computable infinitary Σ c 1 formulas with parameters d (the uniformity comes from a small modification to the same proof -see, for example, [Mon12]). See [AK00] for a background on computable infinitary logic L c ω 1 ω .…”

Section: Preliminariesmentioning

confidence: 99%

Independence in computable algebra

2015

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We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.1991 Mathematics Subject Classification. 03D45, 03C57, 12Y05.

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“…Here, we use the latter. (We refer the reader to [, Definition 5.1] for the history of the different definitions explained in more detail.) This is an important concept because, as it turned out, many constructions in the area can be better understood using the notion of jump.…”

Section: Introductionmentioning

confidence: 99%

Jump inversions of algebraic structures and Σ‐definability

Faĭzrahmanov

Kach

Kalimullin

et al. 2019

Mathematical Logic Qtrly

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It is proved that for every countable structure A and a computable successor ordinal α there is a countable structure A −α which is ≤ -least among all countable structures C such that A is -definable in the αth jump C (α) . We also show that this result does not hold for the limit ordinal α = ω. Moreover, we prove that there is no countable structure A with the degree spectrum {d : a ≤ d (ω) } for a > 0 (ω) .

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Rice sequences of relations

Cited by 12 publications

References 33 publications

Strong jump inversion

Strong jump inversion

Independence in computable algebra

Jump inversions of algebraic structures and Σ‐definability

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