Classifying model-theoretic properties

Conidis, Chris J.

doi:10.2178/jsl/1230396753

Cited by 10 publications

(10 citation statements)

References 8 publications

(20 reference statements)

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“…Thus Lemma 5.2 reduces to showing that for every uniformly ∅ -computable sequence of dense sets of binary strings, every non-GL 2 degree bounds a real meeting every set in the sequence. This is a consequence of a combination of results in computable model theory and genericity arguments with GL 2 from [20,9].…”

Section: Then It Is Met (And Feels Satisfied At Every Stage T S)mentioning

confidence: 90%

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Barmpalias

Downey

Greenberg

2009

Trans. Amer. Math. Soc.

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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every left-c.e. real of that degree is reducible in a computable Lipschitz way to a random left-c.e. real (an Ω-number).

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Section: Then It Is Met (And Feels Satisfied At Every Stage T S)mentioning

confidence: 90%

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Barmpalias

Downey

Greenberg

2009

Trans. Amer. Math. Soc.

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“…It follows from computability-theoretic work of Csima, Hirschfeldt, Knight, and Soare [17] and Conidis [16] that Π 0 1 G implies AMT, and that the two are equivalent over RCA 0 together with Σ 0 2 induction. Hirschfeldt, Shore, and Slaman [32] showed that AMT does not imply Π 0 G over RCA 0 alone.…”

Section: Rmyc-mmentioning

confidence: 99%

“…As another consequence, we get a proof of Mycielski's Theorem 2.13, as noticed earlier by Miller and Yu [46], who gave their own direct proof of Corollary 2. 16, though without a bound on the complexity of the perfect tree: Let M 0 , M 1 , . .…”

Section: Cauchy Sequences In (S δ)mentioning

confidence: 99%

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

Hirschfeldt¹,

Jockusch²,

Schupp³

2021

Preprint

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For A ⊆ ω, the coarse similarity class of A, denoted by [A], is the set of all B ⊆ ω such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric δ on the space S of coarse similarity classes defined by letting δ([A], [B]) be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form {[A] : A ∈ U } where U is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of U .We then define a distance between Turing degrees based on Hausdorff distance in the metric space (S, δ). We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, 1/2, and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance 1/2 from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability.Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramseytheoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic.Finally, we study the completeness of (S, δ) from the perspectives of computability theory and reverse mathematics.

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“…Proof. Csima et al [4] and Conidis [3] proved that AMT is computably equivalent to the statement "For every ∆ 0 2 function f : ω → ω, there is a function not dominated by f ." Apply Theorem 5.18.…”

Section: The Gap Principlementioning

confidence: 99%

Thin set theorems and cone avoidance

Cholak

Patey

2020

Trans. Amer. Math. Soc.

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The thin set theorem RT n <∞,ℓ asserts the existence, for every kcoloring of the subsets of natural numbers of size n, of an infinite set of natural numbers, all of whose subsets of size n use at most ℓ colors. Whenever ℓ = 1, the statement corresponds to Ramsey's theorem. From a computational viewpoint, the thin set theorem admits a threshold phenomenon, in that whenever the number of colors ℓ is sufficiently large with respect to the size n of the tuples, then the thin set theorem admits strong cone avoidance.Let d 0 , d 1 , . . . be the sequence of Catalan numbers. For n ≥ 1, RT n <∞,ℓ admits strong cone avoidance if and only if ℓ ≥ dn and cone avoidance if and only if ℓ ≥ d n−1 . We say that a set A is RT n <∞,ℓ -encodable if there is an instance of RT n <∞,ℓ such that every solution computes A. The RT n <∞,ℓencodable sets are precisely the hyperarithmetic sets if and only if ℓ < 2 n−1 , the arithmetic sets if and only if 2 n−1 ≤ ℓ < dn, and the computable sets if and only if dn ≤ ℓ.

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Classifying model-theoretic properties

Cited by 10 publications

References 8 publications

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Working with strong reducibilities above totally $\omega $-c.e. and array computable degrees

Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

Thin set theorems and cone avoidance

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