Highness properties close to PA-completeness

Greenberg, Noam; Miller, Joseph Scott; Nies, André

doi:10.48550/arxiv.1912.03016

Cited by 3 publications

(16 citation statements)

References 18 publications

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“…(The fact that PSUB is provable in WKL 0 is also implicit in an earlier proof by Barmpalias, Lewis, and Ng [5]; see also [20,proof of Theorem 8.8.8].) These facts also follow from the results of Greenberg, Miller, and Nies [25], since WSWWKL clearly implies PSUB. Indeed, it implies the following principle, which was shown by Barmpalias and Wang [6] (who denote it by P + ) to be strictly stronger than PSUB.…”

Section: Barmpalias and Wangmentioning

confidence: 63%

“…If we weaken Theorem 7.5 to say only that each individual path on T is 1-random, then it has an obvious proof, since we can take a computable binary tree R all of whose paths are 1-random, and use ∅ ′ to find a perfect subtree of R, using the fact that R has no isolated paths. But in this case we can improve the theorem from A ′ to any set that has PA degree relative to A, by work of Greenberg, Miller, and Nies [25] and of Chong, Li, Wang, and Yang [15]. Say that a binary tree has positive measure if the set of paths through T does.…”

Section: Barmpalias and Wangmentioning

confidence: 99%

“…Corresponding to their computability-theoretic work mentioned above, Greenberg, Miller, and Nies [25] defined the following principles, whose strength they showed to be strictly intermediate between WWKL 0 and WKL 0 .…”

Section: Barmpalias and Wangmentioning

confidence: 99%

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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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Section: Barmpalias and Wangmentioning

confidence: 63%

Section: Barmpalias and Wangmentioning

confidence: 99%

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

Hirschfeldt¹,

Jockusch²,

Schupp³

2021

Preprint

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“…It it known that High(CR,MLR) implies the ability to compute a 1-random real [4]. While conversely, computing a 1-random real is far from being High(CR,MLR) (see [5] figure 1). Meanwhile, [4] observed that every PA degree is High(CR,MLR).…”

Section: Introductionmentioning

confidence: 99%

“…In Theorem 3.1 we construct such a martingale that does not compute any PA degree, thus answer the question positively. On the other hand, [5] proved that every D ∈High(CR,MLR) compute a DNR h where h is some slow growing computable increasing function. Thus the current upper bound and lower bound of High(CR,MLR) are pretty close.…”

Section: Introductionmentioning

confidence: 99%

Upper bound on some hightness notions

Liu

2020

Preprint

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We give upper bound for several highness properties in computability randomness theory. First, we prove that discrete covering property does not imply the ability to compute a 1-random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily extract a 1-random real. Second, we prove that given a homogeneous binary tree that does not admit an infinite computable path, a sequence of bounded martingale whose initial capital tends to zero, there exists a martingale S majorizing infinitely many of them such that S does not compute an infinite path of the tree. This implies that 1) High(CR,MLR) does not imply PA-completeness, answering a question of Miller; 2) ≤ CR does not imply ≤ T , answering a question of Nies. The proof of the second result suggests that the coding power of the universal c.e. martingale lies in its infinite variance.

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A computable analysis of majorizing martingales

Liu

2021

Bull. London Math. Soc.

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We give upper bounds for several highness properties in computability randomness theory. First, we prove that a certain discrete covering property (which requires to almost cover all sets in a certain class) does not imply the ability to compute a 1‐random real, answering a question of Greenberg, Miller and Nies. This also implies that an infinite set of incompressible strings does not necessarily compute a 1‐random real. Second, we prove that given a homogeneous binary tree that does not admit an infinite computable path, a sequence of bounded martingales whose initial capitals tend to zero (where a martingale is bounded if its range is a bounded set of reals), there exists a martingale S majorizing infinitely many of them such that S does not compute an infinite path of the tree. This implies that: (1) a certain highness notion (which degenerates non 1‐random into noncomputably random) does not imply PA‐completeness, answering a question of Miller; (2) the computably random reducibility is not equivalent to Turing reducibility, answering a question of Nies. The proof of the second result suggests that the coding power of the universal computably enumerable martingale lies in its infinite variance.

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Highness properties close to PA-completeness

Cited by 3 publications

References 18 publications

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

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

Upper bound on some hightness notions

A computable analysis of majorizing martingales

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