Highness properties close to PA completeness

Greenberg, Noam; Miller, Joseph S.; Nies, André

doi:10.1007/s11856-021-2200-7

Cited by 14 publications

(18 citation statements)

References 29 publications

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“…These properties follow from the results of §1.1 as well as their more specific forms given in §2, §3, §4. The overlap with the recent works of Hirschfeldt et al (2021) and Greenberg et al (2021) is clarified in §1.3.…”

Section: Our Results On Pathwise Randomnessmentioning

confidence: 63%

“…Chong et al (2019) started an investigation of measure-theoretic versions of the perfect set theorem and their line of research gained traction almost instantly. Greenberg et al (2021) studied the discrete and continuous covering properties and their relationship with PA degrees and WKL. Through their covering properties they have obtained the first part of Theorem 1.10 and clause (b) of Theorem 1.11, independently.…”

Section: Background and Recent Work On Models Of Compactnessmentioning

confidence: 99%

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Pathwise-random trees and models of second-order arithmetic

Barmpalias¹,

Wang²

2021

Preprint

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A tree is pathwise-random if all of its paths are algorithmically random, with a fixed upper bound on their randomness deficiency. We show that: (a) an algorithmically random real computes a pathwise-random tree with infinitely many paths if and only if it computes the halting problem; it follows that the class of pathwise-random trees with unbounded width is null, with respect to any computable measure; (b) there exists a positive-measure pathwise-random tree which does not compute any complete extension of Peano arithmetic; and (c) there exists a perfect pathwise-random tree which does not compute any positive pathwise-random tree. We use these results to obtain models of second-order arithmetic that separate compactness principles below weak Königs lemma, answering questions by Chong et al.(2019).

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Section: Our Results On Pathwise Randomnessmentioning

confidence: 63%

Section: Background and Recent Work On Models Of Compactnessmentioning

confidence: 99%

Pathwise-random trees and models of second-order arithmetic

Barmpalias¹,

Wang²

2021

Preprint

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“…The computational properties of Theorem 1 have been studied extensively via second-order representations, namely in e.g. [23,28,29,48]. The same holds for constructive analysis by [5,7,14,39], involving different (but related) constructive enrichments.…”

Section: Theorem 1 (Jordan Decomposition Theorem)mentioning

confidence: 99%

Betwixt Turing and Kleene

Normann

Sanders

2021

Logical Foundations of Computer Science

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Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any finite type. Various research programs have been proposed in which higher-order objects, like functions on the real numbers, are represented/coded as real numbers, so as to make them amenable to the Turing framework. It is then a natural question whether there is any significant difference between the Kleene approach or the Turing-approach-via-codes. Continuous functions being well-studied in this context, we study functions of bounded variation, which have at most countably many points of discontinuity. A central result is the Jordan decomposition theorem that a function of bounded variation on [0, 1] equals the difference of two monotone functions. We show that for this theorem and related results, the difference between the Kleene approach and the Turing-approach-via-codes is huge, in that full second-order arithmetic readily comes to the fore in Kleene's approach, in the guise of Kleene's quantifier ∃ 3 .

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“…Greenberg et al [5] proved that there is a computable function f on [0, 1] of bounded variation such that every Jordan decomposition of f in this weak sense is PA-complete. One direction of our argument, 4 ⇒ 1 of Theorem 4.10, is based on their proof; extra effort is required to make it work over RCA 0 as a base theory.…”

Section: Functions Of Bounded Variationmentioning

confidence: 99%

“…Greenberg et al [5,Theorem 1.4 and Section 2.3], going back to unpublished work with Slaman, built a computable function of bounded variation such that any continuous Jordan decomposition computes the halting problem, and every Jordan decomposition allowing discontinuity computes a completion of Peano arithmetic. To prove some of our results above, we adapt their methods to the setting of reverse mathematics.…”

mentioning

confidence: 99%

The Reverse Mathematics of Theorems of Jordan and Lebesgue

Nies¹,

Triplett²,

Yokoyama³

2021

J. symb. log.

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The Jordan decomposition theorem states that every function $f \colon \, [0,1] \to \mathbb {R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function of bounded variation is differentiable almost everywhere in the sense of Lebesgue measure. We analyze the strength of these theorems in the setting of reverse mathematics. Over $\mathsf {RCA}_{0}$ , a stronger version of Jordan’s result where all functions are continuous is equivalent to $\mathsf {ACA}_0$ , while the version stated is equivalent to ${\textsf {WKL}}_{0}$ . The result that every function on $[0,1]$ of bounded variation is almost everywhere differentiable is equivalent to ${\textsf {WWKL}}_{0}$ . To state this equivalence in a meaningful way, we develop a theory of Martin–Löf randomness over $\mathsf {RCA}_0$ .

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Highness properties close to PA completeness

Cited by 14 publications

References 29 publications

Pathwise-random trees and models of second-order arithmetic

Pathwise-random trees and models of second-order arithmetic

Betwixt Turing and Kleene

The Reverse Mathematics of Theorems of Jordan and Lebesgue

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