Lowness, Randomness, and Computable Analysis

Nies, André

doi:10.1007/978-3-319-50062-1_42

Cited by 4 publications

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References 53 publications

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“…See eg Nies [13,Chapter 3]. Many of these notions have shown their importance in particular for the interaction of randomness and analysis such as Nies [14] and Brattka, Miller and Nies [2]. One of them is the following weakening of ML-randomness, which will be of importance in the present paper.…”

Section: Randomness Notionsmentioning

confidence: 98%

Randomness and Solovay degrees

Miyabe

Nies

Stephan

2018

JLA

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We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-Löf's, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-Löf-randomness, we show that Schnorr randomness is not upward closed in the Solovay degrees. Next, some left-r.e. Schnorr random α is the sum of two left-r.e. reals that are far from random. We also show that the left-r.e. reals of effective dimension > r , for some rational r , form a filter in the Solovay degrees.

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Section: Randomness Notionsmentioning

confidence: 98%

Randomness and Solovay degrees

Miyabe

Nies

Stephan

2018

JLA

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“…See [17,Section 7] for more background on the Γ-value. In particular, 1 − Γ(A) can be seen as a Hausdorff pseudodistance between {Y : Y ≤ T A} and the computable sets with respect to the Besicovitch distance ρ(U △V ) between bit sequences U, V (where ρ is the upper density).…”

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confidence: 99%

Muchnik degrees and cardinal characteristics

Monin,

Nies

2017

Preprint

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A mass problem is a set of functions on ω. For mass problems C, D, one says that C is Muchnik reducible to D if each function in C is computed by a function in D. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.Let D(p) be the mass problem of infinite bit sequences y (i.e., 0,1 valued functions) such that for each computable bit sequence x, the bit sequence x ↔ y has asymptotic lower density is at most p (where x ↔ y has a 1 in position n iff x(n) = y(n)). We show that all members of this family of mass problems parameterized by a real p with 0 < p < 1/2 have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems D(p) with the mass problem IOE(2 2 n ); here for an order function h, the mass problem IOE(h) consists of the functions f that agree infinitely often with each computable function bounded by h. This result also yields a new version of the first author's affirmative answer to the "Gamma question", whether Γ(A) < 1/2 implies Γ(A) = 0 for each Turing oracle A.Dual to the problem D(p), let B(p), for 0 ≤ p < 1/2, be the set of bit sequences y such that ρ(x ↔ y) > p for each computable set x. We prove that the Medvedev (and hence Muchnik) complexity of the B(p) is the same for all p ∈ (0, 1/2), by showing that they are Medvedev equivalent to the mass problem of functions bounded by 2 2 n that are almost everywhere different from each computable function.Next we obtain a proper hierarchy: together with Joseph Miller, we show that for any order function g there exists a faster growing order function h such that IOE(h) is strictly Muchnik above IOE(g).We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, d(p) is the least size of a set G of bit sequences so that for each bit sequence x there is a bit sequence y in G so that ρ(x ↔ y) > p. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. Contents BENOIT MONIN AND ANDR É NIES 2. Defining mass problems based on relations 6 3. Coincidences of Muchnik degrees 7 The ∆-value of a Turing oracle 13 4. Analog of Theorem 3.5 for cardinal characteristics 13 5. A proper hierarchy of problems IOE(h) in the weak degrees 18 6. Some open questions 26 References 28

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Muchnik Degrees and Cardinal Characteristics

Monin¹,

Nies²

2020

J. symb. log.

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A mass problem is a set of functions → . For mass problems C,D, one says that C is Muchnik reducible to D if each function in C is computed by a function in D. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.For p ∈ [0,1] let D(p) be the mass problem of infinite bit sequences y (i.e., {0,1}-valued functions) such that for each computable bit sequence x, the bit sequence x ↔ y has asymptotic lower density at most p (where x ↔ y has a 1 in position n iff x(n) = y(n)). We show that all members of this family of mass problems parameterized by a real p with 0 < p < 1/2 have the same complexity in the sense of Muchnik reducibility. We prove this by showing Muchnik equivalence of the problems D(p) with the mass problem IOE(2 2 n ); here for an order function h, the mass problem IOE(h) consists of the functions f that agree infinitely often with each computable function bounded by h. This result also yields a new version of the proof to of the affirmative answer to the "Gamma question" due to the first author: Γ(A) < 1/2 implies Γ(A) = 0 for each Turing oracle A.As a dual of the problem D(p), define B(p), for 0 ≤ p < 1/2, to be the set of bit sequences y such that (x ↔ y) > p for each computable set x. We prove that the Medvedev (and hence Muchnik) complexity of the mass problems B(p) is the same for all p ∈ (0,1/2), by showing that they are Medvedev equivalent to the mass problem of functions bounded by 2 2 n that are almost everywhere different from each computable function.Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type IOE: we show that for any order function g there exists a faster growing order function h such that IOE(h) is strictly above IOE(g) in the sense of Muchnik reducibility.We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, d(p) is the least size of a set G of bit sequences such that for each bit sequence x there is a bit sequence y in G so that (x ↔ y) > p. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. §1. Introduction. It is of fundamental interest in computability theory to determine the inherent computational complexity of an object, such as an infinite bit sequence, or more generally a function f on the natural numbers. To determine this complexity, one can place the object within classes of objects that all have a similar complexity. Among such classes, we will focus on highness properties. They specify a sense in which the object in question is computationally powerful.The Γ-value of an infinite bit sequence A, introduced by Andrews et al.[1], is a real in between 0 and 1 that in a sense measures how well all oracle sets in its Turing degree can be approximated by computable sequences. For each p ∈ (0,1], "Γ(A) < p" is a highness property of A. The values 0,1/2...

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Lowness, Randomness, and Computable Analysis

Abstract: Analytic concepts contribute to our understanding of randomness of reals via algorithmic tests. They also influence the interplay between randomness and lowness notions. We provide a survey.

Cited by 4 publications

References 53 publications

Randomness and Solovay degrees

Randomness and Solovay degrees

Muchnik degrees and cardinal characteristics

Muchnik Degrees and Cardinal Characteristics

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