Randomness and Semimeasures

Bienvenu, Laurent; Hölzl, Rupert; Porter, Christopher P.; Shafer, Paul

doi:10.1215/00294527-3839446

Cited by 7 publications

(10 citation statements)

References 11 publications

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“…However, this plan does not work: as found in [5], this generalized statement is not true. Levin [12] found a way to overcome these difficulties by using a less general statement.…”

Section: Levin's Approachmentioning

confidence: 95%

Gács-Kučera's Theorem Revisited by Levin

Shen¹

2021

Preprint

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“…However, this plan does not work: as found in [5], this generalized statement is not true. Levin [12] found a way to overcome these difficulties by using a less general statement.…”

Section: Levin's Approachmentioning

confidence: 95%

Gács-Kučera's Theorem Revisited by Levin

Shen¹

2021

Preprint

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“…Let C be a deep Π 0 1 class, whose canonical co-c.e. tree T satisfies m(T f(n) ) ≤ 2 −n for some computable function f. By Theorem 9.1 and the symmetry of information, for every Y ∈ C we have K(Ω n) − K Ω n | Y f(n), k n = I Ω n : Y f(n) − O(1) ≥ n − O(log n), (2) where k n stands for K(Y f(n)). Take a Y ∈ C such that Ω is random relative to Y .…”

Section: Mutual Information and Depthmentioning

confidence: 98%

“…Thus, in replacing with , this amounts to restricting the Turing functional Φ that induces to those inputs on which Φ is total. The proof of (i) is straightforward; for a proof of (ii), see [2]. Remark 2.6.…”

Section: Left-ce Semi-measuresmentioning

confidence: 98%

See 1 more Smart Citation

Deep Classes

Bienvenu¹,

Porter²

2016

Bull. symb. log

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A set of infinite binary sequences C ⊆ 2 N is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of Π 0 1 classes. In this paper, we introduce the notion of depth for Π 0 1 classes, which is a stronger form of negligibility. Whereas a negligible Π 0 1 class C has the property that one cannot probabilistically compute a member of C with positive probability, a deep Π 0 1 class C has the property that one cannot probabilistically compute an initial segment of a member of C with high probability. That is, the probability of computing a length n initial segment of a deep Π 0 1 class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep Π 0 1 classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin's Ω.1991 Mathematics Subject Classification. 03D32, 68Q30, 03D80. Key words and phrases. computability theory, algorithmic randomness, Π 0 1 classes, probabilistic computation.Both authors acknowledge the support of the John Templeton Foundation through the grant "Structure and Randomness in the Theory of Computation."Depth can be viewed as a strong form of negligibility. A deep Π 0 1 class C has the property that producing an initial segment of some member of C is in some sense maximally difficult: the probability of obtaining such an initial segment not only converges to 0 as we consider longer initial segments of members of C, but this convergence is fast, as we can effectively bound the rate of convergence. That initial segments of members of deep classes are so difficult to produce via probabilistic computation reflects the fact that members of deep classes are highly structured: initial segments of these sequences cannot be successfully produced by any combination of Turing machine and random oracle.Although the notion of depth is isolated for the first time in the present study, it is implicitly used in the work of both Levin [22] and Stephan [32]. In fact, in each of these papers one can extract a proof that the consistent completions of Peano arithmetic form a deep class, a result that we reprove in Section 7.The primary goals of this paper are (1) to prove a number of basic results about deep Π 0 1 classes, (2) to determine the exact relationship between negligibility and depth (and a related notion we call tt-negligibility), and (3) to provide a number of examples of deep Π 0 1 classes that occur naturally in computability theory and algorithmic randomness.The outline of this paper is as follows: In Section 2, we provide the technical background...

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“…The underlying general topic here is the connection between Turing functionals and semimeasures, which was explored in [ZL70]. For a recent account of this topic the reader is referred to [BHPS16], while [DH10] also contains related material in various sections of Chapters 3,6 and 7.…”

Section: Lower Bounds On the Redundancy In Computation From Random Realsmentioning

confidence: 99%

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

Barmpalias

Lewis-Pye

Teutsch

2016

Information and Computation

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Abstract. The Kučera-Gács theorem [Kuč85, Gác86] is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n + h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate coding procedure which achieves redundancy √ n log n. A similar bound is implicit in the later proof by Merkle and Mihailović [MM04]. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that n 2 −g(n) = ∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies n 2 −g(n) < ∞. Moreover, the class of such reals is comeager and includes a ∆ 0 2 real as well as all weakly 2-generic reals. On the other hand, it has been recently shown in [BLP16] that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that n 2 −g(n) < ∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.

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Randomness and Semimeasures

Cited by 7 publications

References 11 publications

Gács-Kučera's Theorem Revisited by Levin

Gács-Kučera's Theorem Revisited by Levin

Deep Classes

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

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