Algorithmic Randomness of Closed Sets

Barmpalias, George; Brodhead, Paul; Cenzer, Douglas; Dashti, S. Ali; Weber, Rebecca

doi:10.1093/logcom/exm033

Cited by 31 publications

(79 citation statements)

References 25 publications

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“…The randomness cancellation phenomena that play such a large role here have also arisen in other contexts, notably dimension spectra of random closed sets [2,7] and of random translations of the Cantor set [8]. Our work is as much an investigation of these phenomena as it is an analysis of a particular class of fractals.…”

Section: Introductionmentioning

confidence: 74%

Dimension spectra of random subfractals of self-similar fractals

Lutz

Mayordomo

et al. 2014

Annals of Pure and Applied Logic

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The (constructive Hausdorff) dimension of a point x in Euclidean space is the algorithmic information density of x. Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffices to specify x on a general-purpose computer with arbitrarily high precisions 2 −r . The dimension spectrum of a set X in Euclidean space is the subset of [0, n] consisting of the dimensions of all points in X.The dimensions of points have been shown to be geometrically meaningful (Lutz 2003, Hitchcock 2003, and the dimensions of points in self-similar fractals have been completely analyzed (Lutz and Mayordomo 2008). Here we begin the more challenging task of analyzing the dimensions of points in random fractals. We focus on fractals that are randomly selected subfractals of a given self-similar fractal. We formulate the specification of a point in such a subfractal as the outcome of an infinite two-player game between a selector that selects the subfractal and a coder that selects a point within the subfractal. Our selectors are algorithmically random with respect to various probability measures, so our selector-coder games are, from the coder's point of view, games against nature.We determine the dimension spectra of a wide class of such randomly selected subfractals. We show that each such fractal has a dimension spectrum that is a closed interval whose endpoints can be computed or approximated from the parameters of the fractal. In general, the maximum of the spectrum is determined by the degree to which the coder can reinforce the randomness in the selector, while the minimum is determined by the degree to which the coder can cancel randomness in the selector. This constructive and destructive interference between the players' randomnesses is somewhat subtle, even in the simplest cases. Our proof techniques include van Lambalgen's theorem on independent random sequences, measure preserving transformations, an application of network flow theory, a Kolmogorov complexity lower bound argument, and a nonconstructive proof that this bound is tight.

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Section: Introductionmentioning

confidence: 74%

Dimension spectra of random subfractals of self-similar fractals

Lutz

Mayordomo

et al. 2014

Annals of Pure and Applied Logic

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“…The subspace

F (2^{ω}) ∖ {⌀}

can naturally be identified with trees on

{0, 1}

with no dead branches. Barmpalias, Brodhead, Cenzer, Dashti, and Weber gave a natural construction of these trees from ternary sequences in 3 ω . Axon showed that the corresponding map

T : 3^{ω} \to F (2^{ω}) ∖ {⌀}

is a homeomorphism between 3 ω and the Fell topology restricted to

F (2^{ω}) ∖ {⌀}

.…”

Section: Computable Randomness and Isomorphismsmentioning

confidence: 99%

Computable randomness and betting for computable probability spaces

Rute

2016

Mathematical Logic Qtrly

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Abstract. Unlike Martin-Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting "bit-wise" definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. The paper contains a new type of randomness-endomorphism randomness-which the author hopes will shed light on the open question of whether Kolmogorov-Loveland randomness is equivalent to Martin-Löf randomness. The last section contains ideas for future research.

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“…As a warm-up, we provide a new, simpler proof of a known result from Barmpalias, Brodhead, Cenzer, Dashti, and Weber [2].…”

Section: Applications Of Randomness Preservation and No Randomness Exmentioning

confidence: 99%

“…The Interplay of Classes of Algorithmically Random Objects 3 obtained results from [2] and [3], as well as a proof of a conjecture in [3] that every random closed subset of 2 ω is the set of zeros of a random continuous function on 2 ω . We study the support of a certain class of random measures in Section 5 and establish a correspondence between between random closed sets and the random measures studied in [8].…”

Section: Introductionmentioning

confidence: 99%

“…Thus to randomly generate a non-empty closed set, it suffices to randomly generate an infinite tree. Following Barmpalias, Brodhead, Cenzer, Dashti, and Weber [2], we will code infinite trees by reals in 3 ω , thereby reducing the process of randomly generating infinite trees to the process of randomly generating reals. Given x ∈ 3 ω , define a tree T x ⊆ 2 <ω inductively as follows.…”

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confidence: 99%

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The interplay of classes of algorithmically random objects

Culver

Porter

2015

JLA

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Abstract:We study algorithmically random closed subsets of 2 ω , algorithmically random continuous functions from 2 ω to 2 ω , and algorithmically random Borel probability measures on 2 ω , especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-Löf random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply.Our main results are the following. First we answer an open question in Barmpalias, Brodhead, Cenzer, Remmel, and Weber [3] by showing that X ⊆ 2 ω is a random closed set if and only if it is the set of zeros of a random continuous function on 2 ω . As a corollary we obtain the result that the collection of random continuous functions on 2 ω is not closed under composition. Next, we construct a computable measure Q on the space of measures on 2 ω such that X ⊆ 2 ω is a random closed set if and only if X is the support of a Q-random measure. We also establish a correspondence between random closed sets and the random measures studied in Culver [8]. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in (0, 1).

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Algorithmic Randomness of Closed Sets

Cited by 31 publications

References 25 publications

Dimension spectra of random subfractals of self-similar fractals

Dimension spectra of random subfractals of self-similar fractals

Computable randomness and betting for computable probability spaces

The interplay of classes of algorithmically random objects

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