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Randomness and differentiability
, 2014
"... We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a rea ..."
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Cited by 13 (3 self)
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We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a real number z ∈ [0, 1] is weakly 2random if and only if each almost everywhere differentiable computable function [0, 1] → R is differentiable at z. (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z is MartinLöf random if and only if every computable function of bounded variation is differentiable at z, and similarly for absolutely continuous functions. We also use our analytic methods to show that computable randomness of a real is base invariant, and to derive other preservation results for randomness notions.
Computing Ktrivial sets by incomplete random sets
"... Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem. ..."
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Every Ktrivial set is computable from an incomplete MartinLöf random set, i.e., a MartinLöf random set that does not compute the halting problem.
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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Cited by 1 (0 self)
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric
Logic Blog 2012
"... The 2012 logic blog has focussed on the following: Randomness and computable analysis/ergodic theory; Systematizing algorithmic randomness notions; Traceability; Higher randomness; Calibrating the complexity of equivalence relations from computability theory and algeBra. ..."
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The 2012 logic blog has focussed on the following: Randomness and computable analysis/ergodic theory; Systematizing algorithmic randomness notions; Traceability; Higher randomness; Calibrating the complexity of equivalence relations from computability theory and algeBra.
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"... L1computability, layerwise computability and Solovay reducibility ..."
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LOGIC BLOG 2013
"... Abstract. The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of Π11 randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For instance, what is the strength of Jordan’s theorem in a ..."
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Abstract. The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of Π11 randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For instance, what is the strength of Jordan’s theorem in analysis? (His theorem states that each function of bounded variation is the difference of two nondecreasing functions.) 3. Randomness and computable analysis. This focusses on the connection of randomness of a real z and Lebesgue density of effectively closed sets at z. 4. Exploring similarity relations for Polish metric spaces, such as isometry, or having GromovHausdorff distance 0. In particular their complexity was studied. 5. Various results connecting computability theory and randomness. Previous Logig Blogs from 2010 on can be found on Nies ’ web site.