Demuth’s Path to Randomness

Kučera, Antonín; Nies, André; Porter, Christopher P.

doi:10.1017/bsl.2015.24

Cited by 9 publications

(11 citation statements)

References 44 publications

(64 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, strengthening the notion of capturing even when the tests are not nested gives rise to a weaker notion of randomness which turns out to be useful. This definition also goes back to Demuth; see [30] for more background. Definition 6.1.…”

Section: Lowness For Weak Demuth Randomnessmentioning

confidence: 99%

Characterizing Lowness for Demuth Randomness

Bienvenu¹,

Downey²,

Greenberg³

et al. 2014

J. symb. log.

Self Cite

View full text Add to dashboard Cite

Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

show abstract

Section: Lowness For Weak Demuth Randomnessmentioning

confidence: 99%

Characterizing Lowness for Demuth Randomness

Bienvenu¹,

Downey²,

Greenberg³

et al. 2014

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, when defining the sets U n we are allowed to change our mind sometimes about what U n is. In Martin-Löf tests, the function taking n to an index for U n is computable; Demuth allowed effectively approximable functions, with a computable bound on the number of mind-changes; this notion of randomness is now known as weak Demuth randomness (see [26] for background on Demuth's work in randomness). Franklin and Ng showed that a particular class of weak Demuth tests also captured difference randomness; in their tests, the different "versions" for each component U n have to be disjoint.…”

mentioning

confidence: 99%

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Bienvenu¹,

Day²,

Greenberg³

et al. 2014

Bull. symb. log

Self Cite

View full text Add to dashboard Cite

Abstract. Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute H 1 .A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turing degrees. At some level of randomness all interesting interactions cease. Computably enumerable sets are each definable and, in a sense, they are as effective as any noncomputable set can be. As a consequence, the lower and upper cones of noncomputable c.e. sets are definable null sets, and thus if a set is "sufficiently" random, it cannot compute, nor be computed by, a noncomputable c.e. set. However, the most studied notion of algorithmic randomness, Martin-Löf randomness, is not strong enough to support this argument, and in fact, significant interactions between Martin-Löf random sets and c.e. sets occur. The study of these interactions has lead to a number of surprising results that show a remarkably robust relationship between Martin-Löf random sets and the class of K-trivial sets. Interestingly, the significant interaction occurs "at the boundaries": the Martin-Löf random sets in question are close to being non-random (in that they fail fairly simple statistical tests), and K-trivial c.e. sets are close to being computable.The following theorem resolves one of the main open questions in algorithmic randomness, and further strengthens the relationship between the Martin-Löf random sets and the K-trivial sets. Theorem 1. There is an incomplete Martin-Löf random set that computes every K-trivial set.This theorem is essentially a corollary of two recent results, both proved in 2012: the first by Bienvenu, Greenberg, Kučera, Nies and Turetsky [3]; and the second by Day and Miller [10]. In the remainder of this announcement, we will explain the background to the problem behind this theorem, and indicate the main ideas used in the proof.In this announcement, by "random" we will henceforth mean Martin-Löf random. We will give the full definition shortly, but essentially, an element X of Cantor space is Martin-Löf random if it is not an element of a particular kind of effectively

show abstract

“…The following two results show the relevance of avoidability and hyperavoidability to the present discussion. Theorem 3.13 (Demuth and Kučera [19]; see also Kučera, Nies, and Porter [21]). Let X ∈ 2 ω be a non-computable sequence.…”

Section: Definition 311mentioning

confidence: 99%

Randomness for computable measures and initial segment complexity

Hölzl

Porter

2017

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2 ω , the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure µ, some sequence in this family fails to be random with respect to µ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2 ω , namely diminutive measures and trivial measures.

show abstract

Demuth’s Path to Randomness

Cited by 9 publications

References 44 publications

Characterizing Lowness for Demuth Randomness

Characterizing Lowness for Demuth Randomness

COMPUTINGK-TRIVIAL SETS BY INCOMPLETE RANDOM SETS

Randomness for computable measures and initial segment complexity

Contact Info

Product

Resources

About