On the construction of effectively random sets

Merkle, Wolfgang; Mihailovic, Nenad

doi:10.2178/jsl/1096901772

Cited by 16 publications

(12 citation statements)

References 15 publications

(7 reference statements)

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“…Let c ∈ N. There is a functional Θ such that for each A there is a Z / ∈ R c for which Θ Z = A with use function bounded by 2n. [17] for a proof of this result (and an improvement of the bound on the use of Θ).…”

Section: Introductionmentioning

confidence: 95%

Using random sets as oracles

Hirschfeldt

Nies

Stephan

2007

Journal of the London Mathematical Society

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Let R be a notion of algorithmic randomness for individual subsets of N. A set B is a base for R randomness if there is a Z T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets, and discuss several consequences of this result. On the other hand, the bases for computable randomness include every Δ 0 2 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, an n-c.e. set is a base for computable randomness if and only if it is Turing incomplete.

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Section: Introductionmentioning

confidence: 95%

Using random sets as oracles

Hirschfeldt

Nies

Stephan

2007

Journal of the London Mathematical Society

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“…n · log n bottleneck of Gács. The coding methods discussed above can all be seen as derivatives of the method of Gács [17], and this is the reason why they all have the characteristic bottleneck √ n · log n. Gács' method was originally introduced in terms of effectively closed sets, while Merkle and Mihailović [33] presented it in terms of martingales. Here we present a version of Gács' method in terms of shortest descriptions (to our knowledge, the first in the literature), which codes each stream X into some Y with oracle-use K(X ↾ n ) + O √ n .…”

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confidence: 99%

Compression of Data Streams Down to Their Information Content

Barmpalias

Lewis-Pye

2019

IEEE Trans. Inform. Theory

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According to Kolmogorov complexity, every finite binary string is compressible to a shortest code -its information content -from which it is effectively recoverable. We investigate the extent to which this holds for infinite binary sequences (streams). We devise a new coding method which uniformly codes every stream X into an algorithmically random stream Y, in such a way that the first n bits of X are recoverable from the first I(X ↾ n ) bits of Y, where I is any partial computable information content measure which is defined on all prefixes of X, and where X ↾ n is the initial segment of X of length n. As a consequence, if g is any computable upper bound on the initial segment prefix-free complexity of X, then X is computable from an algorithmically random Y with oracle-use at most g. Alternatively (making no use of such a computable bound g) one can achieve an oracle-use bounded above by K(X ↾ n ) + log n. This provides a strong analogue of Shannon's source coding theorem for algorithmic information theory.A fruitful way to quantify the complexity of a finite object such as a string σ in a finite alphabet 1 is to consider the length of the shortest binary program which prints σ. This fundamental idea gives rise to a theory of algorithmic information and compression, which is based on the theory of computation and was pioneered by Kolmogorov [21] and Solomonoff [41]. The Kolmogorov complexity of a binary string σ is the length of the shortest program that outputs σ with respect to a fixed universal Turing machine. The use of prefix-free machines in the definition of Kolmogorov complexity was pioneered by Levin [26] and Chaitin [10], and allowed for the development of a robust theory of incompressibility and algorithmic randomness for streams (i.e. infinite binary sequences). Information content measures, defined by Chaitin [11] after Levin [26], are functions that assign a positive integer value to each binary string, representing the amount of information contained in the string.Definition 1.1 (Information content measure). A partial function I from strings to N is an information content measure if it is right-c.e. 2 and I(σ)↓ 2 −I(σ) is finite.Prefix-free Kolmogorov complexity can be characterized as the minimum (modulo an additive constant) information content measure. If K(σ) denotes the prefix-free Kolmogorov complexity of the string σ, then for c ∈ N we say that σ is c-incompressible if K(σ) ≥ |σ| − c. It is a basic fact concerning Kolmogorov complexity that for some universal constant c: every string σ has a shortest code σ * which is itself c-incompressible.(1)Our goal is to investigate the extent to which the above fact holds in an infinite setting, i.e. for streams instead of strings. In the context of Kolmogorov complexity, algorithmic randomness is defined as incompressibility. 3 So (1) can be read as follows: we can uniformly code each string σ into an algorithmically random string of length K(σ). In order to formalise an infinitary analogue of this statement, we need to make use of oracle-machine ...

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“…One of their approaches is to build indifferent sets for non-autoreducible sets. While this works for Martin-Löf random sets, the technique does not generalise to weaker forms of randomness because recursively random sets may be autoreducible [29]. On the other hand, Franklin and Stephan [17] showed that every complement of a dense simple set is indifferent with respect to Schnorr randomness for all Schnorr random sets.…”

Section: Classes That Can Be Made Into Themselvesmentioning

confidence: 99%

Things that can be made into themselves

Stephan

Teutsch

2014

Information and Computation

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One says that a property P of sets of natural numbers can be made into itself iff there is a numbering α 0 , α 1 , . . . of all left-r.e. sets such that the index set {e : α e satisfies P } has the property P as well. For example, the property of being Martin-Löf random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of the present work is the investigation of the structure of left-r.e. sets under inclusion modulo a finite set. In contrast to the corresponding structure for r.e. sets, which has only maximal but no minimal members, both minimal and maximal left-r.e. sets exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly differs from Friedberg's classical construction of maximal r.e. sets. Finally, we investigate whether the properties of minimal and maximal left-r.e. sets can be made into themselves.Here the additive log factor is used for coding two implicit programs into a single string. On the other hand, by the definition of f , H(A ↾↾ m) > m + 3r(m)for all m, a contradiction. Therefore I is indifferent for A.

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On the construction of effectively random sets

Cited by 16 publications

References 15 publications

Using random sets as oracles

Using random sets as oracles

Compression of Data Streams Down to Their Information Content

Things that can be made into themselves

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