Numberings and Randomness

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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“…. } Brodhead and Kjos-Hanssen [7] showed that there exists a Friedberg numbering, or enumeration without repetition, of the left-r.e. reals.…”

Section: Singleton Classesmentioning

confidence: 99%

“…Numberings for left-r.e. sets were first studied by Brodhead and Kjos-Hanssen [7,23] and provide more expressive possibilities than the traditional numberings for r.e. sets.…”

Section: Introductionmentioning

confidence: 99%

Things that can be made into themselves

Stephan

Teutsch

2014

Information and Computation

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“…reals. Brodhead and Kjos-Hanssen [3] observed that there exists an effective enumeration, or numbering, of the left-r.e. reals, and Chaitin [4] showed that some left-r.e.…”

Section: Effective Randomnessmentioning

confidence: 99%

“…Universal left-r.e. numberings also exist [3]: if ϕ e induces a universal r.e. numbering, then ϕ e induces a universal left-r.e.…”

Section: Sets Reals and Acceptable Numberingsmentioning

confidence: 99%

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Arithmetic complexity via effective names for random sequences

Kjos-Hanssen

Stephan

Teutsch

2012

ACM Trans. Comput. Logic

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We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

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Effectively Infinite Classes of Numberings of Computable Families of Reals

Faizrahmanov

2023

Russ Math.

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Numberings and Randomness

Abstract: We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-Löf random left-computably enumerable reals has a Friedberg numbering, as does the family of all Π

Cited by 4 publications

References 9 publications

Things that can be made into themselves

Things that can be made into themselves

Arithmetic complexity via effective names for random sequences

Effectively Infinite Classes of Numberings of Computable Families of Reals

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