Randomness and initial segment complexity for measures.
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| Title: | Randomness and initial segment complexity for measures. |
|---|---|
| Authors: | Nies, André1 (AUTHOR) andre@cs.auckland.ac.nz, Stephan, Frank1,2,3 (AUTHOR) fstephan@comp.nus.edu.sg |
| Source: | Theoretical Computer Science. Jan2022, Vol. 900, p1-19. 19p. |
| Subjects: | Kolmogorov complexity, Algorithmic randomness, Probability measures, Open-ended questions |
| Abstract: | We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and provide a robustness property related to Solovay tests. The initial segment complexity of a measure μ at a length n is defined as the μ -average over the descriptive complexity of strings of length n , in the sense of either C or K. We relate this weak randomness notion for a measure to the growth of its initial segment complexity. We show that a maximal growth implies the weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss C -triviality and K -triviality for measures and relate these two notions with each other. Here, triviality means that the initial segment complexity grows as slowly as possible. We show that every measure that is Martin-Löf random in the sense of Hoyrup and Rojas is Martin-Löf absolutely continuous; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions. [ABSTRACT FROM AUTHOR] |
| Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 154242572 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Randomness and initial segment complexity for measures. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Nies%2C+André%22">Nies, André</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> andre@cs.auckland.ac.nz</i><br /><searchLink fieldCode="AR" term="%22Stephan%2C+Frank%22">Stephan, Frank</searchLink><relatesTo>1,2,3</relatesTo> (AUTHOR)<i> fstephan@comp.nus.edu.sg</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Theoretical+Computer+Science%22">Theoretical Computer Science</searchLink>. Jan2022, Vol. 900, p1-19. 19p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Kolmogorov+complexity%22">Kolmogorov complexity</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithmic+randomness%22">Algorithmic randomness</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+measures%22">Probability measures</searchLink><br /><searchLink fieldCode="DE" term="%22Open-ended+questions%22">Open-ended questions</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures. We begin with examples, and provide a robustness property related to Solovay tests. The initial segment complexity of a measure μ at a length n is defined as the μ -average over the descriptive complexity of strings of length n , in the sense of either C or K. We relate this weak randomness notion for a measure to the growth of its initial segment complexity. We show that a maximal growth implies the weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss C -triviality and K -triviality for measures and relate these two notions with each other. Here, triviality means that the initial segment complexity grows as slowly as possible. We show that every measure that is Martin-Löf random in the sense of Hoyrup and Rojas is Martin-Löf absolutely continuous; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Theoretical Computer Science is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.tcs.2021.11.003 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 19 StartPage: 1 Subjects: – SubjectFull: Kolmogorov complexity Type: general – SubjectFull: Algorithmic randomness Type: general – SubjectFull: Probability measures Type: general – SubjectFull: Open-ended questions Type: general Titles: – TitleFull: Randomness and initial segment complexity for measures. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Nies, André – PersonEntity: Name: NameFull: Stephan, Frank IsPartOfRelationships: – BibEntity: Dates: – D: 08 M: 01 Text: Jan2022 Type: published Y: 2022 Identifiers: – Type: issn-print Value: 03043975 Numbering: – Type: volume Value: 900 Titles: – TitleFull: Theoretical Computer Science Type: main |
| ResultId | 1 |
