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 Tr. Mat. Inst. Steklova, 2011, Volume 274, Pages 41–102 (Mi tm3322)

Algorithmic tests and randomness with respect to a class of measures

Laurent Bienvenua, Peter Gácsb, Mathieu Hoyrupc, Cristobal Rojasd, Alexander Shenef

a Laboratoire d'Informatique Algorithmique: Fondements et Applications (LIAFA), CNRS UMR 7089 & Université Paris Diderot, Paris Cedex, France
b Department of Computer Science, Boston University, Boston, MA, USA
c Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Vandœuvre-lés-Nancy, France
d Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
e Laboratoire d'Informatique Fondamentale de Marseille (LIF), Université Aix–Marseille, CNRS UMR 6166, Marseille Cedex, France
f Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Moscow, Russia

Abstract: This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability $p$ of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli measure $B_p$. A notion of “uniform test” for Bernoulli sequences is introduced which allows a quantitative strengthening of this result. Uniform tests are then generalized to arbitrary measures. Bernoulli measures $B_p$ have the important property that $p$ can be recovered from each random sequence of $B_p$. The paper studies some important consequences of this orthogonality property (as well as most other questions mentioned above) also in the more general setting of constructive metric spaces.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 274, 34–89

Bibliographic databases:

UDC: 510.5

Citation: Laurent Bienvenu, Peter Gács, Mathieu Hoyrup, Cristobal Rojas, Alexander Shen, “Algorithmic tests and randomness with respect to a class of measures”, Algorithmic aspects of algebra and logic, Collected papers. Dedicated to Academician Sergei Ivanovich Adian on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 274, MAIK Nauka/Interperiodica, Moscow, 2011, 41–102; Proc. Steklov Inst. Math., 274 (2011), 34–89

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bienvenu L., Monin B., “Von Neumann's Biased Coin Revisited”, 2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science (Lics), IEEE Symposium on Logic in Computer Science, IEEE, 2012, 145–154
2. Bienvenu L., Romashchenko A., Shen A., Taveneaux A., Vermeeren S., “The Axiomatic Power of Kolmogorov Complexity”, Ann. Pure Appl. Log., 165:9, SI (2014), 1380–1402
3. Weihrauch K., Tavana N.R., “Representations of Measurable Sets in Computable Measure Theory”, Log. Meth. Comput. Sci., 10:3 (2014), 7
4. Bauwens B., “Prefix and Plain Kolmogorov Complexity Characterizations of 2-Randomness: Simple Proofs”, Arch. Math. Log., 54:5-6 (2015), 615–629
5. Rute J., “Computable randomness and betting for computable probability spaces”, Math. Log. Q., 62:4-5 (2016), 335–366
6. Rute J., When does randomness come from randomness?, Theor. Comput. Sci., 635 (2016), 35–50
7. Andreev M., Kumok A., “The Sum $2^{ KM(x)-K(x)}$ Over All Prefixes $x$ of Some Binary Sequence Can be Infinite”, Theor. Comput. Syst., 58:3, SI (2016), 424–440
8. Bauwens B., “Relating and Contrasting Plain and Prefix Kolmogorov Complexity”, Theor. Comput. Syst., 58:3, SI (2016), 482–501
9. Bauwens B., Shen A., Takahashi H., “Conditional Probabilities and Van Lambalgen'S Theorem Revisited”, Theor. Comput. Syst., 61:4, SI (2017), 1315–1336
10. Bienvenu L., Hoyrup M., Shen A., “Layerwise Computability and Image Randomness”, Theor. Comput. Syst., 61:4, SI (2017), 1353–1375
11. Vereshchagin N., Shen A., “Algorithmic Statistics: Forty Years Later”, Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60Th Birthday, Lecture Notes in Computer Science, 10010, eds. Day A., Fellows M., Greenberg N., Khoussainov B., Melnikov A., Rosamond F., Springer International Publishing Ag, 2017, 669–737
12. Novikov G., “Randomness Deficiencies”, Unveiling Dynamics and Complexity, Cie 2017, Lecture Notes in Computer Science, 10307, eds. Kari J., Manea F., Petre I., Springer International Publishing Ag, 2017, 338–350
13. Rute J., “Schnorr Randomness For Noncomputable Measures”, Inf. Comput., 258 (2018), 50–78
14. Bienvenu L., Figueira S., Monin B., Shen A., “Algorithmic Identification of Probabilities Is Hard”, J. Comput. Syst. Sci., 95 (2018), 98–108
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