Document Open Access Logo

Probabilistic vs Deterministic Gamblers

Authors Laurent Bienvenu , Valentino Delle Rose , Tomasz Steifer

PDF
Thumbnail PDF

File

LIPIcs.STACS.2022.11.pdf
  • Filesize: 0.63 MB
  • 13 pages

Document Identifiers

Author Details

Laurent Bienvenu
  • LaBRI, CNRS & Université de Bordeaux, France
Valentino Delle Rose
  • Dipartimento di Ingegneria Informatica e Scienze Matematiche, University of Siena, Italy
Tomasz Steifer
  • Institute of Fundamental Technological Research, Polish Academy of Sciences, Warszawa, Poland

Funding

  • Steifer, Tomasz: Supported by the National Science Centre Poland grant no. 2018/31/B/ HS1/04018 and the 2021 French Government Scholarship.

Cite As Get BibTex

Laurent Bienvenu, Valentino Delle Rose, and Tomasz Steifer. Probabilistic vs Deterministic Gamblers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.11

Abstract

Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion - almost everywhere computable randomness. A binary sequence X is a.e. computably random if there is no probabilistic computable strategy which is total and succeeds on X for positive measure of oracles. Using the fireworks technique we construct a sequence which is partial computably random but not a.e. computably random. We also prove the separation between a.e. computable randomness and partial computable randomness, which happens exactly in the uniformly almost everywhere dominating Turing degrees.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
Keywords
  • Algorithmic randomness
  • Martingales
  • Probabilistic computation
  • Almost everywhere domination

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Laurent Bienvenu. Game-theoretic characterizations of randomness: unpredictability and stochasticity. PhD thesis, Université de Provence, 2008. Google Scholar
  2. Laurent Bienvenu and Ludovic Patey. Diagonally non-computable functions and fireworks. Information and Computation, 253:64-77, 2017. Google Scholar
  3. Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, and Reed Solomon. On a conjecture of Dobrinen and Simpson concerning almost everywhere domination. Journal of Symbolic Logic, 71(1):119-136, 2006. Google Scholar
  4. Sam Buss and Mia Minnes. Probabilistic algorithmic randomness. Journal of Symbolic Logic, 78(2):579-601, 2013. Google Scholar
  5. Karel de Leeuw, Edward F. Moore, Claude Shannon, and Norman Shapiro. Computability by probabilistic machines. In Automata Studies. Princeton University Press, 1956. Google Scholar
  6. Natasha Dobrinen and Stephen G. Simpson. Almost everywhere domination. Journal of Symbolic Logic, 69(3):914-922, 2004. Google Scholar
  7. Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer New York, New York, NY, 2010. Google Scholar
  8. Stuart Alan Kurtz. Randomness and Genericity in the Degrees of Unsolvability. PhD thesis, University of Illinois at Urbana-Champaign, 1982. Google Scholar
  9. Donald Martin. Measure, category, and degrees of unsolvability. Unpublished manuscript, 1967. Google Scholar
  10. Wolfgang Merkle. The complexity of stochastic sequences. Journal of Computer and System Sciences, 74(3):350-357, 2008. Google Scholar
  11. André Nies. Computability and randomness. Oxford Logic Guides. Oxford University Press, 2009. Google Scholar
  12. André Nies, Frank Stephan, and Sebastiaan Terwijn. Randomness, relativization and Turing degrees. Journal of Symbolic Logic, 70:515-535, 2005. Google Scholar
  13. Andrei Rumyantsev and Alexander Shen. Probabilistic constructions of computable objects and a computable version of Lovász local lemma. Fundamenta Informaticae, 132(1):1-14, 2014. Google Scholar
  14. Claus Schnorr. Zufälligkeit und Wahrscheinlichkeit, volume 218 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York, 1971. Google Scholar
  15. Michiel van Lambalgen. Random sequences. PhD dissertation, University of Amsterdam, Amsterdam, 1987. Google Scholar
  16. Liang Yu. When van Lambalgen’s theorem fails. Proceedings of the American Mathematical Society, 135(3):861-864, 2007. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact