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Agafonov’s Theorem for Probabilistic Selectors

Authors Ulysse Léchine , Thomas Seiller , Jakob Grue Simonsen

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  • 15 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Probabilistic computation
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Normal sequences
  • probabilistic automata
  • Agafonov’s theorem

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Abstract

A normal sequence over {0,1} is an infinite sequence for which every word of length k appears with frequency 2^{-k}. Agafonov’s eponymous theorem states that selection by a finite state selector preserves normality, i.e. if α is a normal sequence and A is a finite state selector, then the subsequence A(α) is either finite or a normal sequence. 
In this work, we address the following question: does this result hold when considering probabilistic selectors? We provide a partial positive answer, in the case where the probabilities involved are rational. More formally, we prove that given a normal sequence α and a rational probabilistic selector P, the selected subsequence P(α) will be a normal sequence with probability 1.

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Ulysse Léchine, Thomas Seiller, and Jakob Grue Simonsen. Agafonov’s Theorem for Probabilistic Selectors. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.67

Author Details

Ulysse Léchine
  • Université Sorbonne Paris Nord, France
Thomas Seiller
  • CNRS, Paris, France
Jakob Grue Simonsen
  • University of Copenhagen, Denmark

Funding

  • Seiller, Thomas: Partially supported by the ANR-22-CE48-0003-01 project DySCo.

References

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