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Asymptotic Divergences and Strong Dichotomy

Authors Xiang Huang , Jack H. Lutz, Elvira Mayordomo, Donald M. Stull

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Xiang Huang
  • Le Moyne College, Syracuse, NY 13214, USA
  • Iowa State University, Ames, IA 50011, USA
Jack H. Lutz
  • Iowa State University, Ames, IA 50011, USA
Elvira Mayordomo
  • Departamento de Informática e Ingeniería de Sistemas, Instituto de Investigación en Ingeniería de Aragón, Universidad de Zaragoza, 50018 Zaragoza, Spain
Donald M. Stull
  • Iowa State University, Ames, IA 50011, USA


We thank students in Iowa State University’s fall, 2017, advanced topics in computational randomness course for listening to a preliminary version of this research. We thank three anonymous reviewers of an earlier draft of this paper for useful comments and corrections.

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Xiang Huang, Jack H. Lutz, Elvira Mayordomo, and Donald M. Stull. Asymptotic Divergences and Strong Dichotomy. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 51:1-51:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


The Schnorr-Stimm dichotomy theorem [Schnorr and Stimm, 1972] concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. The theorem asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S. (2) If S is normal, then any finite-state gambler betting on S loses money at an exponential rate betting on S. In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||α) of a probability measure α on Σ from a sequence S over Σ and the upper asymptotic divergence Div(S||α) of α from S in such a way that a sequence S is α-normal (meaning that every string w has asymptotic frequency α(w) in S) if and only if Div(S||α)=0. We also use the Kullback-Leibler divergence to quantify the total risk Risk_G(w) that a finite-state gambler G takes when betting along a prefix w of S. Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to α-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S. (1') The infinitely-often exponential rate of winning in 1 is 2^{Div(S||α)|w|}. (2') The exponential rate of loss in 2 is 2^{-Risk_G(w)}. We also use (1') to show that 1-Div(S||α)/c, where c= log(1/ min_{a∈Σ} α(a)), is an upper bound on the finite-state α-dimension of S and prove the dual fact that 1-div(S||α)/c is an upper bound on the finite-state strong α-dimension of S.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Formal languages and automata theory
  • finite-state dimension
  • finite-state gambler
  • Kullback-Leibler divergence
  • normal sequences


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