Dimensions of Copeland–Erdös sequences

Gu, Xiaoyang; Lutz, Jack H.; Moser, Philippe

doi:10.1016/j.ic.2006.01.006

“…After the discovery of algorithmic dimensions in the present century [24,25,14,2], the Schnorr-Stimm dichotomy led to the realization [8] that the finite-state world, unlike any other known to date, is one in which maximum dimension is not only necessary, but also sufficient, for randomness. This in turn led to the discovery of nontrivial extensions of classical theorems on normal numbers [11,36] to new quantitative theorems on finite-state dimensions [19,16], a line of inquiry that will certainly continue. It has also led to a polynomial-time algorithm [4] that computes real numbers that are provably absolutely normal (normal in every base) and, via Lempel-Ziv methods, to a nearly linear time algorithm for this [27].…”

Section: Asymptotic Divergences and Strong Dichotomymentioning

confidence: 99%

Asymptotic Divergences and Strong Dichotomy

Huang

¹

,

Lutz

²

,

Mayordomo

³

et al. 2021

IEEE Trans. Inform. Theory

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The Schnorr-Stimm dichotomy theorem [31] 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 RiskG(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.

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“…After the discovery of algorithmic dimensions in the present century [24,25,14,2], the Schnorr-Stimm dichotomy led to the realization [8] that the finite-state world, unlike any other known to date, is one in which maximum dimension is not only necessary, but also sufficient, for randomness. This in turn led to the discovery of nontrivial extensions of classical theorems on normal numbers [11,36] to new quantitative theorems on finite-state dimensions [19,16], a line of inquiry that will certainly continue. It has also led to a polynomial-time algorithm [4] that computes real numbers that are provably absolutely normal (normal in every base) and, via Lempel-Ziv methods, to a nearly linear time algorithm for this [27].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Divergences and Strong Dichotomy

Huang

¹

,

Lutz

²

,

Mayordomo

³

et al. 2019

Preprint

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The Schnorr-Stimm dichotomy theorem [31] 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 infinitelyoften 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

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Effective Dimensions and Relative Frequencies

Gu

¹

,

Lutz

²

Logic and Theory of Algorithms

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Abstract. Consider the problem of calculating the fractal dimension of a set X consisting of all infinite sequences S over a finite alphabet Σ that satisfy some given condition P on the asymptotic frequencies with which various symbols from Σ appear in S. Solutions to this problem are known in cases where (i) the fractal dimension is classical (Hausdorff or packing dimension), or (ii) the fractal dimension is effective (even finite-state) and the condition P completely specifies an empirical distribution π over Σ, i.e., a limiting frequency of occurrence for every symbol in Σ. In this paper we show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of such a set X when the condition P only imposes partial constraints on the limiting frequencies of symbols. Our results automatically extend to less restrictive effective fractal dimensions (e.g., polynomial-time, computable, and constructive dimensions), and they have the classical results (i) as immediate corollaries. Our methods are nevertheless elementary and, in most cases, simpler than those by which the classical results were obtained.

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Dimensions of Copeland–Erdös sequences

Cited by 7 publications

References 16 publications

Asymptotic Divergences and Strong Dichotomy

Asymptotic Divergences and Strong Dichotomy

Asymptotic Divergences and Strong Dichotomy

Effective Dimensions and Relative Frequencies

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