Two Characterizations of Finite-State Dimension

Kozachinskiy, Alexander; Shen, Alexander

doi:10.1007/978-3-030-25027-0_6

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…1 Though Shen and Kozachinskiy proved the equivalence between disjoint block entropies and sliding block entropies for finite state dimension, the same techniques in [KS19] proves the equivalences for finite state strong dimension by replacing lim inf's with lim sup's.…”

Section: Auxiliary Lemmasmentioning

confidence: 97%

A Weyl Criterion for Finite-State Dimension and Applications

Lutz¹,

Nandakumar²,

Pulari³

2021

Preprint

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Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The 1972 Schnorr-Stimm dichotomy theorem gave the first automatatheoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them in 1909. This theorem implies, in present-day terminology, that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the 1916 Weyl criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of the Weyl criterion especially fruitful. This raises the question whether the Weyl criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc.This paper does exactly this. We extend the Weyl criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. In case the exponential sums are convergent, they converge to the Fourier coefficients of a probability measure whose dimension is precisely the finite state dimension of the sequence. We demonstrate the utility of this formulation through applications, including a new, Fourier analytic, proof of Schnorr and Stimm's landmark result.

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Section: Auxiliary Lemmasmentioning

confidence: 97%

A Weyl Criterion for Finite-State Dimension and Applications

Lutz¹,

Nandakumar²,

Pulari³

2021

Preprint

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“…The reviewers of the previous version of this paper pointed out that Doty and Moser [25] were the first who characterized the finite-state dimension of a sequence in terms of automatic complexity based on decompressors. We overlooked this paper (from 2006) and apologize to its authors for not mentioning it in [53,29]. They consider finite-state transducers that have an initial state, transition and output functions.…”

Section: Shallit and Wangmentioning

confidence: 99%

“…The initial version of this paper (that does not deal with finite-state dimension and does not contain the complexity criterion for normality) was presented at Fundamentals of Computation Theory symposium in 2017 [53] and is available in arxiv as [54]. The results about finite-state dimension and the complexity criterion were presented at Fundamentals of Computation Theory symposium in 2019 [29].…”

Section: Acknowledgmentsmentioning

confidence: 99%

Automatic Kolmogorov complexity, normality, and finite-state dimension revisited

Kozachinskiy

Shen

2021

Journal of Computer and System Sciences

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“…For example, Bourke, Hitchcock, and Vinodchandran proved that the lower and upper finite-state dimensions of a sequence S ∈ Σ ∞ are equal to the lower and upper block entropy rates of S, respectively (i.e., the lower and upper limiting normalized entropies of the frequencies of aligned blocks of symbols contained within S) [5]. In a recent paper, Kozachinskiy and Shen show that finite-state dimension can also be characterized in terms of the entropy rates of non-aligned blocks of symbols and in terms of superadditive calibrated functions on strings [15].…”

Section: Introductionmentioning

confidence: 99%

“…Later, Bourke, Hitchcock, and Vindochandran proved a characterization of the lower and upper finite-state dimensions of sequences [5] in terms of (aligned) block entropy rates. Kozachinskiy and Shen recently proved that the lower finite-state dimension can also be characterized using the entropy rates of non-aligned block frequencies [15].…”

mentioning

confidence: 99%

Finite-State Mutual Dimension

Case¹,

Lutz²

2021

Preprint

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In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finitestate dimension (a finite-state version of algorithmic dimension) of a sequence S ∈ Σ ∞ and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences S ∈ Σ ∞ and T ∈ Σ ∞ . In this paper, we propose a definition for the lower and upper finite-state mutual dimensions mdim F S (S : T ) and M dim F S (S : T ) between two sequences S ∈ Σ ∞ and T ∈ Σ ∞ over an alphabet Σ. Intuitively, the finite-state dimension of a sequence S ∈ Σ ∞ represents the density of finite-state information contained within S, while the finite-state mutual dimension between two sequences S ∈ Σ ∞ and T ∈ Σ ∞ represents the density of finite-state information shared by S and T . Thus "finite-state mutual dimension" can be viewed as a "finite-state" version of mutual dimension and as a "mutual" version of finite-state dimension.The main results of this investigation are as follows. First, we show that finite-state mutual dimension, defined using information-lossless finite-state compressors, has all of the properties expected of a measure of mutual information. Next, we prove that finitestate mutual dimension may be characterized in terms of block mutual information rates. Finally, we provide necessary and sufficient conditions for two normal sequences to achieve mdim F S (S : T ) = M dim F S (S : T ) = 0.

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Two Characterizations of Finite-State Dimension

Cited by 5 publications

References 15 publications

A Weyl Criterion for Finite-State Dimension and Applications

A Weyl Criterion for Finite-State Dimension and Applications

Automatic Kolmogorov complexity, normality, and finite-state dimension revisited

Finite-State Mutual Dimension

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