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DOI: 10.4230/LIPIcs.LICS.2026.78

Randomness Extraction Fails for Finite-State Dimension

Authors: Subin Pulari and Akhil S

Published in: LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026)

Abstract

Finite-state dimension, introduced as a finite-state analogue of Hausdorff dimension, quantifies the lower asymptotic density of information in an infinite sequence as perceived by finite-state automata. It admits several equivalent formulations; two particularly useful are via finite-state gambling strategies and via the optimal asymptotic compression ratio achieved by information-lossless finite-state compressors. Normal sequences represent the highest level of algorithmic randomness visible to finite automata, and are exactly those sequences having finite-state dimension equal to 1. This motivates a bounded-memory notion of randomness extraction: can a finite-state transducer, reading a single sequence streamingly, extract a normal output from a single input source? More modestly, can it always transform the input into an output of strictly higher finite-state dimension? Finite-state transducers can perform surprisingly effective one-pass transformations: even with constant memory they can implement variable-length coding schemes including Shannon-Fano coding, remove local redundancy, and increase the apparent randomness rate on many structured or stochastic inputs. We show randomness extraction using transducers is impossible in a strong, explicit form. For every rational s ∈ (0,1), we construct a near linear-time computable binary sequence X with dim_FS(X) = s such that for every finite-state transducer T, the output satisfies dim_FS(T(X)) ≤ s. Thus, for these sequences, finite-state transduction cannot extract normality - indeed it cannot even improve finite-state dimension. Our proof proceeds by a structural analysis of finite-state transducers together with a dimension-preserving diagonal construction that, for each target s, builds a sequence whose organization defeats every such transducer’s attempt to concentrate randomness. The result is a finite-state analogue of Miller’s non-extractability phenomenon for effective dimension, but its proof relies on substantially different techniques, tailored to the finite-state setting. Furthermore, we show that the impossibility persists even with multiple independent input streams. We treat two notions of independence: (i) Kolmogorov-complexity–based independence (via joint prefix complexity), and (ii) a finite-state notion of relative independence, formulated via relative finite-state dimension. By sharp contrast with the effective-dimension setting - where two independent sources suffice for a uniform effective procedure that boosts randomness rate arbitrarily close to 1 - we show that finite-state dimension exhibits no comparable multi-source extraction phenomenon. Specifically, for every rational s ∈ (0,1) and every fixed k ≥ 2, there exist k independent sources, each of finite-state dimension s, such that for every k-input finite-state transducer T, the output satisfies dim_FS(T(X_1,… ,X_k)) ≤ s. Thus, even independent streams do not allow bounded-memory transduction to output a normal sequence or to increase finite-state dimension.

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Subin Pulari and Akhil S. Randomness Extraction Fails for Finite-State Dimension. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 78:1-78:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{pulari_et_al:LIPIcs.LICS.2026.78,
  author =	{Pulari, Subin and S, Akhil},
  title =	{{Randomness Extraction Fails for Finite-State Dimension}},
  booktitle =	{41st Annual Symposium on Logic in Computer Science (LICS 2026)},
  pages =	{78:1--78:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-434-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{380},
  editor =	{Faggian, Claudia and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.78},
  URN =		{urn:nbn:de:0030-drops-268656},
  doi =		{10.4230/LIPIcs.LICS.2026.78},
  annote =	{Keywords: Finite-state dimension, normal numbers, randomness extraction, finite-state transducers}
}

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DOI: 10.4230/LIPIcs.MFCS.2024.76

Point-To-Set Principle and Constructive Dimension Faithfulness

Authors: Satyadev Nandakumar, Subin Pulari, and Akhil S

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract

Hausdorff Φ-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of Φ-dimension using the notion of constructive Φ-s-supergales. We prove a Point-to-Set Principle for Φ-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive Φ-dimension. A class of covering sets Φ is said to be "faithful" to Hausdorff dimension if the Φ-dimension and Hausdorff dimension coincide for every set. Similarly, Φ is said to be "faithful" to constructive dimension if the constructive Φ-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of "faithfulness" of Cantor coverings at the Hausdorff and constructive levels are equivalent. We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin [Albeverio et al., 2020] to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion.

Cite as

Satyadev Nandakumar, Subin Pulari, and Akhil S. Point-To-Set Principle and Constructive Dimension Faithfulness. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 76:1-76:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2024.76,
  author =	{Nandakumar, Satyadev and Pulari, Subin and S, Akhil},
  title =	{{Point-To-Set Principle and Constructive Dimension Faithfulness}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{76:1--76:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.76},
  URN =		{urn:nbn:de:0030-drops-206321},
  doi =		{10.4230/LIPIcs.MFCS.2024.76},
  annote =	{Keywords: Kolmogorov complexity, Constructive dimension, Faithfulness, Point to set principle, Continued fraction dimension, Cantor series expansion}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2023.70

Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals

Authors: Satyadev Nandakumar, Akhil S, and Prateek Vishnoi

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract

We establish that constructive continued fraction dimension originally defined using s-gales [Nandakumar and Vishnoi, 2022] is robust, but surprisingly, that the effective continued fraction dimension and effective (base-b) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. In the process, we construct an optimal lower semi-computable s-gale for continued fractions. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that feasible dimension is invariant with respect to base conversion [Hitchcock and Mayordomo, 2013]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [Nandakumar and Vishnoi, 2022]. In contrast, for any 0 < ε < 0.5, we prove the existence of a real whose effective Hausdorff dimension is less than ε, but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the "non-faithfulness" of certain families of covers, investigated by Peres and Torbin [Peres and Torbin] and by Albeverio, Ivanenko, Lebid and Torbin [Albeverio et al., 2020]. We also establish that for any real, the constructive Hausdorff dimension is at most its effective continued fraction dimension.

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Satyadev Nandakumar, Akhil S, and Prateek Vishnoi. Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 70:1-70:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2023.70,
  author =	{Nandakumar, Satyadev and S, Akhil and Vishnoi, Prateek},
  title =	{{Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{70:1--70:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.70},
  URN =		{urn:nbn:de:0030-drops-186041},
  doi =		{10.4230/LIPIcs.MFCS.2023.70},
  annote =	{Keywords: Algorithmic information theory, Kolmogorov complexity, Continued fractions, Effective Hausdorff dimension}
}

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Documents authored by S, Akhil

Randomness Extraction Fails for Finite-State Dimension

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Point-To-Set Principle and Constructive Dimension Faithfulness

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Effective Continued Fraction Dimension Versus Effective Hausdorff Dimension of Reals

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