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Finite-State Dimension

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Book cover Automata, Languages and Programming (ICALP 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2076))

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Abstract

Classical Hausdorff dimension was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multi-account finite-state gamblers to develop the finite-state dimensions of sets of binary sequences and individual binary sequences. Every rational sequence (binary expansion of a rational number) has finite-state dimension 0, but every rational number in [0; 1] is the finite-state dimension of a sequence in the low-level complexity class AC0. Our main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite-state compressors.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Jack J. Dai

  2. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    James I. Lathrop & Jack H. Lutz

  3. Departamento de Informática e Ing. de Sistemas, Universidad de Zaragoza, 50015, Zaragoza, SPAIN

    Elvira Mayordomo

Authors
  1. Jack J. Dai
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  2. James I. Lathrop
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  3. Jack H. Lutz
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  4. Elvira Mayordomo
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Editor information

Editors and Affiliations

  1. Departament de Llenguatges i Sistemes Informátics, Univ. Politècnica de Catalunya, C/Jordi Girona Salgado 1-3, 08034, Barcelona, Spain

    Fernando Orejas

  2. Computer Technology Institute (CTI), University of Patras, 61 Riga Feraiou street, 26221, Patras, Greece

    Paul G. Spirakis

  3. Institute of Information and Computing Sciences, Utrecht University, Padualaan 14, 3584, CH, Utrecht, The Netherlands

    Jan van Leeuwen

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© 2001 Springer-Verlag Berlin Heidelberg

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Dai, J.J., Lathrop, J.I., Lutz, J.H., Mayordomo, E. (2001). Finite-State Dimension. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_83

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  • DOI: https://doi.org/10.1007/3-540-48224-5_83

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42287-7

  • Online ISBN: 978-3-540-48224-6

  • eBook Packages: Springer Book Archive

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