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Automatic maps in exotic numeration systems

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Abstract

We generalize the classical notion of ab-automatic sequence for a sequence indexed by the natural numbers. We replace the integers by a semiring and use a numeration system consisting of the powers of a baseb and an appropriate set of digits. For example, we define (−3)-automatic sequences (indexed by the ordinary integers or by the rational integers) and (−1 +i)-automatic sequences (indexed by the Gaussian integers). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties that permit the replacement of one numeration system by another.

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Authors and Affiliations

  1. CNRS, LRI, Bâtiment 490, F-91405, Orsay Cedex, France

    J. P. Allouche

  2. Département de Mathématiques, Université Bordeaux I, F-33405, Talence Cedex, France

    E. Cateland

  3. Department of Pure Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    W. J. Gilbert

  4. Institut für Dynamische Systeme, Universität Bremen, Kitzbühler Str. 2, D-28359, Bremen, Germany

    H. O. Peitgen & G. Skordev

  5. Department of Computer Science, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    J. O. Shallit

Authors
  1. J. P. Allouche
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  2. E. Cateland
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  3. W. J. Gilbert
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  4. H. O. Peitgen
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  5. J. O. Shallit
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  6. G. Skordev
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Allouche, J.P., Cateland, E., Gilbert, W.J. et al. Automatic maps in exotic numeration systems. Theory of Computing Systems 30, 285–331 (1997). https://doi.org/10.1007/BF02679463

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