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Theory of Computing Systems

, Volume 30, Issue 3, pp 285–331 | Cite as

Automatic maps in exotic numeration systems

  • J. P. Allouche
  • E. Cateland
  • W. J. Gilbert
  • H. O. Peitgen
  • J. O. Shallit
  • G. Skordev
Article

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.

Keywords

Automatic Sequence Finite Subset Finite Automaton Numeration System Double Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • J. P. Allouche
    • 1
  • E. Cateland
    • 2
  • W. J. Gilbert
    • 3
  • H. O. Peitgen
    • 4
  • J. O. Shallit
    • 5
  • G. Skordev
    • 4
  1. 1.CNRS, LRIOrsay CedexFrance
  2. 2.Département de MathématiquesUniversité Bordeaux ITalence CedexFrance
  3. 3.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  4. 4.Institut für Dynamische SystemeUniversität BremenBremenGermany
  5. 5.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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