Automatic maps in exotic numeration systems

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...

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Bibliographic Details
Published in:Theory of computing systems 1997-06, Vol.30 (3), p.285-331
Main Authors: Allouche, J. P., Cateland, E., Gilbert, W. J., Peitgen, H. O., Shallit, J. O., Skordev, G.
Format: Article
Language:English
Subjects:
Automata theory
Computation
Digits
Gaussian
Integers
Rings (mathematics)
Sequences
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Summary: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.
ISSN:1432-4350
1433-0490
DOI:10.1007/BF02679463