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Representations of numbers and finite automata

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Abstract

Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a particular case of normalization. The same questions are discussed for the representation of real numbers in basis θ, where θ is a real number > 1, in connection with symbolic dynamics. In particular it is shown that if θ is a Pisot number, then the normalization and the addition in basis θ are computable by a finite automaton.

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Authors and Affiliations

  1. Institut Blaise Pascal, Université Paris VIII and LITP, 4 place Jussieu, 75252, Paris Cedex 05, France

    Christiane Frougny

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  1. Christiane Frougny
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This work has been supported by the PRC Mathématiques et Informatique.

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Frougny, C. Representations of numbers and finite automata. Math. Systems Theory 25, 37–60 (1992). https://doi.org/10.1007/BF01368783

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  • DOI: https://doi.org/10.1007/BF01368783

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