Negative bases and automata

Abstract : We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.
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Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 1 (1), pp.75--94
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Christiane Frougny, Anna Chiara Lai. Negative bases and automata. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 1 (1), pp.75--94. 〈hal-00990484〉

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