Negative bases and automata

We study expansions in non-integer negative base -β introduced by Ito and Sadahiro. Using countable automata associated with (-β)-expansions, we characterize the case where the (-β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the (-β)-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 β to negative base -β. When β is a Pisot number, the conversion can be realized by a finite on-line transducer.

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