Negative bases and automata
Formal Languages and Automata Theory
2010-12-17 v1 Discrete Mathematics
Dynamical Systems
Number Theory
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.
Cite
@article{arxiv.1012.3721,
title = {Negative bases and automata},
author = {Christiane Frougny and Anna Chiara Lai},
journal= {arXiv preprint arXiv:1012.3721},
year = {2010}
}