Negative bases and automata

TL;DR

This paper explores expansions in negative non-integer bases, characterizing their automata representations, especially for Pisot numbers, and provides algorithms for base conversion using finite transducers.

Contribution

It introduces automata characterizations for negative base expansions, proves the sofic property for Pisot bases, and develops finite transducer algorithms for base conversion.

Findings

(-β)-shift is a system of finite type under certain conditions.

For Pisot β, the (-β)-shift is a sofic system.

Finite transducers can realize addition and conversion algorithms.

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.