Addition in Dumont-Thomas Numeration Systems in Theory and Practice

Abstract

Given a fixpoint of a substitution, the associated Dumont-Thomas numeration system provides a convenient immediate way to describe the fixpoint as an automatic sequence. In order to study first-order properties of these fixpoints using B\"uchi-Bruy\`ere characterization, it is necessary for the numeration system to be addable. This means that its addition relation must be computable by a finite automaton. Dumont-Thomas numeration systems can be regarded as an extension of positional numeration systems with states where the greediness is state-dependent. We introduce sequence automata as a tool to extend the results of Bruy\`ere-Hansel and Frougny-Solomyak on the regularity of the addition of Bertrand numeration systems to the case of Dumont-Thomas numeration systems related to some Pisot number. We present a practical implementation of the addition compatible with the Walnut computation tool, along with some experimental results