On the enumerating series of an abstract numeration system

TL;DR

This paper proves that rational abstract numeration systems can be effectively represented by N-rational series, enabling simple computation and recognition of sets, and also shows the decidability of their correspondence.

Contribution

It provides a simple proof linking rational abstract numeration systems with N-rational series and establishes the decidability of identifying such systems.

Findings

Representation of rational systems by N-rational series

Efficient computation of word values in these systems

Decidability of recognizing rational abstract numeration systems

Abstract

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple computation of the value of words in this system and easy constructions for the recognition of recognisable sets of numbers. It is also shown that conversely it is decidable whether an N-rational series corresponds to a rational abstract numeration system.