Ostrowski numeration systems, addition and finite automata

TL;DR

This paper introduces an efficient algorithm for addition in Ostrowski numeration systems and shows that for quadratic bases, this addition is recognizable by finite automata, leading to decidability results.

Contribution

It provides a three-pass algorithm for addition and proves recognizability of Ostrowski representations by finite automata for quadratic bases, enabling decidability of related theories.

Findings

Addition in Ostrowski systems can be computed efficiently.

For quadratic bases, addition is recognizable by finite automata.

Decidability of the theory of $( abla ext{N},+,V_a)$ is established.

Abstract

We present an elementary three pass algorithm for computing addition in Ostrowski numeration systems. When $a$ is quadratic, addition in the Ostrowski numeration system based on $a$ is recognizable by a finite automaton. We deduce that a subset of $X\subseteq \mathbb{N}^n$ is definable in $(\mathbb{N},+,V_a)$, where $V_a$ is the function that maps a natural number $x$ to the smallest denominator of a convergent of $a$ that appears in the Ostrowski representation based on $a$ of $x$ with a non-zero coefficient, if and only if the set of Ostrowski representations of elements of $X$ is recognizable by a finite automaton. The decidability of the theory of $(\mathbb{N},+,V_a)$ follows.