Robustness of Pisot-regular sequences

TL;DR

This paper investigates the robustness of Pisot-regular sequences in numeration systems, demonstrating that their definitional properties are invariant under different representations, with proofs based on automata theory.

Contribution

It proves that for Pisot numeration systems, the definition of regular sequences via formal series is independent of representation greediness, using automata constructions.

Findings

The definition of $(oldsymbol{U},oldsymbol{K})$-regular sequences is representation-independent for Pisot systems.

Normalization can be realized by a $2d$-tape finite automaton.

A novel automaton operation combines $2d$-tape and $oldsymbol{K}$-automata to establish invariance.

Abstract

We consider numeration systems based on a $d$-tuple $\mathbf{U}=(U_1,\ldots,U_d)$ of sequences of integers and we define $(\mathbf{U},\mathbb{K})$-regular sequences through $\mathbb{K}$-recognizable formal series, where $\mathbb{K}$ is any semiring. We show that, for any $d$-tuple $\mathbf{U}$ of Pisot numeration systems and any commutative semiring $\mathbb{K}$, this definition does not depend on the greediness of the $\mathbf{U}$-representations of integers. The proof is constructive and is based on the fact that the normalization is realizable by a $2d$-tape finite automaton. In particular, we use an ad hoc operation mixing a $2d$-tape automaton and a $\mathbb{K}$-automaton in order to obtain a new $\mathbb{K}$-automaton.