Robustness of Pisot-regular sequences

06/19/2020

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by   Émilie Charlier, et al.

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We consider numeration systems based on a d-tuple 𝐔=(U_1,…,U_d) of sequences of integers and we define (𝐔,𝕂)-regular sequences through 𝕂-recognizable formal series, where 𝕂 is any semiring. We show that, for any d-tuple 𝐔 of Pisot numeration systems and any commutative semiring 𝕂, this definition does not depend on the greediness of the 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 𝕂-automaton in order to obtain a new 𝕂-automaton.