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Automaticity of rational functions. (English) Zbl 0895.68093
Summary: We study double sequences with values in a finite commutative ring $${\mathcal R}$$. We discuss under which conditions a double sequence, which is given by a rational function, is definable in $$\langle \mathbb{N},+ \rangle$$. Moreover, we give a complete characterization of the automaticity properties of double sequences defined by rational functions with coefficients in the ring $$\mathbb{Z}/(m)= \mathbb{Z}/m \mathbb{Z}$$. Thus answering a question posed by O. Salon in: Propriétés arithmétiques des automates multidimensionneles [Thèse, Université Bordeaux I (1989), p. II–9].

##### MSC:
 68Q80 Cellular automata (computational aspects) 11B85 Automata sequences 68Q45 Formal languages and automata 13M05 Structure of finite commutative rings 68R15 Combinatorics on words
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