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Automata finiteness criterion in terms of van der Put series of automata functions

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Abstract

In the paper we develop the p-adic theory of discrete automata. Every automaton \(\mathfrak{A}\) (transducer) whose input/output alphabets consist of p symbols can be associated to a continuous (in fact, 1-Lipschitz) map from p-adic integers to p-adic integers, the automaton function \(f_\mathfrak{A} \). The p-adic theory (in particular, the p-adic ergodic theory) turned out to be very efficient in a study of properties of automata expressed via properties of automata functions. In the paper we prove a criterion for finiteness of the number of states of automaton in terms of van der Put series of the automaton function. The criterion displays connections between p-adic analysis and the theory of automata sequences.

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Correspondence to V. Anashin.

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Dedicated to Professor Igor Volovich on the occasion of his 65-th birthday

The text was submitted by the author in English.

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Anashin, V. Automata finiteness criterion in terms of van der Put series of automata functions. P-Adic Num Ultrametr Anal Appl 4, 151–160 (2012). https://doi.org/10.1134/S2070046612020070

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