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arXiv · 1112.5089

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

Vladimir Anashin
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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 $p$ 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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Vladimir Anashin. 2011-12-21. Automata finiteness criterion in terms of van der Put series of automata functions. https://doi.org/10.1134/s2070046612020070

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