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

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

2011-12-21

arxiv.org/abs/1112.5089v1

Computer Science

Formal Languages and Automata Theory

Scientific paper

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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