06105109

j

06105109 Anashin, V. Automata finiteness criterion in terms of van der Put series of automata functions. $p$-Adic Numbers Ultrametric Anal. Appl. 4, No. 2, 151-160 (2012). 2012 MAIK Nauka/Interperiodica, Moscow; Springer, Heidelberg EN automaton finiteness conditions $p$-adic numbers van der Put series automata sequences

doi:10.1134/S2070046612020070

Summary: In the paper we develop the p-adic theory of discrete automata. Every automaton $\frak{A}$ (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.