Geodesic
Ergodicity of the probabilistic converter, a serial connection of two automata
Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 38-48.

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The paper provides necessary and sufficient conditions for the the ergodicity of a serial connection of automata under which the output sequence of a substitution Mealy automaton is fed to the input of an output-free substitution automaton. It is shown that the condition of complete indecomposability of the state transition probability matrix of a Mealy automaton provides a sufficient condition for ergodicity of the probabilistic converter as a serial connection of automata. It is also shown that if the partial state transition functions of a Mealy automaton commute, then the condition of ergodicity of a serial connection is equivalent to that of both original probabilistic converters.
Keywords: Mealy automaton, output-free automaton, automaton extension, substitution automaton, serial connection of automata, probabilistic converter, ergodicity of probabilistic converter, indecomposable and aperiodic matrices, completely indecomposable matrices. 
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I. A. Kruglov. Ergodicity of the probabilistic converter, a serial connection of two automata. Diskretnaya Matematika, Tome 32 (2020) no. 3, pp. 38-48. http://geodesic.mathdoc.fr/item/DM_2020_32_3_a2/

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