Description of Cascade Networks of Finite Automata In Terms of Codings and Coverings

The authors introduce the concepts of consistent codings and consistent systems of coverings of the set of states of a finite automaton. In terms of this concept, they characterize cascade networks of automata representing a given automaton, and particular cases of them, namely parallel-sequential, parallel, and sequential networks. They establish necessary and sufficient existence conditions for exact representations of an automaton by sequential and parallel-sequential networks, formulated in terms of embedded sequences of partitions of the set of states of the automaton.