Modeling and decomposition of automata by means of calculators

We describe the functioning of a finite automaton in the following manner. The next letter of the input sequence is input to the automaton only after the automaton reaches one of the specially identified final states at some stage of the process. We consider the output letter at this stage to be the response to the given input letter.
We consider methods for identifying sets of final states for which the resulting dictionary mapping remains an automaton mapping and show how the given construction can be applied to the decomposition of finite automata.