Can finite state machines be encoded as input/output for a neural network?
The better question might be - Why you do want to mix deep neural networks (DNN) and a finite state machine (FSM)? 

There is research showing that a DNN can simulate any FSM. Since there are more frameworks for DNN and DNN can perform more tasks than FSM, it appears to be more useful just to forgo FSM altogether. 

More specifically, "Neural network for synthesizing deterministic finite automata" shows how a relatively simple neural network (NN) can quickly and automatically learn the correct deterministic finite automaton (DFA).

There is a strong trend towards end-to-end DNN.