In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton "A" with states "Q" and input set "I", output alphabet "U", then one can expand the states to "Q' " such that the new automaton "A' " embeds into a cascade of "simple", irreducible automata: In particular, "A" is emulated by a feed-forward cascade of (1) automata whose transitions semigroups are finite simple groups and (2) automata that are banks of flip-flops running in parallel. The new automaton "A' " has the same input and output symbols as "A". Here, both the states and inputs of the cascaded automata have a very special hierarchical coordinate form.