This second part is concerned with the definability of sets of infinite sequences of symbols using finite automata. The fact that an automaton should process an infinite word may seem unnatural at first, especially when using the intuition that the automaton works through the input symbols one by one starting from the beginning of the word. However, when we think of acceptance as of the existence of a run, i.e. a consistent decoration of input positions with states, then the generalisation from finite to infinite inputs is rather natural.